2.1 The log normal space and pointed diagrams

2.1.1 The log normal space at a point

We recall the classical definition of tangential basepoints. For a pair $(\underline {X},\underline {D})$ given by a smooth scheme $\underline {X}$ over k endowed with a simple normal crossings divisor $\underline {D} = \bigcup _i \underline {D}_i$ with $\underline {D}_i$ its irreducible components, $\underline {X}$ decomposes into

$$ \begin{align*}\underline{X} = \underline{X}_0 \sqcup \underline{X}_1 \sqcup \cdots \sqcup \underline{X}_j \sqcup \cdots\end{align*} $$

where $\underline {X}_j$ is the locus of intersection of exactly j irreducible components of $\underline {D}$ . For $x \in \underline {X}_j$ , the normal space to $\underline {D}$ at x is defined as the k-vector space

(3) $$ \begin{align} \mathrm{N}_x(\underline{X},\underline{D}):= \mathrm{T}_x\underline{X}/\mathrm{T}_x\underline{X}_j \end{align} $$

where $\mathrm {T}_x\underline {X}, \mathrm {T}_x \underline {X}_j$ are the tangent (k-vector) spaces of $\underline {X},\underline {X}_j$ at x.

For each $\underline {D}_i$ containing x, we let

$$ \begin{align*}\mathrm{N}_x(\underline{X},\underline{D}_i):= \mathrm{T}_x\underline{X}/\mathrm{T}_x\underline{D}_i\end{align*} $$

so that there is a decomposition

$$ \begin{align*}\mathrm{N}_x(\underline{X},\underline{D}) = \bigoplus_i \mathrm{N}_x(\underline{X},\underline{D}_i).\end{align*} $$

The definition of tangential basepoints we use is the following from [Reference DeligneDel89]:

Definition 2.1.1. A tangential basepoint of $(\underline {X},\underline {D})$ is the datum $(x,v)$ of a rational point $x\in \underline {X}(k)$ and a vector $v \in \mathrm {N}_x(\underline {X},\underline {D})$ such that its projection on $\mathrm {N}_x(\underline {X},\underline {D}_i)$ for each $\underline {D}_i$ containing x is nonvanishing. We denote this tangential basepoint by $\mathbf {x}:= (x,v)$ .

Note that there is a unique tangential basepoint at a rational point x of $\underline {X}\setminus \underline {D}$ ; it is $(x,0)$ . A tangential basepoint at a point lying on a single irreducible component of $\underline {D}$ is the same thing as the datum of a nonzero normal vector to this component at this point.

We describe a logarithmic translation of this definition. Let $\mathbf {Div}_k$ be the category of divisorial log schemes over k. This is nothing but the category of log schemes $(\underline {X},\underline {D})$ with $\underline {X}$ a smooth scheme over k with a simple normal crossings divisor $\underline {D}$ , whose morphisms $f: (\underline {X},\underline {D}) \rightarrow (\underline {Y},\underline {E})$ are given by morphisms of k-schemes $\underline {f}: \underline {X} \rightarrow \underline {Y}$ that satisfy $\underline {f}^{-1}(\underline {E}) \subset \underline {D}$ . To simplify, we will often denote by X the whole datum of $(\underline {X},\underline {D})\in \mathbf {Div}_k.$ Recall that by [Reference Binda, Park and Arne ØstværBPØ22, A.5.10], the category $\mathbf {Div}_k$ is equivalent to the category of log smooth log schemes whose underlying scheme is smooth over k.

Let $X \in \mathbf {Div}_k$ , and $x: \mathrm {Spec}(k) \rightarrow X$ a rational point. We denote by

$$ \begin{align*}x^{\mathrm{log}}:= (\mathrm{Spec}(k), \mathcal{M}_{X}|_x)\end{align*} $$

the log scheme whose underlying scheme is $\mathrm {Spec}(k)$ with the log structure pulled back from X via x.

Remark 2.1.2. Note that the structure monoid $\mathcal {M}_{X}|_x \neq \mathcal {M}_{X,x}$ in general. Indeed, if we let $\beta $ denote the composition $\mathcal {M}_{X,x} \rightarrow \mathcal {O}_{X,x} \rightarrow k$ , then by definition $\mathcal {M}_{X}|_x$ is obtained as the pushout of the diagram

It is proved in [Reference Dupont, Panzer and PymDPP24, Theorem 3.5] that for $X:= (\underline {X},\underline {D}) \in \mathbf {Div}_k$ and $x\in \underline {X}(k)$ , there is a bijection between the set of tangential basepoints at x and the set of group morphisms

$$ \begin{align*}(\mathcal{M}_{X}|_x)^{\mathrm{gp}} \rightarrow k^\times\end{align*} $$

which restrict to the identity on $k^\times $ . In coordinates, if we let $x_1,\dots , x_j$ be local coordinates of $\underline {D}$ at x, then there is a (non-canonical) isomorphism

$$ \begin{align*}\mathcal{M}_{X}|_x \simeq k^\times x_1^{\mathbb{N}}\cdots x_j^{\mathbb{N}}\end{align*} $$

and the datum of a morphism $(\mathcal {M}_{X}|_x)^{\mathrm {gp}} \rightarrow k^\times $ which restricts to the identity on $k^\times $ is equivalent to the data of $\lambda _1,\dots , \lambda _j \in k^\times ,$ where $\lambda _i$ is the image of $x_i$ . This datum is associated to the tangential basepoint

$$ \begin{align*}\bigg(x,\lambda_1 \frac{\partial}{\partial x_1}\bigg|_x + \dots + \lambda_j \frac{\partial}{\partial x_j}\bigg|_x\bigg).\end{align*} $$

This monoid is related to the schematic normal space in the following way:

Lemma 2.1.3. There is a canonical isomorphism of schemes over k

$$ \begin{align*}\mathrm{N}_x(\underline{X},\underline{D}) \xrightarrow{\sim} \mathrm{Spec}(k[\mathcal{M}_{X}|_x]\otimes_{k[k^\times]} k)\end{align*} $$

where $\mathrm {N}_x(\underline {X},\underline {D})$ is seen as the affine space associated with (3).

Proof. The normal space is defined by

$$ \begin{align*}\mathrm{N}_x(\underline{X},\underline{D}):= \mathrm{Spec(Sym}((\mathrm{T}_xX/\mathrm{T}_xX_j)^\vee)),\end{align*} $$

and the k-algebra $k[\mathcal {M}_{X}|_x]\otimes _{k[k^\times ]} k$ is identified with

$$ \begin{align*}k[x_1,\dots, x_j]\end{align*} $$

where $x_1,\dots , x_j$ is any choice of local coordinates of D at x. There is a morphism of schemes over k

$$ \begin{align*}\mathrm{N}_x(\underline{X},\underline{D}) \rightarrow \mathrm{Spec}(k[\mathcal{M}_{X}|_x]\otimes_{k[k^\times]} k)\end{align*} $$

defined by the morphism of k-algebras

$$ \begin{align*}k[\mathcal{M}_{X}|_x]\otimes_{k[k^\times]} k \rightarrow \mathrm{Sym}(\mathrm{T}_xX/\mathrm{T}_xX_j)^\vee\end{align*} $$

which associates to $x_i$ the element $dx_i|_x \in (\mathrm {T}_xX/\mathrm {T}_xX_j)^\vee .$ Note that this construction does not depend on the choice of coordinates we made since another choice $\{x^{\prime }_i\}$ would satisfy $x^{\prime }_i = \lambda _i x_{i}$ in $\mathcal {M}_{X}|_x$ with $\lambda _i \in k^\times $ . The normal space satisfies

$$ \begin{align*}\mathrm{N}_x(\underline{X},\underline{D}) \simeq \mathrm{Spec}(k[dx_1|_x,\dots ,dx_j|_x])\simeq \mathbb{A}^j_k,\end{align*} $$

so that we indeed defined an isomorphism.

This lemma motivates the following definition:

Definition 2.1.4. For $(\underline {X},\underline {D}) \in \mathbf {Div}_k$ and $x \in \underline {X}(k)$ , we call log normal space to D at x the log scheme $\mathrm {N}^{\mathrm {log}}_x(\underline {X},\underline {D}) \in \mathbf {lSch}_k$ whose underlying scheme is defined by

$$ \begin{align*}\underline{\mathrm{N}}^{\mathrm{log}}_x(\underline{X},\underline{D}):= \mathrm{Spec}(k[\mathcal{M}_{X}|_x]\otimes_{k[k^\times]} k),\end{align*} $$

and whose log structure is induced by the morphism of monoids

$$ \begin{align*}\mathcal{M}_{X}|_x \rightarrow k[\mathcal{M}_{X}|_x] \rightarrow k[\mathcal{M}_{X}|_x]\otimes_{k[k^\times]} k.\end{align*} $$

Remark 2.1.5. From the description given in the proof of lemma 2.1.3, we see that the log structure induced on $\mathrm {N}_x(\underline {X},\underline {D})$ is a divisorial log structure given by

$$ \begin{align*}dx_1|_x\dots dx_j|_x = 0.\end{align*} $$

The log scheme obtained is simply $\mathbb {A}_{\mathbb {N}^j}.$ Hence, note that $\mathrm {N}^{\mathrm {log}}_x(\underline {X},\underline {D}) \in \mathbf {Div}_k$ .

Example 2.1.6.

• • For $(\underline {X},\underline {D}) = \mathbb {A}_{\mathbb {N}}$ , $x = 0 \in \mathbb {A}^1$ , then

  $$ \begin{align*}\mathrm{N}^{\mathrm{log}}_x(\underline{X},\underline{D}) = \mathbb{A}_{\mathbb{N}}\end{align*} $$

• • For $(\underline {X},\underline {D}) = \mathbb {A}_{\mathbb {N}^2}$ , $x = (1,0)$ , then

  $$ \begin{align*}\mathrm{N}^{\mathrm{log}}_x(\underline{X},\underline{D}) = \mathbb{A}_{\mathbb{N}}.\end{align*} $$

• • For any $(\underline {X},\underline {D})$ , if $x\in X\setminus D$ ,

  $$ \begin{align*}\mathrm{N}^{\mathrm{log}}_x(\underline{X},\underline{D}) = *.\end{align*} $$

The constructions $x^{\mathrm {log}}$ and $\mathrm {N}^{\mathrm {log}}_x(\underline {X},\underline {D})$ are related by a functorial (strict) closed immersion of log schemes over k

$$ \begin{align*}x^{\mathrm{log}} \hookrightarrow \mathrm{N}^{\mathrm{log}}_x(\underline{X},\underline{D})\end{align*} $$

defined as follows: its underlying morphism of schemes is given by the morphism of k-algebras

$$ \begin{align*}k[\mathcal{M}_{X}|_x] \otimes_{k[k^\times]} k \rightarrow k\end{align*} $$

coming from the morphism of monoids defining the log structure on $x^{\mathrm {log}}$

$$ \begin{align*}\mathcal{M}_{X}|_x \rightarrow k,\end{align*} $$

and the morphism of log structures is described by

which shows that the morphism of log schemes thus defined is strict.

The log scheme $\mathrm {N}^{\mathrm {log}}_x(\underline {X},\underline {D})$ is the correct logarithmic version of the schematic normal space to D at x associated with the normal space (3). In this direction, we show that the set of morphisms from the log scheme $\mathrm {Spec}(k)$ with trivial log structure to the log normal space to D at x are in canonical bijection with the set of tangential basepoints of $(\underline {X},\underline {D})$ at x.

Lemma 2.1.7. For $(\underline {X},\underline {D}) \in \mathbf {Div}_k$ and $x \in \underline {X}(k)$ , there is a bijection between the set of tangential basepoints at x and the set of logarithmic rational points of $\mathrm {N}^{\mathrm {log}}_x(\underline {X},\underline {D})$ , that is the set of morphisms of logarithmic schemes

$$ \begin{align*}\mathrm{Spec}(k) \rightarrow \mathrm{N}^{\mathrm{log}}_x(\underline{X},\underline{D}).\end{align*} $$

Proof. For X an fs log scheme, we denote by $\partial X$ the boundary of X, i.e. the closed subscheme of X such that for $x\in X, x\in \partial X$ if and only if $\mathcal {M}_{X,x} \neq \mathcal {O}_{X,x}^\times $ . It is the locus of points at which the log structure is non-trivial. For Y a log scheme with trivial log structure, any morphism $Y \rightarrow X$ to an fs log scheme lands in the open locus of points with trivial log structure $X\setminus \partial X$ . Thus there is a bijection between the set of logarithmic rational point of $\mathrm {N}^{\mathrm {log}}_x(\underline {X},\underline {D})$ and the set of rational points of $\underline {\mathrm {N}}^{\mathrm {log}}_x(\underline {X},\underline {D})\setminus \partial \underline {\mathrm {N}}^{\mathrm {log}}_x(\underline {X},\underline {D}).$

To identify the open locus with trivial log strucure of the log normal space to D at x, note that it is the open subset on which any function $m \in \mathcal {M}_{X}|_x$ is inverted. Hence it is simply described as the spectrum of the localization of $k[\mathcal {M}_{X}|_x]\otimes _{k[k^\times ]} k$ at these elements (or at a set of generators of $\mathcal {M}_{X}|_x$ ). Therefore the open locus is

$$ \begin{align*}\mathrm{Spec}(k[(\mathcal{M}_{X}|_x)^{\mathrm{gp}}]\otimes_{k[k^\times]} k).\end{align*} $$

A rational point of this scheme is given in a unique way by a group morphism

$$ \begin{align*}(\mathcal{M}_{X}|_x)^{\mathrm{gp}} \rightarrow k^\times\end{align*} $$

which restricts to the identity on $k^\times $ . By the discussion above definition 2.1.4, this is what we wanted to prove.

Remark 2.1.8. If we let $\mathrm {N}_x(\underline {X},\underline {D}_i):= \mathrm {Spec}(\mathrm {Sym}(\mathrm {T}_x\underline {X}/\mathrm {T}_x\underline {D}_i)^\vee )$ there is a decomposition

$$ \begin{align*}\mathrm{N}_x(\underline{X},\underline{D}) = \prod_i \mathrm{N}_x(\underline{X},\underline{D}_i),\end{align*} $$

so that the log structure is obtained via the product

$$ \begin{align*}\prod_i (\mathrm{N}_x(\underline{X},\underline{D}_i),0).\end{align*} $$

Since the locus with non-trivial log structure of $(\mathrm {N}_x(\underline {X},\underline {D}_i),0)$ is reduced to the point $\{0\}$ , this point of view explains in another way that a logarithmic rational point of $\mathrm {N}^{\mathrm {log}}_x(\underline {X},\underline {D})$ is the same thing as a tangential basepoint at x: it is the datum of a vector $v \in \mathrm {N}_x(\underline {X},\underline {D})$ whose projection on each $\mathrm {N}_x(\underline {X},\underline {D}_i)$ does not vanish.

2.1.2 The functoriality of tangential basepoints

Since the construction $\mathrm {N}^{\mathrm {log}}_x(\underline {X},\underline {D})$ is functorial, it allows us to describe the functoriality for tangential basepoints.

Definition 2.1.9. A morphism $f : (\underline {X},\underline {D}) \rightarrow (\underline {Y},\underline {E})$ of divisorial log schemes over k satisfies $f(\mathbf {x}) = \mathbf {y}$ , with $\mathbf {x}$ and $\mathbf {y}$ tangential basepoints on $ (\underline {X},\underline {D})$ and $(\underline {Y},\underline {E})$ respectively if $f(x) = y$ , and if the induced morphism of log schemes $\mathrm {N}_x^{\mathrm {log}}(\underline {X},\underline {D}) \rightarrow \mathrm {N}_y^{\mathrm {log}}(\underline {Y},\underline {E})$ satisfies the condition that the following diagram commutes:

where the morphisms from $\mathrm {Spec}(k)$ are the log rational points associated with $\mathbf {x}$ and $\mathbf {y}$ by the lemma 2.1.7.

