1.1.1

We denote by $\Xi =\Xi (\mathbf {A})$ the class of all exact sequences

(1)\begin{equation} X_2\stackrel{p}{\to} X_1\stackrel{q}{\to} X_0\to 0 \end{equation}

in $\mathbf {A}$, that is, all sequences (1) where $q$ is the cokernel of $p$, which is equivalent to

\[ 0\to \mathbf{A}(X_0,A)\xrightarrow{-{\circ} q} \mathbf{A}(X_1,A)\xrightarrow{-{\circ} p} \mathbf{A}(X_2,A) \]

being exact in $\mathsf {Ab}$ for each $A\in \mathbf {A}$.

1.1.2

A morphism $f:X\to Y$ in $\mathbf {A}$ is split if there exists $g:Y\to X$ such that ${f\circ g\circ f =f}$. Note that this implies that $f\circ g$ and $g\circ f$ are idempotents. If $\mathbf {A}$ is Karoubi (idempotent complete) it thus follows that $f$ is split if and only if we have $X\simeq A\oplus X_0$ and $Y\simeq A\oplus Y_0$ and $f$ is the composition of these isomorphisms with $(\mathrm {id}_A,0)$.

1.2.1

By a $K$-linear symmetric monoidal ($K$-LSM or LSM) category $(\mathbf {C},\otimes , {\mathbb {1}},\sigma )$, we mean a monoidal category $(\mathbf {C},\otimes ,{\mathbb {1}})$ with a symmetric braiding $\sigma$ with a fixed $K$-linear structure on $\mathbf {C}$ for which $-\otimes -$ is $K$-linear in each variable. As is customary, we suppress the associativity constraints and unitors from all notation. Correspondingly we do not place brackets in iterated tensor products. Furthermore, in order to keep long expressions legible, the functor $X\otimes -$, for $X\in \mathbf {C}$, will sometimes be shortened to $X-$. So we might write $XY$ or $Xf$ for an object $Y$ or morphism $f$ in $\mathbf {C}$.

1.2.2

An LSM functor between two $K$-LSM categories is a $K$-LSM functor. Usually we will denote the LSM functor simply by the underlying functor. For two $K$-LSM categories $(\mathbf {C},\otimes , {\mathbb {1}},\sigma )$ and $(\mathbf {C}',\otimes ', {\mathbb {1}}',\sigma ')$, we denote by $\mathsf {LSM}(\mathbf {C},\mathbf {C}')$ the category of LSM functors $\mathbf {C}\to \mathbf {C}'$ and use $\mathsf {LSM}^{{\rm faith}}$ for the subcategory of faithful functors.

1.2.3

For a $K$-LSM category $(\mathbf {C},\otimes , {\mathbb {1}},\sigma )$ and $X\in \mathbf {C}$, a dual of $X$ is a triple $(X^\vee ,\mathrm {ev}_X,\mathrm {co}_X)$ of an object $X^\vee \in \mathbf {C}$ and morphisms $\mathrm {ev}_X:X^\vee \otimes X\to {\mathbb {1}}$ and $\mathrm {co}_X:{\mathbb {1}}\to X\otimes X^\vee$, such that

(2)\begin{equation} \mathrm{id}_X=(X\otimes \mathrm{ev}_X)\circ (\mathrm{co}_X\otimes X)\quad\text{and}\quad \mathrm{id}_{X^\vee}=(\mathrm{ev}_X\otimes X^\vee)\circ (X^\vee\otimes \mathrm{co}_X). \end{equation}

An object which admits a dual is called rigid. If every object in $\mathbf {C}$ admits a dual, then $\mathbf {C}$ is called rigid. The dimension $\dim (X)\in \mathrm {End}({\mathbb {1}})$ of a rigid object is given by $\mathrm {ev}_X\circ \sigma _{XX^\vee }\circ \mathrm {co}_X$.

1.2.4

A tensor ideal $\mathcal {J}$ in a $K$-LSM category $(\mathbf {C},\otimes , {\mathbb {1}},\sigma )$ is an assignment of $K$-submodules $\mathcal {J}(X,Y)\subset \mathbf {C}(X,Y)$ for each $X,Y\in \mathbf {C}$ such that the corresponding class of morphisms is closed under composing or taking the tensor product with any morphism in $\mathbf {C}$. For a tensor ideal $\mathcal {J}$, the quotient category $\mathbf {C}/\mathcal {J}$ has by definition the same objects as $\mathbf {C}$ and as morphism sets the quotient $K$-modules $\mathbf {C}(X,Y)/\mathcal {J}(X,Y)$. By construction, $\mathbf {C}/\mathcal {J}$ is again $K$-LSM, such that $\mathbf {C}\to \mathbf {C}/\mathcal {J}$ is a LSM functor. We can therefore alternatively define tensor ideals as the kernels of LSM functors.