In a simple case, suppose $(\underline {X},\underline {D}) = (\mathbb {A}^n,x_1\dots x_j= 0), (\underline {Y},\underline {E}) = (\mathbb {A}^m,y_1\dots y_k = 0)$ and let $f : (\underline {X},\underline {D}) \rightarrow (\underline {Y},\underline {E})$ be a morphism of divisorial log schemes such that $f(0) = 0$ . There are monomials in the coordinates $x_1,\dots , x_j$ denoted by $M_1, \dots , M_k \in K[x_1,\dots , x_j]$ such that

$$ \begin{align*}f^*y_i = M_i(x_1,\dots, x_j).\end{align*} $$

If we let

$$ \begin{align*}\mathbf{0}:= \bigg(0,\lambda_1 \frac{\partial}{\partial x_1}\bigg|_0 + \dots + \lambda_j \frac{\partial}{\partial x_j}\bigg|_0\bigg)\end{align*} $$

be a tangential basepoint of $(\underline {X},\underline {D})$ at $0$ , then it is easy to see that the tangential basepoint defined by the composition

is

$$ \begin{align*}f(\mathbf{0}):=\bigg(0,M_1(\lambda_1,\dots, \lambda_j) \frac{\partial}{\partial y_1}\bigg|_{0} + \dots + M_k(\lambda_1,\dots,\lambda_j) \frac{\partial}{\partial y_k}\bigg|_{0}\bigg).\end{align*} $$

Remark 2.1.10. Note that in general, for $f: (\underline {X},\underline {D}) \rightarrow (\underline {Y},\underline {E})$ and $\mathbf {x} := (x,t)$ a tangential basepoint on $(\underline {X},\underline {D})$ , then $f(\mathbf {x}) \neq (f(x), df_x(t))$ . For instance if $f: (\mathbb {A}^1,0) \rightarrow (\mathbb {A}^1,0)$ is defined by $f(x) = x^n$ with $ n \geq 2$ , then $df_0(t) = 0$ while if $t = \lambda \partial _x |_0$ , then $f(\mathbf {0}) = (0, \lambda ^n \partial _x |_0)$ .

Remark 2.1.11. This functoriality seems to be aligned with, and was partly inspired by the point of view which consists in viewing tangential basepoints via morphisms $\mathrm {Spec}(k((x))) \rightarrow X$ , see for instance [Reference PetrovPet21, Lemma 10.1]. We will see in the next subsection another motivation for this functoriality.

2.1.3 The category of pointed diagrams

Let us now define the category of pointed diagrams which will be our framework when we define motivic tangential basepoints in subsection 3.3. Note that the diagram of log schemes over k

(4)

is functorial with respect to morphisms of divisorial log schemes over k.

Definition 2.1.12. We call the category of pointed diagrams the category whose objects are diagrams of log schemes over k of the form

where $(\underline {X},\underline {D}) \in \mathbf {Div}_k$ , and $\mathbf {x}: \mathrm {Spec}(k) \rightarrow \mathrm {N}^{\mathrm {log}}_x(\underline {X},\underline {D})$ is a tangential basepoint at a rational point x. To simplify, we denote such diagrams by $(\underline {X},\underline {D},\mathbf {x})$ .

A morphism of pointed diagrams $(\underline {X},\underline {D},\mathbf {x}) \rightarrow (\underline {Y},\underline {E},\mathbf {y})$ is a morphism of diagrams of log schemes over k given by a morphism of divisorial log schemes $f: (\underline {X},\underline {D}) \rightarrow (\underline {Y},\underline {E})$ sending x to y, the induced morphism of log schemes $x^{\mathrm {log}} \rightarrow y^{\mathrm {log}}$ , the induced morphism $\mathrm {N}^{\mathrm {log}}_x(\underline {X},\underline {D}) \rightarrow \mathrm {N}^{\mathrm {log}}_y(\underline {Y},\underline {E})$ and the identity on $\mathrm {Spec}(k)$ . In particular the diagram

must commute, which just means that $f(\mathbf {x}) = \mathbf {y}$ .

Hence, it is clear that the category of pointed diagrams is equivalent to the category of tangentially pointed smooth schemes over k with simple normal crossings divisor, that is the category whose objects are the datum of a divisorial log scheme $(\underline {X},\underline {D})$ over k and a tangential basepoint for $(\underline {X},\underline {D})$ , and whose morphisms are morphisms of divisorial log schemes over k which are compatible with the fixed tangential basepoints in the sense described above. The reason why we focus on the category of pointed diagrams lies in the fact that pointed diagrams are a higher dimensional and functorial analogue of Deligne’s gluing procedure to view tangential basepoints as points. Indeed if we apply the Kato–Nakayama space functor to the diagram (4) with $X:= (\underline {X},\underline {D})$ a smooth curve over k, we get

where, as in the introduction of this article, $\mathrm {Bl}^{\mathbb {R}}$ stands for the real-oriented blow-up. Deligne’s picture is simply the topological pushout of this diagram.

2.2 Virtual points

Now we connect our approach of tangential basepoints via logarithmic geometry to the point of view developed in [Reference Dupont, Panzer and PymDPP24], based on the notion of virtual morphisms of log schemes first defined in [Reference HowellHow17] and [Reference ShuklinShu22]. This class of maps between log schemes enlarge the class of classical morphisms of logarithmic schemes and introduces for instance, maps from a log scheme with trivial log structure which do not land in the locus of points with trivial log structure (see the proof of lemma 2.1.7). This, as we will see is the case for virtual points.

Definition 2.2.1 [Reference Dupont, Panzer and PymDPP24, Definition 2.6]

Let $X,Y \in \mathbf {lSch}_k$ . A virtual morphism $X \rightarrow Y$ is a pair $(\underline {\phi },\phi ^\flat )$ consisting of a morphism of schemes $\underline {\phi }: \underline {X} \rightarrow \underline {Y}$ , and a morphism $\phi ^\flat : \underline {\phi }^{-1}\mathcal {M}^{\mathrm {gp}}_Y \rightarrow \mathcal {M}^{\mathrm {gp}}_X$ of sheaves of groups, making the following diagram commute:

We denote by $\mkern1mu^{\mathbf {v}}\mathbf {lSch}_k$ the category of fs log schemes with virtual morphisms between them.

A morphism of log schemes induces in a unique way a virtual morphism of log schemes so that $\mathbf {lSch}_k$ is a subcategory of $\mkern1mu^{\mathbf {v}}\mathbf {lSch}_k$ . In order to distinguish morphisms in $\mathbf {lSch}_k$ from those in $\mkern1mu^{\mathbf {v}}\mathbf {lSch}_k$ we call the former ordinary morphisms of log schemes.

We can relate the category of pointed diagrams and the category of virtually pointed divisorial log schemes over k:

Hence, tangential basepoints can be described by both virtual morphisms on divisorial log schemes over k and pointed diagrams. Nevertheless, pointed diagrams offer the advantage of being constructed using ordinary morphisms only, which is what we will need to work in a log motivic category. We have the functorial version of the theorem 2.2.3:

Based on the previous proposition, we will simply denote by the thick letter $\mathbf {x}$ the equivalent datum of:

• • a tangential basepoint at $x\in \underline {X}(k),$

• • a pointed diagram $(\underline {X},\underline {D},\mathbf {x}),$

• • a virtual morphism $\mathrm {Spec}(k) \rightarrow X.$

There is a last (partial) representation of tangential basepoints when $k \subset \mathbb {C}$ . If $\mathbf {x}: * \rightarrow X$ is a virtual point, then, since the Kato–Nakayama space construction is functorial with respect to virtual morphisms [Reference Dupont, Panzer and PymDPP24, Proposition 4.2], this datum determines a point on $X^{\mathrm {KN}}$ . More precisely since $X^{\mathrm {KN}}$ is the real-oriented blow-up of $\underline {X}$ along $\underline {D}$ , this determines a point of $(S^1)^j$ where j is the number of irreducible components of $\underline {D}$ at x. Each coordinate of $(S^1)^j$ determines a direction normal to one of the irreducible components of $\underline {D}$ .

In the previous example, we note that two virtual morphisms determined by two constants $\lambda , a\lambda $ where a is a strictly positive constant have the same associated point on the Kato–Nakayama space. Hence a point on the Kato–Nakayama is not as such equivalent to the datum of a tangential basepoint. This is actually related to a classical fact in the theory of the fundamental group. The Betti side of the Hodge structure on the prounipotent completion of the fundamental group only sees the direction of the tangential basepoints, since a change of parametrization of any path based at a tangential basepoint can change the tangent vector by multiplication by an arbitrary strictly positive constant. In section 5.3, we will recall from [Reference Dupont, Panzer and PymDPP23, Section 8.4] how the structure of manifold with log corners of the Kato–Nakayama space allows to complete this approach so that the missing datum is encoded in the manifold with log corners version of virtual morphisms.

3.1 Log motivic categories

Definition 3.1.1 [Reference Cisinski and DégliseCD19, Definition 1.4.2]

Let $\mathbf {C}$ be a (1-)category with $\mathcal {P}$ a class of morphisms which contains all isomorphisms and which is stable by pullback. A $\mathcal {P}$ -premotivic $\infty $ -category over $\mathbf {C}$ is a functor

$$ \begin{align*}\mathcal{T}: \mathrm{N}(\mathbf{C})^{\mathrm{op}} \rightarrow \mathrm{CAlg}(\mathcal{P}\mathrm{r}^{\mathrm{L}})\end{align*} $$

such that

• • For all $f\in \mathcal {P}$ , $f^*:= \mathcal {T}(f)$ has a left adjoint $f_\sharp $ ,

• • for every cartesian square

  
  the natural transformation
  $$ \begin{align*}f^{\prime}_\sharp g^{\prime*} \rightarrow g^*f_\sharp\end{align*} $$
  is an equivalence,

• • for every morphism $f \in \mathcal {P}$ , the $\mathcal {P}$ -projection

  $$ \begin{align*}f_\sharp((-)\otimes f^*(-)) \rightarrow f_\sharp(-) \otimes (-)\end{align*} $$
  is an equivalence.

Definition 3.1.2. For $\mathcal {T}$ any $\mathcal {P}$ -premotivic $\infty $ -category over $\mathbf {C}$ , and for $S\in \mathbf {C}$ we let

$$ \begin{align*}M_S:\mathcal{P}/S \rightarrow \mathcal{T}(S)\end{align*} $$

be the homological motive functor where we denote by $\mathcal {P}/S$ the full subcategory of $\mathbf {C}/S$ with objects the morphisms $X \rightarrow S \in \mathcal {P}$ . We also denote by

$$ \begin{align*}h_S: (\mathbf{C}/S)^{\mathrm{op}} \rightarrow \mathcal{T}(S)\end{align*} $$

the cohomological motive functor. They are defined for $f: X\rightarrow S \in \mathcal {P}$ by

$$ \begin{align*}M_S(X):= f_\sharp f^* \mathbf{1}_S\end{align*} $$

and for $f \in \mathbf {C}/S$ , if we let $f_*$ denote a right adjoint of $f^*$ (which exists since $f^*$ is colimit preserving by hypothesis) by

$$ \begin{align*}h_S(X):= f_*f^*\mathbf{1}_S,\end{align*} $$

where $\mathbf {1}_S \in \mathcal {T}(S)$ is the unit for the monoidal structure.

These two constructions are dual to each other in the sense that

$$ \begin{align*} h_S(X) \simeq \underline{\operatorname{Map}}(M_S(X),\mathbf{1}_S). \end{align*} $$

Let us give the definitions leading to the premotivic $\infty $ -categories with which we will work.

The strict étale and Nisnevich topology are natural log versions of the classical étale and Nisnevich topology on $\mathbf {Sm}_S$ and the dividing topology encompasses log blow-ups. When S has trivial log structure, the $\mathrm {ver}_S$ topology is generated by open immersions $U\setminus \partial U \rightarrow U$ so that sheaves in this topology are canonically equivalent to presheaves on $\mathbf {Sm}_S$ .

Definition 3.1.5 [Reference RobaloRob12, Remark 4.12]

Let $\mathcal {C}$ be a presentable symmetric monoidal $\infty $ -category, and X an object of $\mathcal {C}$ . The stabilization $\mathrm {Stab}_X(\mathcal {C})$ of $\mathcal {C}$ by X is defined as

$$ \begin{align*}\mathrm{lim}(\cdots \mathcal{C} \xrightarrow{\Omega_X} \mathcal{C} \xrightarrow{\Omega_X} \mathcal{C}) \simeq \mathrm{colim}(\mathcal{C} \xrightarrow{\Sigma_X} \mathcal{C} \xrightarrow{\Sigma_X} \mathcal{C} \cdots),\end{align*} $$

where $\Omega _X:= \underline {\operatorname {Map}}(X,-)$ is the internal mapping space, where $\Sigma _X = - \otimes X$ and where the limit is taken in $\mathcal {C}\mathrm {at}_\infty $ and the colimit in $\mathcal {P}\mathrm {r}^{\mathrm {L}}$ .

The recent work of Park culminated in [Reference ParkPar24b], where he managed to extend the classical six-functor formalism derived from the $\mathbf {Sm}$ -premotivic $\infty $ -category

$$ \begin{align*}\mathbf{Sch}^{\mathrm{op}} \rightarrow \mathcal{DA}(-,\Lambda):= \mathrm{Stab}_{\mathbb{G}_m/1}(\mathrm{L}_{\mathbb{A}^1}\mathcal{S}\mathrm{h}_{\mathrm{Nis}}(\mathbf{Sm}_-,\mathcal{M}\mathrm{od}_\Lambda)),\end{align*} $$

where $\mathbb {G}_m/1 := \mathrm {cofib}(M_S(*) \xrightarrow {1} M_S(\mathbb {G}_m))$ and $\mathrm {L}_{\mathbb {A}^1}$ denotes the $\mathbb {A}^1$ localization, into a full six-functor formalism derived from a $\mathbf {lSm}$ -premotivic $\infty $ -category over $\mathbf {lSch}$ . This structure will be the main framework for our constructions.

Definition 3.1.6 [Reference ParkPar22, Definition 2.5.5]

The $\mathbf {lSm}$ -premotivic $\infty $ -category of motivic sheaves on $\mathbf {lSch}$ with coefficients in $\Lambda $ is defined by (the $\otimes $ highlights the symmetric monoidal structure)

$$ \begin{align*}\mathcal{DA}^{\mathrm{log}}(-,\Lambda)^\otimes:= \mathrm{Stab}_{\mathbb{G}_m^{\mathrm{log}}/1}(\mathrm{L}_{\mathbb{A}^1}\mathrm{L}_{\mathrm{ver}}\mathcal{S}\mathrm{h}_{\mathrm{dNis}}(\mathbf{lSm}_-,\mathcal{M}\mathrm{od}_\Lambda))^\otimes \in \mathrm{CAlg}(\mathcal{P}\mathrm{r}^{\mathrm{L}})\end{align*} $$

and the $\mathbf {lSm}$ -premotivic stable homotopy $\infty $ -category of motivic sheaves by

$$ \begin{align*}\mathcal{SH}^{\mathrm{log}}(-)^\otimes:= \mathrm{Stab}_{\mathbb{G}_m^{\mathrm{log}}/1}(\mathrm{L}_{\mathbb{A}^1}\mathrm{L}_{\mathrm{ver}}\mathcal{S}\mathrm{h}_{\mathrm{dNis}}(\mathbf{lSm}_{-},\mathcal{S}\mathrm{pt}))^\otimes \in \mathrm{CAlg}(\mathcal{P}\mathrm{r}^{\mathrm{L}})\end{align*} $$

where $\mathbb {G}_m^{\mathrm {log}}/1:= \mathrm {cofib}(M_S(*) \xrightarrow {1} M_S(\mathbb {G}_m^{\mathrm {log}})) \in \mathrm {L}_{\mathbb {A}^1}\mathrm {L}_{\mathrm {ver}}\mathcal {S}\mathrm {h}_{\mathrm {dNis}}(\mathbf {lSm}_-,\mathcal {M}\mathrm {od}_\Lambda )$ , with $\mathbb {G}^{\mathrm {log}}_m:= (\mathbb {P}^1,\{0,\infty \})$ .

From [Reference ParkPar22, Construction 2.5.4], when $S \in \mathbf {Sch}$ , there is a natural adjunction

defined by $\lambda (X) = (X,\mathcal {O}^\times _X)$ for $X\in \mathbf {Sm}_S$ and $\omega (X) = X \setminus \partial X$ for $X\in \mathbf {lSm}_S$ . It is easy to see that since the induced functor

$$ \begin{align*}\lambda^*: \mathcal{P}\mathrm{sh}(\mathbf{lSm_S},\ \mathcal{M}\mathrm{od}_\Lambda) \rightarrow \mathcal{P}\mathrm{sh}(\mathbf{Sm}_S,\ \mathcal{M}\mathrm{od}_\Lambda)\end{align*} $$

satisfies $\lambda ^*y_X \simeq y_{\omega (X)}$ where y denotes the Yoneda functor, so that it induces a functor

$$ \begin{align*}\begin{array}{ccccc} \lambda^* & : & \mathcal{DA}^{\mathrm{log}}(S,\Lambda) & \rightarrow & \mathcal{DA}(S,\Lambda) \\ & & M_S(X) & \mapsto & M_S(X\setminus \partial X) \end{array}\end{align*} $$

Proposition 3.1.7 [Reference ParkPar22, Construction 2.5.4]

The functor $\lambda ^*$ is an equivalence

$$ \begin{align*}\lambda^*: \mathcal{DA}^{\mathrm{log}}(S,\Lambda) \xrightarrow{\sim} \mathcal{DA}(S,\Lambda).\end{align*} $$

We keep separate notations $\mathcal {DA}^{\mathrm {log}}$ and $\mathcal {DA}$ even on log schemes with trivial log structures to distinguish the realization functors on each of these categories, see section 6. We will also refer to the étale version of the previous construction in the case of rational coefficients:

Definition 3.1.9. For $S\in \mathbf {lSch}$ , the $\infty $ -category of étale motivic sheaves on $\mathbf {lSm}_S$ with $\mathbb {Q}$ coefficients is defined by

$$ \begin{align*}\mathcal{DA}_{\text{t}}^{\mathrm{log}}(S,\mathbb{Q})^\otimes:= \mathrm{Stab}_{\mathbb{G}_m^{\mathrm{log}}/1}(\mathrm{L}_{\mathbb{A}^1}\mathrm{L}_{\mathrm{ver}}\mathcal{S}\mathrm{h}_{\text{dt}}(\mathbf{lSm}_-,\mathcal{M}\mathrm{od}_{\mathbb{Q}}))^\otimes \in \mathrm{CAlg}(\mathcal{P}\mathrm{r}^{\mathrm{L}}).\end{align*} $$

When $S\in \mathbf {Sch}$ , the same proof as the proposition 3.1.7 gives the same result in this context, so that there is a commutative diagram in $\mathrm {CAlg}(\mathcal {P}\mathrm {r}^{\mathrm {L}})$

with the functor induced by the étale sheafification on the horizontal arrows.