1.3.1

A $k$-LSM category $(\mathbf {D},\otimes ,{\mathbb {1}},\sigma )$ is a pseudo-tensor category over $k$ if:

1. (i) $\mathbf {D}$ is essentially small;

2. (ii) $k\to \mathrm {End}({\mathbb {1}})$ is an isomorphism;

3. (iii) $(\mathbf {D},\otimes ,{\mathbb {1}},\sigma )$ is rigid;

4. (iv) $\mathbf {D}$ is pseudo-abelian (additive and Karoubi).

A pseudo-tensor subcategory of such $\mathbf {D}$ is a full monoidal subcategory closed under taking duals, direct sums and summands. It is thus again a pseudo-tensor category. The quotient of a pseudo-tensor category with respect to a non-trivial tensor ideal is again pseudo-tensor.

Occasionally we will use categories as above except that the field $k$ is replaced by some commutative ring $R$. We will use the same terminology ‘tensor category over $R$’.

If only (i)–(iii) are satisfied, we can take the pseudo-abelian envelope (see [Reference André and KahnAK02, § 1.2]) by formally adjoining direct sums and summands, to obtain a pseudo-tensor category.

1.3.3

Following [Reference DeligneDel90, Reference DeligneDel02], a tensor category over $k$ is a pseudo-tensor category which is abelian (i.e. assumption 1.3.1(iv) is strengthened). In such a category, ${\mathbb {1}}$ is automatically a simple object. A tensor functor between tensor categories is an exact LSM functor. Categories of tensor functors will be denoted by $\mathsf {Tens}$. Following [Reference Coulembier, Entova-Aizenbud and HeidersdorfCEH19, Reference Entova-Aizenbud, Hinich and SerganovaEHS20], we use the following terminology.

Definition 1.3.4 For a pseudo-tensor category $\mathbf {D}$ over $k$, a pair $(F,\mathbf {T})$ of a tensor category $\mathbf {T}$ over $k$ and a faithful LSM functor $F:\mathbf {D}\to \mathbf {T}$ constitutes an abelian envelope of $\mathbf {D}$ if, for each tensor category $\mathbf {T}_1/k$, composition with $F$ induces an equivalence

\[ \mathsf{Tens}(\mathbf{T},\mathbf{T}_1)\;\simeq\; \mathsf{LSM}^{\textrm{faith}}(\mathbf{D},\mathbf{T}_1). \]

We will indulge in the usual abuse of terminology, by referring to the tensor category $\mathbf {T}$ of a pair $(F,\mathbf {T})$ as in Definition 1.3.4 as ‘the abelian envelope of $\mathbf {D}$’. The use of the definite article is justified by obvious uniqueness up to equivalence.

1.3.6

For a tensor category $\mathbf {T}$, the ind-completion $\mathrm {Ind}\mathbf {T}$ has an essentially unique LSM structure such that $-\otimes -$ is exact and cocontinuous and $\mathbf {T}$ is a monoidal subcategory; see [Reference DeligneDel90, § 7]. Since $\mathbf {T}$ is assumed to be essentially small, we can define $\mathrm {Ind}\mathbf {T}$ also as the category of left exact functors $\mathbf {T}^{\mathrm {op}}\to \mathsf {Vec}$.

Proof. We need to show that any rigid object $N\in \mathrm {Ind}\mathbf {T}$ is compact. Take therefore a filtered colimit $\varinjlim Y_\alpha$ in $\mathrm {Ind}\mathbf {T}$. As the tensor product in $\mathrm {Ind}\mathbf {T}$ is cocontinuous, we find

\[ \mathrm{Ind}\mathbf{T}(N,\varinjlim Y_\alpha)\,\simeq\,\mathrm{Ind}\mathbf{T}({\mathbb{1}},\varinjlim (N^\vee\otimes Y_\alpha)). \]

Since ${\mathbb {1}}\in \mathbf {T}\subset \mathrm {Ind}\mathbf {T}$ is compact, we thus find indeed that $\mathrm {Ind}\mathbf {T}(N,-)$ commutes with filtered colimits.

The following lemma is straightforward, but it will be useful to have it spelled out.

Proof. That $\mathbf {D}_1$ is closed under taking direct sums and summands follows from the corresponding property of $\mathbf {D}_0$. Clearly ${\mathbb {1}}\in \mathbf {D}_1$. Now if $V,W\in \mathbf {D}_1$, then by definition

\[ V\otimes X_0\otimes W\otimes X_0\in\mathbf{D}_0\ \Rightarrow \ V\otimes W\otimes (X_0\otimes X_0^\vee\otimes X_0)\in\mathbf{D}_0. \]

Since $X_0$ is a direct summand of $X_0\otimes X_0^\vee \otimes X_0$, it follows that $V\otimes W\otimes X_0$ is a direct summand of an object in $\mathbf {D}_0$ and hence also in $\mathbf {D}_0$. In conclusion, $V\otimes W\in \mathbf {D}_1$. That $\mathbf {D}_1$ is closed under taking duals follows similarly.