3.2 log-cdh-descent for log motives

For $S \in \mathbf {lSch}$ , in [Reference ParkPar24a], Park shows that $\mathcal {DA}^{\mathrm {log}}(S,\Lambda )$ satisfies a log version of cdh-descent:

Definition 3.2.1. A commutative square of log-schemes over S

is a log-cdh distinguished square if it is a cartesian square, i is a strict closed immersion, p is proper, and the induced morphism $p^{-1}(X\setminus i(Z)) \rightarrow X \setminus i(Z)$ is an isomorphism.

Proposition 3.2.2 [Reference ParkPar24a, Corollary 3.6.8]

For any log-cdh distinguished square as in definition 7.1, the commutative square of functors

with $q:= i \circ p'$ , is cartesian.

From this proposition, if we denote by $f: X \rightarrow S$ , $g: X' \rightarrow S$ , $h: Z' \rightarrow S$ the structure morphisms, knowing that the pushforward and the pullback along a log smooth morphism are right adjoints, and hence preserve cartesian squares, the following square is cartesian:

Evaluating at $\mathbf {1}_S$ , we thus have that the square

is cartesian. This implies the following in the case where $S = \mathrm {Spec}(k)$ :

Corollary 3.2.3. The inclusion of the boundary of $\mathbb {A}_{\mathbb {N}}$ induces an equivalence in $\mathcal {DA}^{\mathrm {log}}(k,\Lambda )$

$$ \begin{align*}h(\mathbb{A}_{\mathbb{N}})\xrightarrow{\sim} h(*_{\mathbb{N}}).\end{align*} $$

Proof. We apply the previous discussion to the following log-cdh square:

with i the inclusion of the zero point, and p the morphism forgetting the log structure. Therefore we see that the square

is cartesian (and cocartesian by stability of $\mathcal {DA}^{\mathrm {log}}(k,\Lambda )$ ). Since the top arrow is an equivalence, the lemma is proved.

It is not clear in general that the functor h satisfies Künneth’s isomorphism. It obviously does on log smooth log schemes over k since h is the dual of M, but we will need it for the log point $*_{\mathbb {N}^{n+m}} \simeq *_{\mathbb {N}^{n}} \times _k *_{\mathbb {N}^{m}}$ which is non-log-smooth on k. This is the content of the following lemma.

Lemma 3.2.4. Let $n,m$ be two nonnegative integers. The natural morphism

$$ \begin{align*}h(*_{\mathbb{N}^n}) \otimes h(*_{\mathbb{N}^m}) \rightarrow h(*_{\mathbb{N}^{n+m}})\end{align*} $$

is an isomorphism.

Proof. Let us consider the following cartesian square:

The classical proof of Künneth’s isomorphism in the context of a six functors formalism (see [Reference ScholzeSch22, Theorem 1.2]) relies on the projection formulae for the pushforwards

$$ \begin{align*}(p_1)_*((-) \otimes p_1^* (-)) \simeq (p_1)_*(-) \otimes (-)\end{align*} $$

and

$$ \begin{align*}(f_1)_*((-) \otimes f_1^* (-)) \simeq (f_1)_*(-) \otimes (-)\end{align*} $$

and on the base change isomorphism

$$ \begin{align*}f_1^*(f_2)_* \simeq (p_1)_*p_2^*.\end{align*} $$

Since all of the maps in the cartesian square are proper (i.e. their underlying morphism of schemes is proper), the three previous isomorphisms hold by [Reference ParkPar24a, Corollary 3.7.6, Theorem 3.7.8, Theorem 1.2.2].

Corollary 3.2.5. Let $(\underline {X},\underline {D}) \in \mathbf {Div}_k$ , and $x\in \underline {X}(k)$ , the natural closed immersion $x^{\mathrm {log}} \rightarrow \mathrm {N}_x^{\mathrm {log}}(\underline {X},\underline {D})$ induces a natural equivalence

$$ \begin{align*}h(\mathrm{N}_x^{\mathrm{log}}(\underline{X},\underline{D})) \xrightarrow{\sim} h(x^{\mathrm{log}})\end{align*} $$

3.3 Motivic tangential basepoints

We can now build a functor

$$ \begin{align*}\mathbf{Div}_{k}^{\mathbf{v}*/,op} \rightarrow \mathrm{CAlg}(\mathcal{DA}^{\mathrm{log}}(k,\Lambda))^{/\Lambda}\end{align*} $$

allowing an extension of the classical construction of the augmentation $h(X) \rightarrow \Lambda $ associated to a point $x\in \underline {X}(k)$ to the case of tangential basepoints and thus enhancing the functor induced by the cohomological motive functor at the level of arrows

$$ \begin{align*}\operatorname{Arr}(\mathbf{Div}_{k}^{\mathrm{op}}) \rightarrow \operatorname{Arr}(\mathrm{CAlg}(\mathcal{DA}^{\mathrm{log}}(k,\Lambda))) := \mathrm{CAlg}(\mathcal{DA}^{\mathrm{log}}(k,\Lambda))^{\Delta[1]}.\end{align*} $$

Remark 3.3.2. There is a sequence of functors

$$ \begin{align*}\Delta[1] \rightarrow \Delta[2] \xleftarrow[]{\sim} \widetilde{\Delta[2]}\end{align*} $$

where the first morphism is the $1$ -face map and the second contracts the middle morphism.

The category $\mathbf {Div}_{k}^{\mathbf {v}*/}$ , viewed as the category of pointed diagrams (proposition 2.2.4), is a subcategory of $\mathbf {Div}_k^{\overset {\leftarrow }{\Delta [3]}}$ , so that there is a natural functor induced by the cohomological motive functor

$$ \begin{align*}\mathbf{Div}^{\mathbf{v}*/,op}_{k} \rightarrow \mathbf{Div}_k^{\overset{\leftarrow}{\Delta[3]},op} \rightarrow \mathrm{CAlg}(\mathcal{DA}^{\mathrm{log}}(k,\Lambda))^{\overset{\leftarrow}{\Delta[3]}}.\end{align*} $$

Since the image of this composition lands in $\mathcal {DA}^{\mathrm {log}}(k,\Lambda )^{\widetilde {\Delta [2]}}$ , there is an induced morphism

$$ \begin{align*}\mathbf{Div}_{k}^{\mathbf{v}*/,op} \rightarrow \mathrm{CAlg}(\mathcal{DA}^{\mathrm{log}}(k,\Lambda))^{\widetilde{\Delta[2]}}\end{align*} $$

by the following lemma:

Lemma 3.3.3. The functor

$$ \begin{align*}\mathrm{CAlg}(\mathcal{DA}^{\mathrm{log}}(k,\Lambda))^{\widetilde{\Delta[2]}} \rightarrow \mathrm{CAlg}(\mathcal{DA}^{\mathrm{log}}(k,\Lambda))^{\overset{\leftarrow}{\Delta[3]}}\end{align*} $$

induced by the inclusion $\overset {\leftarrow }{\Delta [3]} \rightarrow \widetilde {\Delta [2]}$ is fully faithful.

Proof. The functor $\overset {\leftarrow }{\Delta [3]} \rightarrow \widetilde {\Delta [2]}$ is the $1$ -categorical localization of the category $\overset {\leftarrow }{\Delta [3]}$ by the middle morphism. Note that this is given by the pushout of the following diagram

where the horizontal morphism selects the middle morphism and $\Delta [1]^\simeq $ is the groupoid associated to $\Delta [1]$ .

The category $\widetilde {\Delta [2]}$ is still the pushout of this diagram in $\mathcal {C}\mathrm {at}_\infty $ after application of the nerve functor. This is not straightforward since the nerve functor does not preserve pushouts in general, see [Reference Hackney, Ozornova, Riehl and RovelliHORR24, Figure 1.1] for a counterexample, but it can be checked by computing the pushout explicitly.

More precisely, if we denote by $\mathcal {C}$ the pushout of the diagram above, there is a canonical arrow $\mathcal {C} \rightarrow \widetilde {\Delta [2]}$ . Note that the objects of $\mathcal {C}$ are given by the pushouts of objects of the categories involved in the diagram, as can be checked by applying the homotopy category functor $ho(-)$ to it (note that as a left adjoint to the nerve functor it preserves colimits). Now, since the functor $\displaystyle \Delta [1] \rightarrow \overset {\leftarrow }{\Delta [3]}$ is fully faithful, we can describe all the mapping spaces of $\mathcal {C}$ using [Reference Haine, Ramzi and SteinebrunnerHRS25, Theorem 0.1] which are all straightforwardly shown to be weakly equivalent to the corresponding mapping space in $\widetilde {\Delta [2]}$ (they are actually homeomorphic since they are reduced to a point, or empty). Hence the functor $\mathcal {C}\rightarrow \widetilde {\Delta [2]}$ is an equivalence of $\infty $ -categories.

This shows that, as $\infty $ -categories, $\widetilde {\Delta [2]}$ is the Dwyer–Kan localization of $\overset {\leftarrow }{\Delta [3]}$ along the middle morphism, see [Reference LandLan21, Proposition 2.4.8]. The lemma now comes from the universal property of localization [Reference LandLan21, Definition 2.4.2].

Hence, the cohomological motive functor defines a functor after composing with the sequence from remark 3.3.2:

$$ \begin{align*} \mathbf{Div}_{k}^{\mathbf{v}*/,op} & \rightarrow \mathrm{CAlg}(\mathcal{DA}^{\mathrm{log}}(k,\Lambda))^{\widetilde{\Delta[2]}}\\ & \simeq \mathrm{CAlg}(\mathcal{DA}^{\mathrm{log}}(k,\Lambda))^{\Delta[2]} \\ & \rightarrow \mathrm{CAlg}(\mathcal{DA}^{\mathrm{log}}(k,\Lambda))^{\Delta[1]} \simeq \mathrm{Arr}(\mathrm{CAlg}(\mathcal{DA}^{\mathrm{log}}(k,\Lambda))). \end{align*} $$

Since $\mathrm {CAlg}(\mathcal {DA}^{\mathrm {log}}(k,\Lambda ))^{/\Lambda }$ is the faithful subcategory of $\mathrm {Arr}(\mathrm {CAlg}(\mathcal {DA}^{\mathrm {log}}(k,\Lambda )))$ consisting of arrows whose codomain is $\Lambda $ , and since this category is obtained as a fibre product of $\mathrm {CAlg}(\mathcal {DA}^{\mathrm {log}}(k,\Lambda ))^{\Delta [1]}$ and the point category on $\mathrm {CAlg}(\mathcal {DA}^{\mathrm {log}}(k,\Lambda ))$ , this defines a functor

$$ \begin{align*}\mathbf{Div}^{\mathbf{v}*/,op}_{k} \rightarrow \mathrm{CAlg}(\mathcal{DA}^{\mathrm{log}}(k,\Lambda))^{/\Lambda}.\end{align*} $$

Definition 3.3.5. We call motivic tangential augmentation (or motivic tangential basepoint) functor the functor

given by the composition of the sequence of arrows on the right. We denote by $h(\mathbf {x}): h(X) \rightarrow \Lambda $ the resulting arrow.

Remark 3.3.6. In a more direct fashion, from a diagram

by a choice of homotopy inverse of the central equivalence, we get a diagram

that we compose to get an augmentation $h(X) \rightarrow \Lambda $ .

This construction actually works in the same way in any ‘category of log motives’ in which the cohomological motives of the standard log point and of $\mathbb {A}_{\mathbb {N}}$ are equivalent, and in which Künneth’s isomorphism holds for the cohomological motives of products of the log point.

Remark 3.3.7. Note that if $\mathbf {x}$ is just given by a rational point $x \in \underline {X}(k)\setminus \underline {D}(k)$ , then the motivic augmentation defined by $\mathbf {x}$ is the classical augmentation $h(x)$ . Indeed the pointed diagram associated with $\mathbf {x}$ is

so that the morphism

is the expected one.

4.1 The bar construction

Let us first summarize facts about the bar construction. In this section, nothing is new except the description of the homotopy Hopf algebroid structure in an $\infty $ -categorical context. The main reference here is [Reference LurieLur17, Sections 4.4.2, 5.2.2].

Let $\mathcal {C}$ be a cocomplete symmetric monoidal $\infty $ -category such that the tensor product $\otimes : \mathcal {C} \times \mathcal {C} \rightarrow \mathcal {C}$ commutes with colimits in each variable.

Proposition 4.1.1 [Reference LurieLur17, Construction 4.4.2.7, Example 4.4.2.11]

For $A \in \mathrm {CAlg}(\mathcal {C})$ , $M,N\in \mathrm {Mod}_A(\mathcal {C})$ , there is a canonical simplicial diagram $\mathrm {Bar}_A(M,N)_\bullet $ in $\mathrm {Mod}_A(\mathcal {C})$ informally given by:

such that its colimit is the relative tensor product of M and N over A:

$$ \begin{align*}M\otimes_A N:= \mathrm{Bar}_A(M,N):= \mathrm{colim}_\Delta \mathrm{Bar}_A(M,N)_\bullet.\end{align*} $$

It is the bar construction of A on M and N.

Degreewise $\mathrm {Bar}_A(M,N)_n \simeq M \otimes A^n \otimes N$ with face maps induced by the product

and the actions on M and N

The degeneracies are given by the unit map

For $A \in \mathrm {CAlg}(\mathcal {C})$ , the relative tensor product over A endows the category $\mathrm {Mod}_A(\mathcal {C})$ with a symmetric monoidal $\infty $ -category structure [Reference LurieLur09, Theorem 4.5.2.1].

We will apply the relative tensor product in a more restrictive case, that is when A is a biaugmented commutative algebra object in $\mathcal {C}$ with augmentations (i.e. algebra morphisms) . We denote by $\mathbf {1}_x, \mathbf {1}_y$ the unit element of $\mathcal {C}^\otimes $ viewed as commutative A-algebras through the maps x and y.

Definition 4.1.2. We call bar construction of A based at x and y the object

$$ \begin{align*} \;_x\mathrm{Bar}_y(A):= \mathrm{Bar}_A(\mathbf{1}_y,\mathbf{1}_x) \in \mathrm{Mod}_A(\mathcal{C}). \end{align*} $$

We recall that this object comes with a natural structure of commutative A-algebra object in $\mathcal {C}$ . Indeed, by applying [Reference LurieLur17, Proposition 3.2.4.7, Corollary 3.2.4.8] to $\mathrm {Mod}_A(\mathcal {C})$ , we see that $\mathrm {CAlg}_A(\mathcal {C}) = \mathrm {CAlg}(\mathrm {Mod}_A(\mathcal {C}))$ has coproducts and that these coproducts coincide with iterated relative tensor products.

In particular,

$$ \begin{align*} \;_x\mathrm{Bar}_y(A) = \mathbf{1}_y\otimes_A \mathbf{1}_x = \mathbf{1}_y\sqcup_A \mathbf{1}_x \in \mathrm{CAlg}_A(\mathcal{C}). \end{align*} $$

This way, it is easy to see that the bar construction defines a functor

$$ \begin{align*} \mathrm{Bar}:\mathrm{CAlg}(\mathcal{C})^{//\mathbf{1}} \rightarrow \mathrm{CAlg}(\mathcal{C}) \end{align*} $$

Note that the A-algebra structure is defined by the map $\mkern1mu_x\nabla _y$ described by

$$ \begin{align*} \;_x\mathrm{Bar}_y(A) \otimes_A \,_x\mathrm{Bar}_y(A) & = (\mathbf{1}_y \otimes_A \mathbf{1}_x) \otimes_A (\mathbf{1}_y \otimes_A \mathbf{1}_x)\\ & \simeq (\mathbf{1}_y \otimes_A \mathbf{1}_y) \otimes_A (\mathbf{1}_x \otimes_A \mathbf{1}_x)\\ & \rightarrow \mathbf{1}_y \otimes_A \mathbf{1}_x\\ & = \!_x\mathrm{Bar}_y(A) \end{align*} $$

and by the natural map

$$ \begin{align*} \!_x\eta_y:\mathbf{1} \rightarrow A \rightarrow \mathbf{1}_y \sqcup_A \mathbf{1}_x. \end{align*} $$

Let us recall the definition of a homotopy Hopf algebra in an $\infty $ -category in the sense of [Reference ErgusErg22, Definition 4.0.1, Corollary 4.1.4], which is a bialgebra with a homotopy antipode. We remind the reader that the category of bialgebra objects in $\mathcal {C}$ is defined to be $\mathrm {Alg}(\mathrm {CAlg}(\mathcal {C})^{\mathrm {op}}).$

Definition 4.1.4. A homotopy Hopf algebra object $H \in \mathcal {C}$ is a bialgebra object endowed with a map

$$ \begin{align*}s: H \rightarrow H\end{align*} $$

such that H is a Hopf algebra in the homotopy category of $\mathcal {C}$ with antipode s.

We now prove a general structure result for the bar construction:

This theorem shows that the data of the pointed bar constructions of A, taken over all augmentations of A, carries a structure dual to that of a groupoid in the homotopy category of $\mathcal {C}$ . Here, by a groupoid structure we mean the one carried for instance by the system of pointed path spaces of a topological space X: namely, the collection of all pointed path spaces of X, together with the associative composition of paths, the existence of inverse paths, and the unit path at each point of X. It is in this sense that we will interpret and work with the notion of a homotopy Hopf algebroid in the following sections. In particular, in subsection 4.3 we will define the motivic analogue of the fundamental groupoid in the presence of a motivic t-structure.

Note that we have defined the structure of a homotopy Hopf algebroid only in a weak homotopical sense. In an $\infty $ -categorical framework, one would usually formalize such a structure by verifying all possible higher coherences, so our notion is not as rigorous as it could be in that setting. However, lifting this definition to an $\infty $ -categorical level lies beyond the scope of this article, as we only require the structure to be well defined in the homotopy category of $\mathcal {C}$ .

The rest of this subsection is dedicated to the proof of theorem 4.1.5. This proof consists in the lemmas 4.1.6, 4.1.7, 4.1.8, 4.1.9 and 4.1.10.

Let us first describe the $\infty $ -categorical structure of bialgebra that completes the structure of commutative algebra object of the pointed bar construction:

The complete and careful proof of this fact is [Reference LurieLur17, Theorem 5.2.2.17], where the bar construction is enhanced into a functor

$$ \begin{align*} \mathrm{Bar}: \mathrm{CAlg}^{/\mathbf{1}}(\mathcal{C})^{\mathrm{op}} \rightarrow \mathrm{Alg}(\mathcal{C}^{\mathrm{op}}) \end{align*} $$

which shows that the bar construction over a single augmentation has a natural coalgebra structure. The commutative algebra and coalgebra structures are compatible since the commutative algebra structure can be directly derived from the previous functor: Since

$$ \begin{align*}\mathrm{CAlg}^{/\mathbf{1}}(\mathrm{CAlg}(\mathcal{C})) \simeq \mathrm{CAlg}^{/\mathbf{1}}(\mathcal{C})\end{align*} $$

[Reference LurieLur17, Corollary 2.4.3.10], we can apply the bar construction directly in $\mathrm {CAlg}(\mathcal {C})$ which gives a functor

$$ \begin{align*}\mathrm{Bar}': \mathrm{CAlg}^{/\mathbf{1}}(\mathcal{C}) \simeq \mathrm{CAlg}^{/\mathbf{1}}(\mathrm{CAlg}(\mathcal{C}))^{\mathrm{op}} \rightarrow \mathrm{Alg}(\mathrm{CAlg}(\mathcal{C})^{\mathrm{op}}).\end{align*} $$

Since the forgetful functor $\mathrm {CAlg}(\mathcal {C}) \rightarrow \mathcal {C}$ commutes with colimits of simplicial diagrams [Reference LurieLur17, Corollary 3.2.3.2], we have that $\mathrm {Bar} = \mathrm {Bar}'$ .

The coproduct inducing the bialgebra structure of the bar construction can be derived from the more general deconcatenation map:

Lemma 4.1.7. Suppose A has three augmentations $x,y,z$ . Then there is a natural deconcatenation map in $\mathrm {CAlg}(\mathrm {Mod}_A(\mathcal {C}))$

Proof. The map is built as follows:

$$ \begin{align*} \!_x\mathrm{Bar}_z(A) & = \mathbf{1}_z\otimes_A \mathbf{1}_x\\ & \simeq \mathbf{1}_z \otimes_A A \otimes_A \mathbf{1}_x\\ & \rightarrow \mathbf{1}_z \otimes_A \mathbf{1}_y \otimes_A \mathbf{1}_x\\ & \simeq (\mathbf{1}_z \otimes_A \mathbf{1}_y) \otimes_{\mathbf{1}} (\mathbf{1}_y \otimes_A \mathbf{1}_x)\\ & = \!_y\mathrm{Bar}_z(A) \otimes \!_x\mathrm{Bar}_y(A). \end{align*} $$

Note that the line 4 uses both the associativity and commutativity of the relative tensor product and a change of base algebra [Reference LurieLur17, Section 4.5.3].

Over a single augmentation x, the deconcatenation map is the coproduct map on the bar construction of A over x. Note that there is as well a canonical augmentation given by the natural morphism

$$ \begin{align*}\epsilon_x: \mathbf{1}_x \sqcup_A \mathbf{1}_x \rightarrow \mathbf{1}_x.\end{align*} $$

We can readily prove the counit axiom:

Lemma 4.1.8. For $x,y$ augmentations of A, the following equality

$$ \begin{align*}(\epsilon_y \otimes \mathrm{Id})\circ \mkern1mu^{}_x\Delta_y^y = \mathrm{Id}\end{align*} $$

and

$$ \begin{align*}(\mathrm{Id} \otimes \epsilon_x)\circ \mkern1mu^{}_x\Delta_y^x = \mathrm{Id}\end{align*} $$

hold in the homotopy category of $\mathcal {C}$ .

Proof. Let us treat the first case, as the second one is similar. The composition is

$$ \begin{align*} \mathbf{1}_y\otimes_A \mathbf{1}_x & \simeq \mathbf{1}_y \otimes_A A \otimes_A \mathbf{1}_x\\[-1pt] & \rightarrow \mathbf{1}_y \otimes_A \mathbf{1}_y \otimes_A \mathbf{1}_x\\[-1pt] & \simeq (\mathbf{1}_y \otimes_A \mathbf{1}_y) \otimes (\mathbf{1}_y \otimes_A \mathbf{1}_x)\\[-1pt] & \rightarrow \mathbf{1}_y \otimes (\mathbf{1}_y \otimes_A \mathbf{1}_x)\\[-1pt] & \simeq \mathbf{1}_y \otimes_A \mathbf{1}_x \end{align*} $$

This morphism is given by the canonical morphism

$$ \begin{align*}\mathbf{1}_x \rightarrow \mathbf{1}_y \otimes_A \mathbf{1}_x\end{align*} $$

and by the morphism on the left component

$$ \begin{align*}\mathbf{1}_y \rightarrow \mathbf{1}_y \otimes_A \mathbf{1}_y \rightarrow \mathbf{1}_y\end{align*} $$

which is the identity by the algebra structure of $\mathbf {1}_y \otimes _A \mathbf {1}_y$ .

Let us now consider the involution

$$ \begin{align*}\!_xs_y: \;_x\mathrm{Bar}_y(A) = \mathbf{1}_y \otimes_A \mathbf{1}_x \rightarrow \mathbf{1}_x \otimes_A \mathbf{1}_y = \;_y\mathrm{Bar}_x(A)\end{align*} $$

exchanging the two augmentations. Then, we can check the coinverse axiom. This proves in particular that $\mkern1mu_x\mathrm {Bar}_x(A)$ is a homotopy Hopf algebra for all augmentations x of A.

Lemma 4.1.9. The commutative algebras $\mkern1mu_x\mathrm {Bar}_x(A)$ , $\mkern1mu_x\mathrm {Bar}_y(A)$ and $\mkern1mu_y\mathrm {Bar}_x(A)$ equipped with the deconcatenation $\mkern1mu_x^{}\Delta _x^y$ , product $\mkern1mu_x\nabla _y$ , unit $\mkern1mu_x\eta _y: \mathbf {1} \rightarrow \mkern1mu_x\mathrm {Bar}_y(A)$ , counit $\epsilon _x: \mkern1mu_x\mathrm {Bar}_x(A) \rightarrow \mathbf {1}$ and involution $\mkern1mu_xs_y$ fit in the homotopy commutative diagrams:

and

Proof. We only prove the lemma in the upper diagram since the other case works in the same way. The upper composition can be described by the sequence

$$ \begin{align*} \mathbf{1}_x\otimes_A \mathbf{1}_x & \simeq \mathbf{1}_x \otimes_A A \otimes_A \mathbf{1}_x\\[-1pt] & \rightarrow \mathbf{1}_x \otimes_A \mathbf{1}_y \otimes_A \mathbf{1}_x\\[-1pt] & \simeq (\mathbf{1}_x \otimes_A \mathbf{1}_y) \otimes (\mathbf{1}_y \otimes_A \mathbf{1}_x)\\[-1pt] & \simeq (\mathbf{1}_y \otimes_A \mathbf{1}_x) \otimes (\mathbf{1}_y \otimes_A \mathbf{1}_x)\\[-1pt] & \rightarrow \mathbf{1}_y \otimes_A \mathbf{1}_x \end{align*} $$

where the last arrow is identified with the componentwise product. Note that this last arrow can be described by the change of base ring equivalence [Reference LurieLur17, 4.5.3]:

$$ \begin{align*}(\mathbf{1}_y \otimes_A \mathbf{1}_x) \otimes (\mathbf{1}_y \otimes_A \mathbf{1}_x) \simeq \mathbf{1}_y \otimes_A (\mathbf{1}_x \otimes_A \mathbf{1}_x) \rightarrow \mathbf{1}_y \otimes_A \mathbf{1}_x\end{align*} $$

where the last arrow is induced by the product on $\mathbf {1}_x$ . Then the full composition is homotopic to the composition

$$ \begin{align*} \mathbf{1}_x \otimes_A \mathbf{1}_x & \simeq A \otimes_A \mathbf{1}_x \otimes_A \mathbf{1}_x\\ & \rightarrow \mathbf{1}_y \otimes_A \mathbf{1}_x \otimes_A \mathbf{1}_x\\ & \rightarrow \mathbf{1}_y \otimes_A \mathbf{1}_x \end{align*} $$

where the last arrow is again induced by the product on $\mathbf {1}_x$ . This is itself identified with the map

$$ \begin{align*} \mathbf{1}_x \otimes_A \mathbf{1}_x & \rightarrow \mathbf{1}_x\\ & \simeq A \otimes_A \mathbf{1}_x\\ & \rightarrow \mathbf{1}_y \otimes_A \mathbf{1}_x \end{align*} $$

and this composition is homotopic to $\mkern1mu_y\eta _x\circ \epsilon _x$ .

Finally it remains to check the coassociativity axiom:

Lemma 4.1.10. Let $x,y,z,t$ four augmentations of A. Then the equality

$$ \begin{align*}(\mkern1mu^{}_y\Delta_t^z \otimes \mathrm{Id})\circ\mkern1mu^{}_x\Delta_t^y = (\mathrm{Id} \otimes \mkern1mu^{}_x\Delta_z^y)\circ\mkern1mu^{}_x\Delta_t^z\end{align*} $$

holds in the homotopy category of $\mathcal {C}$ .

Proof. The lemma is proved by checking that the two compositions are described (up to associativity of the tensor product) by the composition

$$ \begin{align*}\mathbf{1}_t \otimes_A \mathbf{1}_x \simeq \mathbf{1}_t \otimes_A A \otimes_A A \otimes_A \mathbf{1}_x \rightarrow \mathbf{1}_t \otimes_A \mathbf{1}_z \otimes_A \mathbf{1}_y \otimes_A \mathbf{1}_x\end{align*} $$

and then two successive changes of base algebras.

Remark 4.1.12. Since it is defined as the colimit of a simplicial object in $\mathcal {C}$ , the bar construction is a filtered object in $\mathcal {C}$ [Reference LurieLur17, Remark 1.2.4.2]. The filtration is given by the geometric realization of skeletons that we denote for $n \in \mathbb {N}$ by

$$ \begin{align*}\mathrm{Bar}_{\leq n}:= \mathrm{colim}_\Delta\;\mathrm{sk}_n\mathrm{Bar}_\bullet.\end{align*} $$

In a $1$ -categorical context, $\mkern1mu_x\mathrm {Bar}_x(A)$ has a structure of filtered Hopf algebra since the operations we introduced are compatible with the filtration, see [Reference WojtkowiakWoj93], [Reference Ignacio Burgos Gil and FresánBGF25, Section 3.5] for a full description. Furthermore, $\mkern1mu_x\mathrm {Bar}_y(A)$ has the structure of a filtered comodule under the coactions of $\mkern1mu_x\mathrm {Bar}_x(A)$ and $\mkern1mu_y\mathrm {Bar}_y(A)$ . We do not know if such a result can be extended to an $\infty $ -categorical context, but we will not need it in the rest of this paper, since we will only describe the filtered structure at the level of realizations.

Example 4.1.13. Let us detail the construction in the classical case of a commutative differential graded algebra A, viewed as an object of the $\infty $ -category of chain complexes over a field K, endowed with two augmentations $x,y: A \rightarrow K$ , since this will come in handy in the computation of the realizations of the motivic fundamental groupoid. By [Reference ArakawaAra23, Corollary 3.13], the geometric realization of a simplicial object $X_\bullet $ in $\mathcal {M}\mathrm {od}_K$ is given by the total complex associated with the bicomplex $\mathrm {M}(X_\bullet )$ where $\mathrm {M}$ is the Moore complex functor. Hence,

$$ \begin{align*}\!_x\mathrm{Bar}_y(A) \simeq \mathrm{Tot}\Big(\mathrm{M}(\mathrm{Bar}_A(\mathbf{1}_x,\mathbf{1}_y)_\bullet)\Big)\end{align*} $$

If we use cohomological conventions, then

$$ \begin{align*}\!_x\mathrm{Bar}^n_y(A) \simeq \bigoplus_{\substack{(p,q) \\ p-q = n}} \bigoplus_{i_1 + \dots + i_q = p} A^{i_1} \otimes \dots \otimes A^{i_q}.\end{align*} $$

Hence, a degree n element of the bar complex is denoted by

$$ \begin{align*}[\omega_1|\cdots|\omega_q], \text{ with } \omega_i \in A, \text{ and } \sum_i \mathrm{deg}(\omega_i) = n + q.\end{align*} $$

In this construction, the filtration on the bar complex is interpreted as the classical length filtration, that is $\;_x\mathrm {Bar}_y(A)_{\leq k}$ is the subcomplex spanned by elements of the form given above, with $q \leq k$ .

4.2 The motivic pointed path space based at tangential basepoints

We now have all the tools in hand to build the motivic pointed path space based at tangential basepoints. Let $X\in \mathbf {Div}_k$ , $\mathbf {x}, \mathbf {y}$ two tangential basepoints of X and $h(\mathbf {x}), h(\mathbf {y}): h(X) \rightarrow \Lambda $ the corresponding motivic tangential augmentations. Then $h(X)$ has the structure of a biaugmented commutative algebra object in $\mathcal {DA}^{\mathrm {log}}(k,\Lambda )$ . We can then apply the bar construction to this datum.

Definition 4.2.1. We call the motivic pointed path space of X based at $\mathbf {x}$ and $\mathbf {y}$ the commutative algebra object

$$ \begin{align*}\mkern1mu^{}_{\mathbf{x}}\mathrm{P}^{\mathrm{m}}_{\mathbf{y}}X:= h(\mathbf{1_y}) \otimes_{h(X)} h(\mathbf{1_x}):= \;_{h(\mathbf{x})}\mathrm{Bar}_{h(\mathbf{y})}(h(X))\end{align*} $$

This determines a functor

$$ \begin{align*}\mathbf{Div}^{\mathbf{v}*//}_{k} \rightarrow \mathrm{CAlg}(\mathcal{DA}^{\mathrm{log}}(k,\Lambda))\end{align*} $$

By the results of the previous subsection, the motivic based path spaces are a highly structured algebra in $\mathcal {DA}^{\mathrm {log}}(k,\Lambda )$ . Precisely, it is a straightforward consequence of theorem 4.1.9 that the following proposition holds:

4.2.1 The cosimplicial pointed path space

When $\mathbf {x}, \mathbf {y}$ are ordinary basepoints, as mentioned in the introduction of this article, the definition of the motivic pointed path space in the case of ordinary basepoints as explained by Deligne and Goncharov in [Reference Deligne and GoncharovDG05] or by Levine and H. Esnault in [Reference Esnault and LevineEL11] relies on a translation to the language of algebraic geometry, due to Wojtkowiak in [Reference WojtkowiakWoj93], of the topological definition of the pointed path space. We generalize this definition in our context, since we will need it to compute the realization of the motivic pointed path space in the next section:

For $X \in \mathbf {lSch}_k$ let us consider the cosimplicial log scheme given by

$$ \begin{align*} X^{[0,1]}:= X^{\Delta[1]}. \end{align*} $$

For $n \in \mathbb {N}$ ,

$$ \begin{align*}(X^{\Delta[1]})^{\mathbf{n}} = \prod_{s\in \Delta[1]_n}X = X \times X^n \times X,\end{align*} $$

the i-th coface map for $0 \leq i \leq n+1$ is given by

$$ \begin{align*}(x_0,\dots, x_i, \dots,x_{n+1}) \longmapsto (x_0, \dots, x_i,x_i,x_{i+1}, \dots, x_{n+1})\end{align*} $$

and the i-th codegeneracy map for $0 \leq i \leq n-1$ by the projection

$$ \begin{align*}X^{i+1} \times X \times X^{n-i} \rightarrow X^{i+1} \times X^{n-i}.\end{align*} $$

Remark 4.2.5. $X^{[0,1]}$ comes equipped with a map

$$ \begin{align*}X^{[0,1]} \rightarrow X^{\partial\Delta[1]} = X \times X\end{align*} $$

induced by the inclusion $\partial \Delta [1] \subset \Delta [1]$ , where the right term is the constant cosimplicial log scheme. This map is degreewise the projection on the first and last factor.

Definition 4.2.6. For $\mathbf {x},\mathbf {y}$ two tangential basepoints on $X\in \mathbf {Div}_k$ , we call cosimplicial path space of X based at $\mathbf {x},\mathbf {y}$ the cosimplicial object of $\mkern1mu^{\mathbf {v}}\mathbf {lSch}_k$ defined by the pullback

Let us explain how this fibre product is defined in the category of cosimplicial diagrams of log schemes with virtual morphisms between them. Its n-cosimplices are $X^n$ , the degeneracies are all the projections $X^{n} \rightarrow X^{n-1}$ , and the cofaces are the same as the ones in the upper description, except for the $0$ -th and the $(n+1)$ -th, defined by the product of the virtual point $\mathbf {y}$ with the identity of $X^{n+1}$ and by the product of the identity of $X^{n+1}$ with the virtual point $\mathbf {x}$ .

Since we cannot view this object in a log motivic formalism in a canonical way (there is no category of virtual log motives) we cannot compare the bar construction of $h(X)$ based at $\mathbf {x},\mathbf {y}$ with a simplicial motive associated with the cosimplicial path space based at $\mathbf {x},\mathbf {y}$ , as in the case with ordinary basepoints. Nevertheless we can check that our definition coincides with the latter in the case with ordinary basepoints.

Proposition 4.2.7. When $\mathbf {x},\mathbf {y}$ are rational points of $\underline {X}\setminus \underline {D}$ , there is a natural equivalence in $\mathrm {CAlg} (\mathcal {DA}^{\mathrm {log}}(k,\Lambda ))$ :

$$ \begin{align*}\displaystyle\mkern1mu^{}_{\mathbf{x}}\mathrm{P}^m_{\mathbf{y}}X \simeq \mathrm{colim}_{\Delta} h(\mkern1mu_{\mathbf{x}}^{}\mathrm{P}^{\bullet}_{\mathbf{y}} X).\end{align*} $$

Proof. Once we check that the motivic augmentations associated with $\mathbf {x},\mathbf {y}$ are the usual augmentation associated with the corresponding rational points $x,y$ which is done in remark 3.3.7, this is a classical fact, proved in [Reference IwanariIwa20, Theorem A.7] for instance, or see the same proof applied in the Betti case in the discussion following theorem 5.2.11.

4.3 The motivic fundamental groupoid at tangential basepoints

We can now define the motivic fundamental groupoid based at tangential basepoints. Although we work in the context of mixed Tate motives, what follows works in a similar fashion in the general case of the existence of a motivic t-structure.

Recall that the triangulated category of mixed Tate motives over a field k is classically defined to be the smallest triangulated full subcategory of the triangulated category of mixed motives à la Voevodsky $\mathbf {DM}(k,\mathbb {Q})$ [Reference VoevodskyVoe00] generated by the Tate objects $\mathbb {Q}(n)$ for $n\in \mathbb {Z}$ . In the language of $\infty $ -categories, the right definition is (see [Reference IwanariIwa20, Section 6.2] for details):

Definition 4.3.1. Let $\mathcal {DM}(k,\mathbb {Q})$ be the stable $\infty $ -category of mixed motives over k with $\mathbb {Q}$ coefficients (see for instance [Reference IwanariIwa20, Reference ParkPar22] for a definition). Then we let $\mathcal {DMT}_{\mathrm {gm}}(k,\mathbb {Q})$ be the smallest stable subcategory of $\mathcal {DM}(k,\mathbb {Q})$ generated by the Tate objects $\mathbb {Q}(n)$ . The stable $\infty $ -category of mixed Tate motives over k is the full subcategory of $\mathcal {DM}(k,\mathbb {Q})$ defined by

$$ \begin{align*}\mathcal{DMT}(k,\mathbb{Q}):= \mathrm{Ind}\mathcal{DMT}_{\mathrm{gm}}(k,\mathbb{Q}).\end{align*} $$

Suppose now that k satisfies the Beilinson–Soulé vanishing conjecture, $\mathcal {DMT}(k,\mathbb {Q})$ and $\mathcal {DMT}_{\mathrm {gm}}(k,\mathbb {Q})$ come with a t-structure [Reference LevineLev93], [Reference IwanariIwa14, Section 7] (defined on the homotopy categories see [Reference LurieLur17, Definition 1.2.1.4]). The heart of $\mathcal {DMT}_{\mathrm {gm}}(k,\mathbb {Q})$ is the neutral tannakian (1-)category of mixed Tate motives over k, denoted by $\mathbf {MT}(k,\mathbb {Q})$ . There is a $0$ -cohomology functor

$$ \begin{align*}\mathrm{H}_{\mathrm{mot}}^0 : \mathcal{DMT}_{\mathrm{gm}}(k,\mathbb{Q}) \rightarrow \mathbf{MT}(k,\mathbb{Q}).\end{align*} $$

The heart of the t-structure on $\mathcal {DMT}(k,\mathbb {Q})$ is $\mathrm {Ind}\mathbf {MT}(k,\mathbb {Q})$ and the corresponding $0$ -cohomology functor $\mathcal {DMT}(k,\mathbb {Q})$ is the indization of $\mathrm {H}^0_{\mathrm {mot}}$ .

Recall as well from [Reference Cisinski and DégliseCD19, Theorem 16.2.18, Theorem 16.1.4] that there is an equivalence of symmetric monoidal $\infty $ -categories

$$ \begin{align*}\mathcal{DA}_{\text{t}}(k,\mathbb{Q})^\otimes \simeq\mathcal{DM}(k,\mathbb{Q})^\otimes\end{align*} $$

so that $\mathcal {DMT}(k,\mathbb {Q})$ can be viewed as a full subcategory of $\mathcal {DA}_{\text {\'{e}t}}(k,\mathbb {Q})$ .

If $X \in \mathbf {Div}_k$ then the structure of homotopy Hopf algebroid of the family of motivic path spaces at tangential basepoint is preserved after étale localization, i.e. after applying the functor induced by étale sheafification. Furthermore, for $\mathbf {x},\mathbf {y}$ tangential basepoints on X, the motivic path space based at $\mathbf {x},\mathbf {y}$ is an element of $\mathcal {DMT}(k,\mathbb {Q})$ by definition. We now show that taking its $0$ -th cohomology group carries the whole structure to $\mathrm {Ind}\mathbf {MT}(k,\mathbb {Q})$ :

Proof. The only potential obstruction to this lies in the fact that the functor $\mathrm {H}^0_{\mathrm {mot}}$ is not symmetric monoidal. Let us show that for $\mathbf {x,x',y,y'}$ tangential basepoints on X it satisfies

$$ \begin{align*}\mathrm{H}^0_{\mathrm{mot}}((\mkern1mu^{}_{\mathbf{x}}\mathrm{P}_{\mathbf{x'}}^mX)^{\text{t}}\otimes (\mkern1mu^{}_{\mathbf{y}}\mathrm{P}_{\mathbf{y'}}^mX)^{\text{t}}) \simeq \mathrm{H}^0_{\mathrm{mot}}((\mkern1mu^{}_{\mathbf{x}}\mathrm{P}_{\mathbf{x'}}^mX)^{\text{t}})\otimes \mathrm{H}^0_{\mathrm{mot}}((\mkern1mu^{}_{\mathbf{y}}\mathrm{P}_{\mathbf{y'}}^mX)^{\text{t}}).\end{align*} $$

This naturally implies the theorem. We know from [Reference BiglariBig07, Example 3.3.2] that the t-structure on $\mathcal {DMT}_{\mathrm {gm}}(k,\mathbb {Q})$ is compatible with the tensor structure, which implies that the family of cohomology functors $\mathrm {H}^i_{\mathrm {mot}}$ satisfies the Künneth formula.

It is now clear that it suffices to show that for $\mathbf {x},\mathbf {y}$ tangential baspoints on X, the cohomology in negative degrees of $\mkern1mu^{}_{\mathbf {x}}\mathrm {P}_{\mathbf {y}}^mX$ is zero. It suffices to check this for the filtered pieces $\mkern1mu^{}_{\mathbf {x}}\mathrm {P}_{\mathbf {y}}^mX_{\leq n}$ . We only need to check that the Betti realization of this motive is concentrated in nonpositive degree because the Betti realization of mixed Tate motives is conservative, see [Reference Deligne and GoncharovDG05, Proposition 2.14] (the fully faithfulness of the Hodge realization implies its conservativity, and hence the conservativity of the Betti realization). This is trivial from the description at the end of section 5.2.

From this, we are able to give the following definition of the motivic fundamental groupoid at tangential basepoints:

Definition 4.3.4. Let $X \in \mathbf {Div}_k$ , with tangential basepoints $\mathbf {x},\mathbf {y}$ , and suppose X is of mixed Tate type. Then the motivic fundamental torsor at $\mathbf {x},\mathbf {y}$ is defined by

$$ \begin{align*}\pi_1^m(X,\mathbf{x},\mathbf{y}):= \operatorname{Spec}\mathrm{H}^0_{\mathrm{mot}}((\mkern1mu^{}_{\mathbf{x}}\mathrm{P}_{\mathbf{y}}^mX)^{\text{t}}) \in \mathrm{Pro}\mathbf{MT}(k,\mathbb{Q})-\mathbf{Sch}\end{align*} $$

Where $\mathrm {Pro}\mathbf {MT}(k,\mathbb {Q})-\mathbf {Sch}$ is the opposite of the category of commutative algebra objects in $\mathrm {Ind}\mathbf {MT}(k,\mathbb {Q}).$

The following proposition is trivial from proposition 4.2.3

Precisely, for $\mathbf {x},\mathbf {y},\mathbf {z}$ three tangential basepoints of X, there is an operation of composition of paths

$$ \begin{align*}\pi^m_1(X,\mathbf{y},\mathbf{z}) \times \pi^m_1(X,\mathbf{x},\mathbf{y}) \rightarrow \pi^m_1(X,\mathbf{x},\mathbf{z})\end{align*} $$

which is associative in the obvious sense, an isomorphism of inversion of paths

$$ \begin{align*}\pi_1^m(X,\mathbf{x},\mathbf{y}) \simeq \pi_1^m(X,\mathbf{y},\mathbf{x}),\end{align*} $$

which yields an inverse for the composition of paths, and each

$$ \begin{align*}\pi_1^m(X,\mathbf{x}):= \pi_1^m(X,\mathbf{x},\mathbf{x})\end{align*} $$

has the structure of a pro- $\mathbf {MT}(k,\mathbb {Q})$ group scheme. All these data satisfy the axioms dual to the axioms given in theorem 4.1.9.

We call the family $\{\pi _1^m(X,\mathbf {x},\mathbf {y})\}_{\mathbf {x},\mathbf {y}}$ equipped with the operations described above the motivic fundamental groupoid at tangential basepoints.

5.1 Realization functors

In this subsection, k is a field of characteristic $0$ . We can naturally generalize the definition of mixed Weil theories given in [Reference DrewDre18] to a log geometric context:

Log mixed Weil theories, modelling log cohomology theories, are the starting point to build realization functors. We prove a logarithmic version of the theorem 3.7 in [Reference DrewDre18]. Note that it is a log and $\Lambda $ -linear version of [Reference RobaloRob15, Theorem 2.9]:

Proposition 5.1.2. The homological motive functor induces a homotopy equivalence of Kan complexes

$$ \begin{align*}\operatorname{Fun}_{\mathrm{CAlg}(\mathcal{P}\mathrm{r}^{\mathrm{L},\otimes})^{\mathcal{M}\mathrm{od}_\Lambda^\otimes/}}(\mathcal{DA}^{\mathrm{log}}(k,\Lambda)^\otimes,\mathcal{M}\mathrm{od}_\Lambda^\otimes) \xrightarrow{\sim} \operatorname{Fun}_{\mathrm{CAlg}(\mathcal{C}\mathrm{at}_\infty^\times)}(\mathrm{N}(\mathbf{lSm}_k)^\times,\mathcal{M}\mathrm{od}_\Lambda^\otimes)_{\mathrm{P}^{\mathrm{log}}}.\end{align*} $$

The subscript $\mathrm {P}^{\mathrm {log}}$ means the subcomplex generated by the functors $\boldsymbol {\Gamma }$ such that $\boldsymbol {\Gamma }^{\mathrm {op}}$ satisfies the three last points of definition 5.1.1.

Remark 5.1.3. The proposition means that the category $\mathcal {DA}^{\mathrm {log}}(k,\Lambda )$ has the universal property that any functor defined from the category of log smooth schemes over k to the category of perfect complexes of $\Lambda $ -modules which satisfies $dNis$ -descent, which trivializes $\mathbb {A}^1$ , $\mathrm {ver}$ -covers and $\mathbb {G}_m^{\mathrm {log}}/1$ factorizes by it. We will use this fact as follows: from a log mixed Weil theory $\boldsymbol {\Gamma }$ , since $\mathrm {Perf}(\Lambda )$ is the sub- $\infty $ -category of dualizable objects in $\mathcal {M}\mathrm {od}_\Lambda $ , we can consider the functor

$$ \begin{align*}(\boldsymbol{\Gamma}^{\mathrm{op}})^\vee: \mathbf{lSm}_k \xrightarrow{\boldsymbol{\Gamma}^{\mathrm{op}}} \mathrm{Perf}(\Lambda)^{\mathrm{op}} \xrightarrow{(-)^\vee} \mathrm{Perf}(\Lambda)\hookrightarrow \mathcal{M}\mathrm{od}_\Lambda.\end{align*} $$

which has the property $\mathrm {P}^{\mathrm {log}}$ so that it induces a realization functor

$$ \begin{align*}\mathrm{R}^\otimes: \mathcal{DA}^{\mathrm{log}}(k,\Lambda)^\otimes \rightarrow \mathcal{M}\mathrm{od}^\otimes_\Lambda\end{align*} $$

satisfying for $X \in \mathbf {lSm}_k$ ,

$$ \begin{align*}\mathbf{R}(M(X)) \simeq \boldsymbol{\Gamma}(X)^\vee.\end{align*} $$

In order to prove the proposition 5.1.2, we first prove the following, based on scattered remarks in [Reference DrewDre18] (see for instance the proof of proposition 3.14):

Lemma 5.1.4. For $S\in \mathbf {lSch}$ , there is a canonical equivalence

$$ \begin{align*}\mathcal{DA}^{\mathrm{log}}(S,\Lambda) \simeq \mathcal{SH}^{\mathrm{log}}(S) \otimes \mathcal{M}\mathrm{od}_\Lambda,\end{align*} $$

where the tensor product is taken in $\mathcal {P}\mathrm {r}^{\mathrm {L}}$ .

Proof. By [Reference DrewDre18, Proposition 2.4 (1)], we have the natural equivalence (note that hypersheaves and sheaves coincide here by [Reference Binda, Park and Arne ØstværBPØ25, Remark 2.4.7])

$$ \begin{align*}\mathcal{S}\mathrm{h}_{\mathrm{dNis}}(\mathbf{lSm}_S,\mathcal{M}\mathrm{od}_\Lambda) \simeq \mathcal{S}\mathrm{h}_{\mathrm{dNis}}(\mathbf{lSm}_S,\mathcal{S}\mathrm{pc})\otimes \mathcal{M}\mathrm{od}_\Lambda.\end{align*} $$

Now, the tensor product of presentable categories commutes with small colimits separately in each variable by [Reference LurieLur17, 4.8.1.24], and the inclusion $\mathcal {P}\mathrm {r}^{\mathrm {L}}_{st} \subset \mathcal {P}\mathrm {r}^{\mathrm {L}}$ preserves small colimits. Therefore, since localization of stable presentable $\infty $ -categories are cofibres in $\mathcal {P}\mathrm {r}^{\mathrm {L}}_{st}$ (Verdier quotients see [Reference Blumberg, Gepner and TabuadaBGT13, Proposition 5.6]), the localization commutes with tensor product with $\mathcal {M}\mathrm {od}_\Lambda $ :

$$ \begin{align*}\begin{array}{ccc} \mathrm{L}_{\mathbb{A}^1}\mathrm{L}_{\mathrm{ver}}\mathcal{S}\mathrm{h}_{\mathrm{dNis}}(\mathbf{lSm}_S,\mathcal{M}\mathrm{od}_\Lambda) & \simeq & (\mathrm{L}_{\mathbb{A}^1}\mathrm{L}_{\mathrm{ver}}\mathcal{S}\mathrm{h}_{\mathrm{dNis}}(\mathbf{lSm}_S,\mathcal{S}\mathrm{pc}))\otimes \mathcal{M}\mathrm{od}_\Lambda\\ &\simeq & (\mathrm{L}_{\mathbb{A}^1}\mathrm{L}_{\mathrm{ver}}\mathcal{S}\mathrm{h}_{\mathrm{dNis}}(\mathbf{lSm}_S,\mathcal{S}\mathrm{pc}_*))\otimes \mathcal{M}\mathrm{od}_\Lambda \end{array}\end{align*} $$

where the last equivalence follows from the fact that $\mathcal {M}\mathrm {od}_\Lambda $ is pointed, see [Reference DrewDre18, Definition 3.1 (4)].

Using again the fact that the tensor product in $\mathcal {P}\mathrm {r}^{\mathrm {L}}$ commutes with small colimits, we have the following natural equivalences:

$$ \begin{align*} \begin{array}{ccl} \mathcal{DA}^{\mathrm{log}}(S,\Lambda) & \simeq & \mathrm{colim}_{\Sigma_{\mathbb{G}_m^{\mathrm{log}}/1}} \mathrm{L}_{\mathbb{A}^1}\mathrm{L}_{\mathrm{ver}}\mathcal{S}\mathrm{h}_{\mathrm{dNis}}(\mathbf{lSm}_S,\mathcal{M}\mathrm{od}_\Lambda)\\ & \simeq & (\mathrm{colim}_{\Sigma_{\mathbb{G}_m^{\mathrm{log}}/1}}\mathrm{L}_{\mathbb{A}^1}\mathrm{L}_{\mathrm{ver}}\mathcal{S}\mathrm{h}_{\mathrm{dNis}}(\mathbf{lSm}_S,\mathcal{S}\mathrm{pc}_*)) \otimes \mathcal{M}\mathrm{od}_\Lambda\\ & \simeq & \mathcal{SH}^{\mathrm{log}}(S) \otimes \mathcal{M}\mathrm{od}_\Lambda \end{array} \end{align*} $$

which concludes the proof.

Let us now check the log version of [Reference RobaloRob15, Theorem 2.39], already proven for the category $\mathrm {log}\mathcal {SH}$ in [Reference Binda, Park and Arne ØstværBPØ25].

Lemma 5.1.5. The homological motive functor induces an equivalence of $\infty $ -categories

$$ \begin{align*}\operatorname{Map}_{\mathrm{CAlg}(\mathcal{P}\mathrm{r}^{\mathrm{L},\otimes})}(\mathcal{SH}^{\mathrm{log}}(k)^\otimes,\mathcal{M}\mathrm{od}_\Lambda^\otimes) \xrightarrow{\sim} \operatorname{Map}_{\mathrm{CAlg}(\mathcal{C}\mathrm{at}_\infty^\times)}(\mathrm{N}(\mathbf{lSm}_k)^\times,\mathcal{M}\mathrm{od}_\Lambda^\otimes)_{\mathrm{P}^{\mathrm{log}}}\end{align*} $$

We can now prove proposition 5.1.2. The proof in [Reference DrewDre18, Theorem 3.7] is directly adaptable:

Proof of proposition 5.1.2

The adjunction $\mathrm {CAlg}(\mathcal {P}\mathrm {r}^{\mathrm {L},\otimes }) \rightleftarrows \mathrm {CAlg}(\mathcal {P}\mathrm {r}^{\mathrm {L},\otimes })^{\mathcal {M}\mathrm {od}_\Lambda ^\otimes /}$ given by $-\otimes \mathcal {M}\mathrm {od}_\Lambda $ and the forgetful functor, together with the equivalence of lemma 5.1.4 show that there is a homotopy commutative diagram

Since the bottom arrow is a homotopy equivalence by lemma 5.1.5, this diagram readily gives the result.

Remark 5.1.6. In the previous proposition, the dividing Nisnevich topology can be straightforwardly replaced by the dividing étale topology, inducing an equivalence

$$ \begin{align*}\operatorname{Map}_{\mathrm{CAlg}(\mathcal{P}\mathrm{r}^{\mathrm{L},\otimes})^{\mathcal{M}\mathrm{od}_\Lambda^\otimes/}}(\mathcal{DA}_{\text{t}}^{\mathrm{log}}(k,\Lambda)^\otimes,\mathcal{M}\mathrm{od}_\Lambda^\otimes) \xrightarrow{\sim} \operatorname{Map}_{\mathrm{CAlg}(\mathcal{C}\mathrm{at}_\infty^\times)}(\mathrm{N}(\mathbf{lSm}_k)^\times,\mathcal{M}\mathrm{od}_\Lambda^\otimes)_{\mathrm{P}^{\mathrm{log}}_{\text{t}}}\end{align*} $$

Where we added strict étale descent in the subscript. Note that this equivalence is compatible with the previous construction through the étale sheafification functor, that is the following diagram commutes:

This means that any étale log mixed Weil theory $\boldsymbol {\Gamma }$ gives rise to compatible realizations that we denote by the same name $\mathrm {R}: \mathcal {DA}_{(\text {\'{e}t})}^{\mathrm {log}}(k,\Lambda ) \rightarrow \mathcal {M}\mathrm {od}_\Lambda $ .

There is also a natural compatibility between log mixed Weil theories and mixed Weil theories: It is easy to see that the functor $\lambda $ from the adjunction

defined in subsection 3.1 exchanges log mixed Weil theories and mixed Weil theories.

The induced equivalence

$$ \begin{align*}\lambda^*: \mathcal{DA}^{\mathrm{log}}(k,\Lambda) \xrightarrow{\sim} \mathcal{DA}(k,\Lambda)\end{align*} $$

defined by $\lambda ^*(M(X)) \simeq M(X\setminus \partial X)$ fits into the following commutative diagram

where the bottom equivalence (and the subscript) is described in [Reference DrewDre18, Theorem 3.7].

We add a useful fact, which will allow us to compute the realization of the cohomological motives of the log point $*_{\mathbb {N}}$ which is a non log smooth log scheme over k: Let $\boldsymbol {\Gamma }$ be a log mixed Weil theory, and $\mathrm {R}$ a realization functor associated with $\boldsymbol {\Gamma }$ . Since k is of characteristic $0$ , compact objects in $\mathcal {DA}^{\mathrm {log}}(k,\Lambda )$ are dualizable so that since R is symmetric monoidal, for all $X\in \mathbf {lSm}_k$ ,

$$ \begin{align*}\begin{array}{ccl} \mathrm{R}(h(X)) & \simeq & \mathrm{R}(\underline{\mathrm{Hom}}(M(X),\mathbf{1}_k))\\ & \simeq & \mathrm{R}(M(X)^\vee)\\ & \simeq & \mathrm{R}(M(X))^\vee\\ & \simeq & \boldsymbol{\Gamma}(X). \end{array}\end{align*} $$

This property can be generalized a little bit: Suppose $\boldsymbol {\Gamma }$ is defined as a functor $\mathbf {lSch}^{\mathrm {op}}_k \rightarrow \mathcal {M}\mathrm {od}_\Lambda $ . And suppose it is such that the inclusion $*_{\mathbf {N}} \hookrightarrow \mathbb {A}_{\mathbb {N}}$ induces an equivalence

$$ \begin{align*}\boldsymbol{\Gamma}(\mathbb{A}_{\mathbb{N}}) \xrightarrow[]{\sim} \boldsymbol{\Gamma}(*_{\mathbf{N}})\end{align*} $$

In this setting,

Lemma 5.1.7.

$$ \begin{align*}\mathrm{R}(h(*_{\mathbb{N}})) \simeq \boldsymbol{\Gamma}(*_{\mathbb{N}})\end{align*} $$

Proof. $h(*_{\mathbb {N}})$ is identified as the pushout of the diagram

by the discussion following proposition 3.2.2. This stays true after applying $\mathrm {R}$ , since it is a colimit preserving functor. Since the following diagram is a pushout diagram by hypothesis,

the claim is proved.

5.2 The Betti/Kato–Nakayama realization

We can now define the Betti fundamental groupoid at tangential basepoints, and after having defined the Betti realization for log motives, compare the Betti realization of the motivic fundamental groupoid based at tangential basepoints with the classical prounipotent completion of the fundamental groupoid at tangential basepoints. We achieve this in the theorem 5.2.11. In this part, we suppose $k\subset \mathbb {C}$ .

5.2.1 The topological fundamental groupoid at tangential basepoints

We first define the torsor of homotopy classes of paths based at tangential basepoints on a general smooth scheme over k with a simple normal crossings divisor. The definition is a higher dimensional version of what is done in [Reference Ignacio Burgos Gil and FresánBGF25, Section 3.8] in the case of smooth projective curves minus a finite number of points.

Definition 5.2.1. Let $(\underline {X},\underline {D}) \in \mathbf {Div}_k$ with $\mathbf {x}:= (x,v), \mathbf {y}:= (y,w)$ be tangential basepoints. The path space of $\underline {X}(\mathbb {C})\setminus \underline {D}(\mathbb {C})$ based at $\mathbf {x},\mathbf {y}$ is the space of smooth maps $\gamma : [0,1] \rightarrow \underline {X}(\mathbb {C})$ such that:

• • $\gamma $ sends the open interval $(0,1)$ to $\underline {X}(\mathbb {C})\setminus \underline {D}(\mathbb {C})$ ,

• • $\gamma (0) = x$ and $\gamma (1) = y$ ,

• • $\gamma '(0) = v$ and $\gamma '(1) = -w$ .

We will call such paths smooth paths of $\underline {X}(\mathbb {C})\setminus \underline {D}(\mathbb {C})$ based at $\mathbf {x},\mathbf {y}$ .

The definition of homotopies of paths is adapted to the tangential datum, by generalizing the definition given in [Reference Ignacio Burgos Gil and FresánBGF25, 3.8.3]:

Definition 5.2.2. Let $\gamma _1,\gamma _2$ be two smooth paths of $\underline {X}(\mathbb {C}) \setminus \underline {D}(\mathbb {C})$ based at tangential basepoints $\mathbf {x},\mathbf {y}$ . A homotopy between $\gamma _1$ and $\gamma _2$ is a smooth map

$$ \begin{align*}H : [0,1]\times [0,1] \rightarrow \underline{X}(\mathbb{C})\end{align*} $$

such that:

• • $\forall t \in [0,1], H(t,0) = \gamma _1(t)$ and $H(t,1) = \gamma _2(t)$ ,

• • $\forall s \in [0,1]$ , $H(0,s) = x$ and $H(1,s) = y$ ,

• • $\forall s \in [0,1]$ , $\frac {\partial H}{\partial t}(0,s) = v$ and $\frac {\partial H}{\partial t}(1,s) = -w$ ,

• • $\forall s \in [0,1], \forall t \in (0,1)$ , $H(s,t) \in \underline {X}(\mathbb {C}) \setminus \underline {D}(\mathbb {C}).$

We can define the torsor of homotopy classes of paths in this situation:

Definition 5.2.3. We call torsor of homotopy classes of paths of $\underline {X}(\mathbb {C}) \setminus \underline {D}(\mathbb {C})$ based at $\mathbf {x},\mathbf {y}$ the quotient of the path space of $\underline {X}(\mathbb {C}) \setminus \underline {D}(\mathbb {C})$ based at $\mathbf {x},\mathbf {y}$ by the equivalence relation generated by homotopies between paths.

As in the classical case, the data of all the torsors of homotopy classes of paths based at tangential basepoints has a structure of a groupoid. We do not describe precisely the operations involved as they are similar to the case of ordinary basepoints. Nevertheless, the product of homotopy classes of paths is carefully defined in [Reference Ignacio Burgos Gil and FresánBGF25, Definition 3.357] in the case of $ \mathbb {P}_k^1\setminus \{0,1,\infty \}$ . In higher dimension, the definition of composition can be deduced from the former by viewing $\underline {X}(\mathbb {C})$ locally as a product of $\mathbb {A}^1_k$ . Recall that for $\gamma _1$ (resp. $\gamma _2$ ) a smooth path of $\mathbb {P}_{\mathbb {C}}^1\setminus \{0,1,\infty \}$ based at tangential basepoints $(x,u), (y,v)$ (resp. $(y,v), (z,w)$ ), the composition of classes $[\gamma _1][\gamma _2]$ is defined locally around y by the following picture:

Figure 2

Figure 2. Long description

The left panel contains a square with two curved arrows starting from the top left and top right corners, both curving downward and meeting at a single black dot near the center. The right panel also has a square with two similar curved arrows, but instead of meeting directly at the central dot, they are joined by a circular loop that encircles the dot before connecting the paths. The central dot is present in both panels, and the loop in the right panel visually represents a continuous path around the dot.

If we take $\gamma $ to be any smoothing of the second composed path, then

$$ \begin{align*}[\gamma_1][\gamma_2] := [\gamma].\end{align*} $$

5.2.2 The Betti fundamental groupoid at tangential basepoints

We now define the Betti fundamental groupoid, and we compute the Betti realization of the motivic fundamental torsor at tangential basepoints. Let $X\in \mathbf {Div}_k$ , endowed with tangential basepoints $\mathbf {x},\mathbf {y}$ . Recall from section 2 that $\mathbf {x}$ and $\mathbf {y}$ induce points on the Kato–Nakayama topological space $X^{\operatorname {KN}}$ . Although as we explained, there is not an equivalence between tangential basepoints and points on $X^{\mathrm {KN}}$ , we will still denote by the same letter $\mathbf {x},\mathbf {y}$ the associated points.

The log Betti cohomology of X is by definition $\mathbf {R}\Gamma (X^{\mathrm {KN}},\underline {\mathbb {Q}}) \in \mathrm {CAlg}(\mathcal {M}\mathrm {od}_{\mathbb {Q}})$ where $\underline {\mathbb {Q}}$ is the constant sheaf with value $\mathbb {Q}$ , and $\mathbf {R}\Gamma $ denotes the derived functor of the global section functor. It is endowed with augmentations defined by the points $\mathbf {x},\mathbf {y} \in X^{\mathrm {KN}}.$ Note that what follows works in the same way if $\mathbb {Q}$ is replaced by $\mathbb {Z}$ .

Definition 5.2.4. The Betti path space of X based at $\mathbf {x},\mathbf {y}$ is defined by

$$ \begin{align*}\mkern1mu^{}_{\mathbf{x}}\mathrm{P}^{\mathrm{B}}_{\mathbf{y}}X:= \mkern1mu_{\mathbf{x}}\mathrm{Bar}_{\mathbf{y}}\textbf{R}\Gamma(X^{\operatorname{KN}},\underline{\mathbb{Q}}) \in \mathrm{CAlg}(\mathcal{M}\mathrm{od}_{\mathbb{Q}}).\end{align*} $$

The Betti fundamental torsor at $\mathbf {x},\mathbf {y}$ of X is defined by

$$ \begin{align*}\pi^{\text{B}}_1(X,\mathbf{x},\mathbf{y}):= \operatorname{Spec}\operatorname{H}^0(\mkern1mu^ {}_{\mathbf{x}}\mathrm{P}^{\text{B}}_{\mathbf{y}}X).\end{align*} $$

Remark 5.2.5. As before, it follows from theorem 4.1.9, that the family $\{\mkern1mu^{}_{\mathbf {x}}\mathrm {P}^{\mathrm {B}}_{\mathbf {y}}X\}_{\mathbf {x},\mathbf {y}}$ is endowed with the structure of a homotopy Hopf algebroid, so that the family $\{\pi ^{\text {B}}_1(X,\mathbf {x},\mathbf {y})\}_{\mathbf {x},\mathbf {y}}$ has the structure of a groupoid. We call it the Betti fundamental groupoid.

Similarly to the classical setting, the Betti fundamental torsor is compared to the prounipotent completion of the fundamental torsor of $X^{\operatorname {KN}}$ , the main tool being Beilinson’s theorem. The comparison that we prove is the following:

Proposition 5.2.6. There is a natural isomorphism of groupoids given by isomorphisms

$$ \begin{align*}\pi_1^{\operatorname{B}}(X,\mathbf{x},\mathbf{y}) \simeq \pi_1^{\operatorname{un}}(\underline{X}(\mathbb{C})\setminus \underline{D}(\mathbb{C}),\mathbf{x},\mathbf{y})\end{align*} $$

for $\mathbf {x},\mathbf {y}$ all tangential basepoints of X.

Proof. First, let us apply Beilinson’s theorem [Reference Deligne and GoncharovDG05, Proposition 3.4] to $X^{\mathrm {KN}}$ . This theorem is proved by Deligne and Goncharov in the context of a locally path connected, locally simply connected topological space Y such that its first homotopy group is of finite type, and such that the functor $\mathbf {R}p_*$ for $p: Y^m\times Y^n \rightarrow Y^n$ commutes with fibres for constant sheaves. Since $X^{\mathrm {KN}}$ is a manifold with boundaries [Reference Kato and NakayamaKN99, Remark 1.5.1] with the homotopy type of a complex algebraic variety, and as such is a paracompact locally contractible space with a finitely generated fundamental group, we can use Beilinson’s theorem to see that there is a natural isomorphism of schemes

$$ \begin{align*}\pi_1^{\operatorname{B}}(X,\mathbf{x},\mathbf{y}) \simeq \pi_1^{\operatorname{un}}(X^{\operatorname{KN}},\mathbf{x},\mathbf{y}).\end{align*} $$

We need to compare the prounipotent completion of the fundamental torsor of $X^{\mathrm {KN}}$ and of $\underline {X}(\mathbb {C})\setminus \underline {D}(\mathbb {C})$ . Note that if $x, y \in \underline {X}(k)\setminus \underline {D}(k)$ , then it is straightforward that we have

$$ \begin{align*}\pi_1^{\text{B}}(X,x,y) \simeq \pi_1^{\text{un}}(X^{\text{KN}},x,y) \simeq \pi_1^{\text{un}}(\underline{X}(\mathbb{C})\setminus \underline{D}(\mathbb{C}),x,y)\end{align*} $$

since $X^{\text {KN}}$ and $\underline {X}(\mathbb {C})\setminus \underline {D}(\mathbb {C})$ are homotopy equivalent. This result is still true for tangential basepoints, as we show in the following lemma concluding the proof of this proposition.

Lemma 5.2.7. For all tangential basepoints $\mathbf {x},\mathbf {y}$ on X, there is a natural isomorphism

$$ \begin{align*}\pi_1(\underline{X}(\mathbb{C})\setminus \underline{D}(\mathbb{C}),\mathbf{x},\mathbf{y}) \xrightarrow{\sim} \pi_1(X^{\operatorname{KN}},\mathbf{x}, \mathbf{y})\end{align*} $$

Proof. The morphism is given by continuation of paths: for $\gamma : (0,1) \rightarrow \underline {X}(\mathbb {C})$ a smooth path of $\underline {X}(\mathbb {C})\setminus \underline {D}(\mathbb {C})$ from $\mathbf {x}$ to $\mathbf {y}$ , we can consider the path $\overline {\gamma }: [0,1] \rightarrow X^{\mathrm {KN}}$ such that $\eta _{|(0,1)} = \gamma $ , $\eta (0) = \mathbf {x}, \eta (1) = \mathbf {y}$ .

To show that this defines an isomorphism between the homotopy groups, we first see that by continuation of homotopies between paths in the same fashion, this operation trivially defines a morphism, $ \pi _1(\underline {X}(\mathbb {C})\setminus \underline {D}(\mathbb {C}),\mathbf {x},\mathbf {y}) \rightarrow \pi _1(X^{\mathrm {KN}},\mathbf {x},\mathbf {y})$ . We now exhibit this morphism as a composition of isomorphisms:

Let $\eta _{\mathbf {x}}:\; (0,1) \; \rightarrow \underline {X}(\mathbb {C})$ (resp. $\eta _{\mathbf {y}}$ ) be any smooth path from $\mathbf {x}$ (resp. $\mathbf {y}$ ) to any point denoted by $x'$ (resp. $y'$ ) that we suppose to be in $\underline {X}(\mathbb {C})\setminus \underline {D}(\mathbb {C})$ .

The morphism we define is given by the following diagram

where the middle isomorphism

$$ \begin{align*}h: \pi_1(\underline{X}(\mathbb{C})\setminus \underline{D}(\mathbb{C}),x',y') \xrightarrow[]{\sim} \pi_1(X^{\mathrm{KN}},x',y')\end{align*} $$

is induced by the homotopy equivalence $\underline {X}(\mathbb {C})\setminus \underline {D}(\mathbb {C}) \hookrightarrow X^{\mathrm {KN}}$ . The reason that this composition is the continuation of path is simple: take $\gamma $ any smooth path of $\underline {X}(\mathbb {C})\setminus \underline {D}(\mathbb {C})$ from $\mathbf {x}$ to $\mathbf {y}$ . The definition of the composition of homotopy classes of smooth paths of $\underline {X}(\mathbb {C})\setminus \underline {D}(\mathbb {C})$ based at tangential basepoints shows that $[\eta _{\mathbf {x}}^{-1}][\gamma ][\eta _{\mathbf {y}}]$ is represented by a smooth path $\lambda $ which locally around x and y looks like picture (5.2.1) and keeps the paths involved unchanged outside of these local pictures. Hence $h([\lambda ]) = [\lambda ]$ , so that the image of $\gamma $ by the composition of morphisms we are studying is the class represented by $\overline {\eta _{\mathbf {x}}}\lambda \overline {\eta _{\mathbf {y}}}^{-1}$ which around x and y looks like (5.2.1) and similarly keeps everything unchanged outside. It is clear that this path is just homotopic to the continuation of $\gamma $ .

Hence, the proposition 5.2.6 is proved. This result will be at the center of the computation of the Betti realization of the motivic pointed path space with tangential basepoints. In order to identify this realization, let us first build the Betti realization functor of $\mathbb {Q}$ -linear log motives.

5.2.3 The Betti realization functor

The Betti realization we obtain is close to the construction for the log motivic homotopy category in [Reference Binda, Park and Arne ØstværBPØ25]. We prove the following:

Proposition 5.2.8. There exists a $\mathbb {Q}$ -linear Betti realization functor

$$ \begin{align*}\mathrm{R}^\otimes_{\mathrm{B}}: \mathcal{DA}^{\mathrm{log}}(k,\mathbb{Q})^\otimes \rightarrow \mathcal{M}\mathrm{od}_{\mathbb{Q}}^\otimes\end{align*} $$

such that for $X \in \mathbf {lSm}_k$ , there is a natural equivalence

$$ \begin{align*}\mathrm{R}_{\mathrm{B}}(h(X)) \simeq \mathbf{R}\Gamma(X^{\mathrm{KN}},\underline{\mathbb{Q}})\end{align*} $$

and such that the following diagram commutes:

The functor $\mathrm {R}_{\mathrm {B}}$ on the right is the classical Betti realization functor as defined in [Reference AyoubAyo10].

In order to show this, we apply the results of subsection 5.1 to the log mixed Weil theory over k

$$ \begin{align*} \begin{array}{ccccc} \boldsymbol{\Gamma}^\otimes_{\mathrm{B}} &: & \mathbf{lSm}^{op,\times}_S & \to & \mathcal{M}\mathrm{od}_{\mathbb{Q}}^\otimes\\ \end{array} \end{align*} $$

defined as follows:

Definition 5.2.9. The log Betti $\mathcal {M}\mathrm {od}_{\mathbb {Q}}$ -valued log mixed Weil theory over k is the symmetric monoidal functor

$$ \begin{align*} \begin{array}{ccccc} \boldsymbol{\Gamma}^\otimes_{\text{B}} &: & \mathbf{lSm}^{op,\times}_S & \to & \mathcal{M}\mathrm{od}_{\mathbb{Q}}^\otimes.\\ \end{array} \end{align*} $$

defined as the composition of symmetric monoidal functors

$$ \begin{align*}\mathbf{lSm}_S^{op,\times} \xrightarrow{(-)^{\text{KN}}} \mathcal{S}\mathrm{pc}^{op,\times} \xrightarrow{\mathbf{R}\Gamma(-,\underline{\mathbb{Q}})} \mathcal{M}\mathrm{od}_{\mathbb{Q}}^\otimes.\end{align*} $$

Note that up to a canonical equivalence, this functor sends $X \in \mathbf {lSm}_k$ to $\operatorname {C}^*(X^{\text {KN}},\mathbb {Q})$ the singular cochain complex of $X^{\text {KN}}$ . Hence, it takes values in $\operatorname {Perf}(\mathbb {Q})$ the sub- $\infty $ -category of perfect complexes of $\mathbb {Q}$ -modules, since the term in each degree of the singular cochain complex is a free $\mathbb {Q}$ -module, and $X^{\text {KN}}$ has the homotopy type of a finite dimensional CW-complex.

Proof of proposition 5.2.8

The functor $(-)^{\text {KN}}$ is symmetric monoidal, satisfies $\mathrm {dNis}$ -descent and sends $\mathbb {G}_m^{\mathrm {log}}$ to a space homotopically equivalent to $S^1$ . It is as well $\operatorname {ver}$ -invariant since $X^{\mathrm {KN}}$ and $\underline {X}(\mathbb {C})\setminus \underline {D}(\mathbb {C})$ are homotopically equivalent [Reference Binda, Park and Arne ØstværBPØ22, Proposition A.9.2].

Since $\mathbf {R}\Gamma (-,\underline {\mathbb {Q}})$ is symmetric monoidal, so is $\boldsymbol {\Gamma }_{\text {B}}$ . Thus it is a $\mathrm {dNis}$ -sheaf, $\text {ver}$ -invariant, it sends $\mathbb {G}_m^{\mathrm {log}}$ on a $\otimes $ -invertible object, and it is trivially $\mathbb {A}^1$ -invariant. Hence, $\boldsymbol {\Gamma }_{\operatorname {B}}$ is a $\mathcal {M}\mathrm {od}_{\mathbb {Q}}$ -valued log mixed Weil theory, from which the proposition follows.

Remark 5.2.10. Note that since $*_{\mathbb {N}}^{\operatorname {KN}}$ and $\mathbb {A}_{\mathbb {N}}^{\operatorname {KN}}$ are homotopically equivalent, the morphism

$$ \begin{align*}\boldsymbol{\Gamma}(\mathbb{A}_{\mathbb{N}}) \rightarrow \boldsymbol{\Gamma}(*_{\mathbb{N}})\end{align*} $$

is an equivalence.

5.2.4 Computation of the Betti realization of the motivic fundamental groupoid

Now that the Betti realization functor is built, we can get back to the motivic fundamental group. We will show the following:

Theorem 5.2.11. Let $X \in \mathbf {Div}_k$ and $\mathbf {x}, \mathbf {y}$ tangential basepoints on X. There is a natural equivalence of algebras

$$ \begin{align*}\mathrm{H}^0(\mathrm{R}_{\mathrm{B}}(\mkern1mu_{\mathbf{x}}^{}\mathrm{P}^{m}_{\mathbf{y}}X)) \simeq \mathcal{O}(\pi^{\mathrm{un}}_1(\underline{X}(\mathbb{C})\setminus \underline{D}(\mathbb{C}), \mathbf{x},\mathbf{y})).\end{align*} $$

In particular, when X is such that $M(X)$ is a mixed Tate motive, and when k satisfies Beilinson-Soulé’s conjecture, there is a natural isomorphism of groupoids induced by

$$ \begin{align*}\mathrm{R}_{\mathrm{B}}\pi_1^m(X,\mathbf{x},\mathbf{y}) \simeq \pi^{\operatorname{un}}_1(\underline{X}(\mathbb{C})\setminus \underline{D}(\mathbb{C}),\mathbf{x},\mathbf{y}).\end{align*} $$

Proof. By the proposition 5.2.6, we only need to show that there is a natural equivalence in $\mathrm {CAlg}(\mathcal {M}\mathrm {od}^\otimes _{\mathbb {Q}})$

$$ \begin{align*}\mathrm{R}_{\mathrm{B}}(\mkern1mu^{}_{\mathbf{x}}\mathrm{P}_{\mathbf{y}}^mX) \simeq \mkern1mu^{}_{\mathbf{x}}\mathrm{P}^{\mathrm{B}}_{\mathbf{y}}X.\end{align*} $$

Since $\mathrm {R}^\otimes _{\mathrm {B}}$ is colimit preserving, it commutes with the bar construction, hence

$$ \begin{align*}\mathrm{R}_{\mathrm{B}}(\mkern1mu^{}_{\mathbf{x}}\mathrm{P}_{\mathbf{y}}^mX) \simeq \!_{\mathrm{R}_{\mathrm{B}}(\mathbf{x})}\mathrm{Bar}_{\mathrm{R}_{\operatorname{B}}(\mathbf{y})}\mathbf{R}\Gamma(X^{\operatorname{KN}},\underline{\mathbb{Q}}).\end{align*} $$

Recall that up to homotopy equivalence, $h(\mathbf {x}): h(X) \rightarrow \mathbb {Q}$ is defined by the choice of a homotopy inverse of

$$ \begin{align*}h(\mathrm{N}^{\mathrm{log}}_x(\underline{X},\underline{D})) \rightarrow h(x^{\mathrm{log}}).\end{align*} $$

The realization of this morphism is

$$ \begin{align*}C^*(\mathrm{N}_x^{\text{log}}(\underline{X},\underline{D})^{\operatorname{KN}},\mathbb{Q}) \rightarrow C^*((x^{\text{log}})^{\operatorname{KN}},\mathbb{Q})\end{align*} $$

(see lemma 5.1.7 and remark 5.2.10) induced by the embedding

$$ \begin{align*}(x^{\mathrm{log}})^{\operatorname{KN}} \hookrightarrow \mathrm{N}_x^{\mathrm{log}}(\underline{X},\underline{D})^{\operatorname{KN}}.\end{align*} $$

A natural homotopy inverse to this morphism is the map induced by the retraction of the real blow-up of $\mathbb {C}^i$ along its axes on the product of infinitesimal spheres at the origin $(S^1)^i$ . Therefore, the augmentation $\mathrm {R}_{\mathrm {B}}(h(\mathbf {x}))$ is up to homotopy the map obtained from the sequence

$$ \begin{align*}* \xrightarrow{v} \mathrm{N}_x^{\mathrm{log}}(\underline{X},\underline{D})^{\mathrm{KN}} \rightarrow (S^1)^i \rightarrow X^{\operatorname{KN}}\end{align*} $$

by applying $C^*(-,\mathbb {Q})$ . The composition of these maps is just the point

$$ \begin{align*}* \rightarrow X^{\operatorname{KN}}\end{align*} $$

defined by $\frac {v}{||v||} \in S^i = (x^{\mathrm {log}})^{\operatorname {KN}}$ . The theorem is proved.

Remark 5.2.12. We remind the reader that the ring of functions of the prounipotent completion of the torsor of paths from $\mathbf {x}$ to $\mathbf {y}$ on $\underline {X}(\mathbb {C})\setminus \underline {D}(\mathbb {C})$ is defined by

$$ \begin{align*}\mathcal{O}(\pi^{\mathrm{un}}_1(\underline{X}(\mathbb{C})\setminus \underline{D}(\mathbb{C}), \mathbf{x},\mathbf{y})) := \mathrm{colim} (\mathbb{Q}[\pi_1(\underline{X}(\mathbb{C})\setminus \underline{D}(\mathbb{C}),\mathbf{x},\mathbf{y})]/J^{n+1})^\vee\end{align*} $$

where $J^n = \mathbb {Q}[\pi _1(\underline {X}(\mathbb {C})\setminus \underline {D}(\mathbb {C}),\mathbf {x},\mathbf {y})]I^n$ where the product is obtained from the right action of $\mathbb {Q}[\pi _1(\underline {X}(\mathbb {C})\setminus \underline {D}(\mathbb {C}),\mathbf {x})]$ restricted to $I^n$ where I is the augmentation ideal.

We identified the Betti realization of the motivic pointed path space. Let us now relate it to the cosimplicial path space construction. Passing to the Betti cohomology allows us to bypass the limitations of proposition 4.2.7. Indeed, we can see that the Betti path space can be classically extracted from the cosimplicial space

after applying the functor $\text {R}\Gamma (-,\underline {\mathbb {Q}})$ and taking the colimit on the simplicial diagram thereby obtained. Let us explain this point further. It is easy to see that

$$ \begin{align*}\mkern1mu^{}_{\mathbf{x}}\mathrm{P}^{\mathrm{B}}_{\mathbf{y}}X \simeq \mathbb{Q} \otimes_{\boldsymbol{\Gamma}_{\mathrm{B}}(X)\otimes \boldsymbol{\Gamma}_{\mathrm{B}}(X)} \boldsymbol{\Gamma}_{\mathrm{B}}(X) \simeq \mathbb{Q} \otimes_{\mathrm{R}\boldsymbol{\Gamma}(X^{\mathrm{KN}},\underline{\mathbb{Q}})\otimes \mathrm{R}\boldsymbol{\Gamma}(X^{\mathrm{KN}},\underline{\mathbb{Q}})} \mathrm{R}\boldsymbol{\Gamma}(X^{\mathrm{KN}},\underline{\mathbb{Q}}).\end{align*} $$

Since $\mathrm {R}\boldsymbol {\Gamma }(X^{\mathrm {KN}},\underline {\mathbb {Q}}) \simeq \mathrm {colim}_\Delta \mathrm {R}\boldsymbol {\Gamma }((X^{\mathrm {KN}})^{[0,1]},\underline {\mathbb {Q}})$ , we see that

$$ \begin{align*}\mkern1mu^{}_{\mathbf{x}}\mathrm{P}^{\mathrm{B}}_{\mathbf{y}}X \simeq \mathrm{colim}_\Delta \mathbb{Q} \otimes_{\mathrm{R}\boldsymbol{\Gamma}(X^{\mathrm{KN}},\underline{\mathbb{Q}})\otimes \mathrm{R}\boldsymbol{\Gamma}(X^{\mathrm{KN}},\underline{\mathbb{Q}})} \mathrm{R}\boldsymbol{\Gamma}((X^{\mathrm{KN}})^{[0,1]},\underline{\mathbb{Q}}).\end{align*} $$

It is now straightforward to see that the simplicial objects $\boldsymbol {\Gamma }_{\mathrm {B}}(\displaystyle \mkern1mu^{}_{\mathbf {x}}\mathrm {P}^{\bullet }_{\mathbf {y}}X^{\mathrm {KN}},\underline {\mathbb {Q}})$ and $\mathbb {Q} \otimes _{\mathrm {R}\boldsymbol {\Gamma }(X^{\mathrm {KN}},\underline {\mathbb {Q}})\otimes \mathrm {R}\boldsymbol {\Gamma }(X^{\mathrm {KN}},\underline {\mathbb {Q}})} \mathrm {R}\boldsymbol {\Gamma }((X^{\mathrm {KN}})^{[0,1]},\underline {\mathbb {Q}})$ are weakly equivalent, so that

$$ \begin{align*}\mkern1mu^{}_{\mathbf{x}}\mathrm{P}^{\mathrm{B}}_{\mathbf{y}}X \simeq \mathrm{colim}_\Delta \boldsymbol{\Gamma}_{\mathrm{B}}( \mkern1mu^{}_{\mathbf{x}}\mathrm{P}^{\bullet}_{\mathbf{y}}X^{\operatorname{KN}},\underline{\mathbb{Q}}).\end{align*} $$

By [Reference ArakawaAra23, Corollary 3.13], we see that there is a natural quasi-isomorphism

$$ \begin{align*}\mkern1mu^{}_{\mathbf{x}}\mathrm{P}^{\text{B}}_{\mathbf{y}}X \simeq \operatorname{Tot}\textbf{R}\Gamma(\mkern1mu^{}_{\mathbf{x}}\mathrm{P}^{\bullet}_{\mathbf{y}}X^{\operatorname{KN}},\underline{\mathbb{Q}}).\end{align*} $$

The previous quasi-isomorphism has a filtered version:

$$ \begin{align*}\mkern1mu^{}_{\mathbf{x}}\mathrm{P}^{\text{B}}_{\mathbf{y}}X_{\leq n} \simeq \operatorname{Tot}\textbf{R}\Gamma(\mkern1mu^{}_{\mathbf{x}}\mathrm{P}^{\bullet}_{\mathbf{y}}X^{\operatorname{KN}}_{\leq n},\underline{\mathbb{Q}}),\end{align*} $$

where the object $\!_{\mathbf {x}}\mathrm {P}^{\bullet }_{\mathbf {y}}X^{\operatorname {KN}}_{\leq n}$ is the diagram obtained from $\!_{\mathbf {x}}\mathrm {P}^{\bullet }_{\mathbf {y}}X^{\operatorname {KN}}$ by restriction to the sets of cardinality $\leq n$ . Based on this result, we will now describe in a closer way to the classical point of view, the Betti realization of the filtered pieces of the path space based at tangential basepoints. This will be useful to compute the periods of the motivic fundamental groupoid. The main reference here is [Reference Ignacio Burgos Gil and FresánBGF25, Section 3], see also [Reference Enriquez and LecomteEL23, Reference Enriquez and LecomteEL21].

Definition 5.2.13. Let n be a non-negative integer, and let $\mathbf {x}, \mathbf {y}$ be two tangential points on X, a divisorial log scheme over k. For $0 \leq i \leq n$ , we denote by $\underline {Y_i}$ the image in $\underline {X}^n$ of the i-th coface map (which is a virtual morphism of log schemes when $i = 0,n$ ) of the cosimplicial based path space $\mkern1mu_{\mathbf {x}}\mathrm {P}_{\mathbf {y}}X$

$$ \begin{align*}\partial_i^n: X^{n-1} \rightarrow X^n.\end{align*} $$

Note that $\underline {\partial ^n_i}$ is an immersion of schemes, so that we can endow $\underline {Y_i}$ with the log structure induced by the isomorphisms $\underline {X}^{n-1} \simeq \underline {Y_i}$ . It comes equipped with a virtual morphism

$$ \begin{align*}Y_i \rightarrow X^n.\end{align*} $$

Remark 5.2.14. The scheme $\underline {Y_i}$ can be described explicitly. If we write $X^n$ in coordinates $(x_1,\dots , x_n)$ , then $\underline {Y_i}$ is

• • $\{x_1 = y \}$ if $i = 0$ ,

• • $\{x_i = x_{i+1}\}$ if $0 < i < n$ ,

• • $\{x_n = x\}$ if $i = n$ .

Naturally, the virtual morphism $Y_i \rightarrow X^n$ is

• • The product of the virtual point $\mathbf {y}$ and the identity on $X^{n-1}$ if $i = 0$ .

• • The ordinary morphism of log schemes induced by the i-th coface of $\mkern1mu_{\mathbf {x}}\mathrm {P}_{\mathbf {y}}X$ .

• • The product of the identity on $X^{n-1}$ and the virtual point $\mathbf {x}$ if $i= n$ .

For $\mathrm {I} \subsetneq \{0,\dots , n\}$ , we let $\underline {Y_{\mathrm {I}}} = \bigcap _{i\in \mathrm {I}} \underline {Y_i}$ . If we denote by $\mathrm {I}^c:= \Delta ^n\setminus \mathrm {I}$ , there are isomorphisms

$$ \begin{align*}\underline{Y_{\mathrm{I}}} \simeq \underline{X}^{|\mathrm{I}^c|-1},\end{align*} $$

obtained by deleting redundant coordinates, which induces a log structure on $\underline {Y_{\mathrm {I}}}$ .

For $i \in \mathbb {N}$ , we define

$$ \begin{align*}Y^i:= \bigsqcup_{\mathrm{I} \subsetneq \{0,\dots,n\}, |\mathrm{I}| = i}Y_{\mathrm{I}}\end{align*} $$

and we denote by $(X^n,Y^{\bullet })$ the family thus defined. Note that for $i>0$ , there are i natural morphisms

given by the diagram for $\mathrm {I} \subsetneq \{0,\dots ,n\}$ , and $k \in \mathrm {I}$

where the right morphism is the k-th coface of $\mkern1mu_{\mathbf {x}}\mathrm {P}_{\mathbf {y}}X$ .

These morphisms endow $(X^n,Y^{\bullet })$ with a natural structure of an augmented symmetric semi-simplicial object in $\!^{\mathbf {v}}\mathbf {Div}_k$ , that is a diagram in $\!^{\mathbf {v}}\mathbf {Div}_k$ indexed by the category of finite sets, with injective morphisms between them (see [Reference Chan, Galatius and PayneCGP21, Section 3]).

Definition 5.2.15. We let $\boldsymbol {\Gamma }_{\mathrm {B}}(X^n,Y^{\bullet })$ be the complex in $\mathcal {M}\mathrm {od}_{\mathbb {Q}}$ associated with the augmented symmetric semi-simplicial variety with log corners $(X^n,Y^{\bullet })$ after applying $\boldsymbol {\Gamma }_{\mathrm {B}}^\otimes $ .

Remark 5.2.16. If we denote by $Y:= \cup _i Y_i$ , then it is a classical fact (see [Reference Deligne and GoncharovDG05]) that the complex $\boldsymbol {\Gamma }_{\mathrm {B}}(X^n,Y^{\bullet })$ is closely related to the relative Betti cohomology of the pair $(X^n,Y)$ , that we denote by $\boldsymbol {\Gamma }_{\mathrm {B}}(X^n,Y)$ . More precisely, there is a fibre sequence

$$ \begin{align*}\underline{\mathbb{Q}}_{|Y_{\Delta^n}}[-n-1] \rightarrow \boldsymbol{\Gamma}_{\mathrm{B}}(X^n,Y) \rightarrow \boldsymbol{\Gamma}_{\mathrm{B}}(X^n,Y^{\bullet}),\end{align*} $$

where $Y_{\Delta ^n} := \cap _{i\in \{0,\dots ,n\}} Y_i$ . Note that it is either a point if $\mathbf {x} = \mathbf {y}$ , or the empty set if $\mathbf {x} \neq \mathbf {y}$ . Hence the fibre sequence induces an isomorphism

$$ \begin{align*}\boldsymbol{\Gamma}_{\mathrm{B}}(X^n,Y) \simeq \boldsymbol{\Gamma}_{\mathrm{B}}(X^n,Y^{\bullet})\end{align*} $$

when $\mathbf {x} \neq \mathbf {y}$ .

In general, the complex $\boldsymbol {\Gamma }_{\mathrm {B}}(X^n,Y^{\bullet })$ is used to compute the filtered pieces of the bar construction. Indeed, with [Reference Ignacio Burgos Gil and FresánBGF25, 3.297, Lemma 3.298] we know that there are natural quasi-isomorphisms of complexes

$$ \begin{align*}\mkern1mu^{}_{\mathbf{x}}\mathrm{P}^{\mathrm{B}}_{\mathbf{y}}X_{\leq n} \simeq \boldsymbol{\Gamma}_{\mathrm{B}}(\mkern1mu^{}_{\mathbf{x}}\mathrm{P}^{\bullet}_{\mathbf{y}}X_{\leq n}) \simeq \boldsymbol{\Gamma}_{\mathrm{B}}(X^n,Y^{\bullet})[n].\end{align*} $$

The quasi-isomorphisms can be described this way: The complex $\boldsymbol {\Gamma }_{\mathrm {B}}(X^n,Y^{\bullet })$ can be represented by the total complex of the double complex

(5) $$ \begin{align} \bigoplus_{\mathrm{I} \subsetneq \{0,\dots, n\}, |\mathrm{I}| = \bullet} C^*(Y_{\mathrm{I}}^{\mathrm{KN}},\mathbb{Q}), \end{align} $$

where the horizontal differentials are given by the semi-simplicial structure of $(X^n,Y^{\bullet })$ (see a full description in [Reference Ignacio Burgos Gil and FresánBGF25, A.8.3]).

The complex $\boldsymbol {\Gamma }_{\mathrm {B}}(\mkern1mu^{}_{\mathbf {x}}\mathrm {P}^{\bullet }_{\mathbf {y}}X_{\leq n})$ is obtained as the total complex of the double complex

(6) $$ \begin{align} C^*(\mkern1mu^{}_{\mathbf{x}}\mathrm{P}^{\bullet}_{\mathbf{y}}X^{\mathrm{KN}}_{\leq n}) \end{align} $$

where the horizontal differentials are given by the cosimplicial structure of $\mkern1mu^{}_{\mathbf {x}}\mathrm {P}^{\bullet }_{\mathbf {y}}X^{\mathrm {KN}}$ .

The complexes (5) and (6) are isomorphic up to a shift of degree n, and the isomorphism is given explicitly in degree k by:

$$ \begin{align*}\begin{array}{ccc} \bigoplus_{0 \leq i \leq n} C^{k+i}((X^{\mathrm{KN}})^i,\mathbb{Q}) & \rightarrow & \bigoplus_{0 \leq i \leq n} \bigoplus_{\mathrm{I} \subsetneq \{0,\dots, n\}, |\mathrm{I}| = i} C^{k+n-i}(Y_{\mathrm{I}}^{\mathrm{KN}},\mathbb{Q}) \\ (\omega_i)_{0 \leq i \leq n} & \mapsto & (\omega_{\mathrm{I}})_{\mathrm{I} \subsetneq \Delta^n} \end{array}\end{align*} $$

where $(\omega _{\mathrm {I}})$ is defined by

$$ \begin{align*}\omega_{\mathrm{I}} = \left\{ \begin{array}{@{}ll} (-1)^{n|\mathrm{I}^c|}\epsilon(\mathrm{I}^c)\omega_{i} & \text{if } \mathrm{I} = \{0,\dots,i\}\\ 0 & \text{else} \end{array} \right.\end{align*} $$

where we let $\epsilon (\mathrm {I}):= \prod _{i \in \mathrm {I}} (-1)^i$ , and where $\omega _i = \omega _{|\mathrm {I}^c|-1}$ as a cochain on $(X^{\mathrm {KN}})^{|\mathrm {I}^c|-1}$ , is viewed as a cochain on $Y_{\mathrm {I}}^{\mathrm {KN}}$ .

Remark 5.2.17. The signs appearing above are conventions, and they are constrained by the signs chosen for the differentials of the complexes involved. We followed here the conventions from [Reference Ignacio Burgos Gil and FresánBGF25].

Using this representation, the isomorphism given by the Beilinson lemma is classically described as follows:

For $i\in \mathbb {N}$ , let us denote by

(7) $$ \begin{align} \Delta^i:= \{(t_1,\dots,t_i)\in \mathbb{R}^i, 0 \leq t_i \leq \dots \leq t_1 \leq 1\} \end{align} $$

the standard i-simplex. Let $\gamma : [0,1] \rightarrow X^{\mathrm {KN}}$ be a continuous path based at $\mathbf {x},\mathbf {y}$ . It induces a singular chain

$$ \begin{align*} \begin{array}{cccc} \sigma_{\gamma,i}: & \Delta^i & \rightarrow & (X^{\mathrm{KN}})^i\\ & (t_1,\dots, t_i) & \mapsto & (\gamma(t_1),\cdots,\gamma(t_i)) \end{array} \end{align*} $$

Moreover, for $\mathrm {I} \subsetneq \{0,\dots , n \}$ , $\sigma _{\gamma ,|\mathrm {I}^c|-1}$ defines a map

$$ \begin{align*}\sigma_{\gamma,\mathrm{I}}: \Delta^{|\mathrm{I}^c|-1} \rightarrow (X^{\mathrm{KN}})^{|\mathrm{I}^c|-1} \simeq Y^{\mathrm{KN}}_{\mathrm{I}}\end{align*} $$

Proposition 5.2.18 [Reference Deligne and GoncharovDG05, Example 3.8],[Reference Ignacio Burgos Gil and FresánBGF25, Section 3.6.4]

The isomorphism

$$ \begin{align*}\mathrm{R}^0_{\mathrm{B}}(\mkern1mu_{\mathbf{x}}^{}\mathrm{P}^{m}_{\mathbf{y}}X)_{\leq n} \simeq (\mathbb{Q}[\pi_1(X^{\mathrm{KN}},\mathbf{x},\mathbf{y})]/J^{n+1})^\vee\end{align*} $$

where J is defined as in Remark 5.2.12, is dual to the map obtained via

$$ \begin{align*}\begin{array}{ccc} \pi_1(X,\mathbf{x},\mathbf{y}) & \rightarrow & (\boldsymbol{\Gamma}_{\mathrm{B}}(X^n,Y^{\bullet})^n)^\vee\\ \gamma & \mapsto & ((-1)^{\frac{|\mathrm{I}|(|\mathrm{I}|+1)}{2}}(-1)^{\frac{n(n-1)}{2}}\epsilon(\mathrm{I}^c)\sigma_{\gamma,\mathrm{I}})_{\mathrm{I}\subsetneq [n]}. \end{array}\end{align*} $$

Remark 5.2.19. The signs are again conventions fixed in a coherent way with the conventions chosen above.