1.3.9

Consider a pseudo-tensor category $\mathbf {D}$ over $k$ and a field extension $K/k$. The naive extension of scalars of $\mathbf {D}$ (see [Reference André and KahnAK02, 5.1.1]) is the $K$-linear category with the same objects as $\mathbf {D}$, but with morphism sets given by $K\otimes _k\mathbf {D}(-,-)$. We define $\mathbf {D}_K$ as the Karoubi envelope of the naive extension of scalars. Note that in [Reference André and KahnAK02, § 5.3], the notation $(\mathbf {D}_K)^\sharp$ is used for what we call $\mathbf {D}_K$. Now $\mathbf {D}_K$ is canonically a pseudo-tensor category over $K$.

1.4.1

Following [Reference DeligneDel07, § 10], we have the category $[\mathrm {GL}_t,R]_0$, which is the free rigid $R$-LSM monoidal category on one object $V_t$ of dimension $t$. Its objects are (up to isomorphism) tensor products of $V_t$ and $V_t^\vee$.

The pseudo-abelian envelope $[\mathrm {GL}_t,R]$ of $[\mathrm {GL}_t,R]_0$ is thus a pseudo-tensor category over $R$. By construction, every object $X$ in $[\mathrm {GL}_t,R]$ is a direct summand of a direct sum of objects $\otimes ^i V_t\otimes \otimes ^jV_t^\vee$. We denote by $\deg X$ the minimal $d\in \mathbb {N}$ such that $X$ is a direct summand of a direct sum of $\otimes ^a V_t \otimes \otimes ^b V^\vee _t$ with $a+b\le d$. By [Reference DeligneDel07, Théorème 10.5], $[\mathrm {GL}_t,k]$ is a semisimple tensor category when $\mathrm {char}(k)=0$ and $t\not \in \mathbb {Z}$.

The following is a reformulation of [Reference DeligneDel07, Proposition 10.3].

1.4.3

Set $\mathbf {D}:=[\mathrm {GL}_t,k]$ for an algebraically closed field $k$ of characteristic $0$, for $t\in \mathbb {Z}\subset k$. Consider the tensor category $\mathsf {svec}$ of finite-dimensional super vector spaces; see [Reference DeligneDel90, § 1.4]. Let $\mathrm {GL}(m|n)$ be the affine group scheme in $\mathsf {svec}$ of automorphisms of the super space $k^{m|n}$ of even dimension $m$ and odd dimension $n$. As in [Reference DeligneDel02, § 0.3], we have the tensor category $\mathsf {Rep}_k\mathrm {GL}(m|n)$ of its representations in $\mathsf {svec}$ which restrict to the canonical $\mathbb {Z}/2$-action along the homomorphism $\mathbb {Z}/2\to \mathrm {GL}(m|n)$ defining the grading on $\mathrm {GL}(m|n)$. As an application of Lemma 1.4.2, there exists an LSM functor

\[ H_{m|n}:\mathbf{D}\to \mathsf{Rep}_k\mathrm{GL}(m|n),\quad V_t\mapsto k^{m|n}, \]

for every $m,n\in \mathbb {N}$ with $m-n=t$.

1.4.5

A rigid object $X$ in a LSM category is symmetrically self-dual if we can take $X^\vee = X$ and we have $\mathrm {ev}_X=\mathrm {ev}_X\circ \sigma _{X,X}$. Following [Reference DeligneDel07, § 9], we have the category $[\mathrm {O}_t,R]_0$, which is the free $R$-LSM category on one symmetrically self-dual object $U_t$ of dimension $t$. Its objects are tensor powers of $U_t$.

The pseudo-abelian envelope $[\mathrm {O}_t,R]$ of $[\mathrm {O}_t,R]_0$ is thus a pseudo-tensor category over $R$. By construction, every object $X$ in $[\mathrm {O}_t,R]$ is a direct summand of a direct sum of objects $\otimes ^i U_t$. We denote by $\deg X$ the minimal $d\in \mathbb {N}$ such that $X$ is a direct summand of a direct sum of $\otimes ^i U_t$ with $i\le d$. By [Reference DeligneDel07, Théorème 9.7], $[\mathrm {O}_t,k]$ is a semisimple tensor category when $\mathrm {char}(k)=0$ and $t\not \in \mathbb {Z}$.

1.4.7

Set $\mathbf {D}:=[\mathrm {O}_t,k]$ for an algebraically closed field $k$ of characteristic $0$, for $t\in \mathbb {Z}\subset k$. Consider a non-degenerate (super)symmetric bilinear form on $k^{m|2n}\in \mathsf {svec}$ and let $\mathrm {OSp}(m|2n)$ be the closed subgroup of $\mathrm {GL}(m|2n)$ which preserves the form. As an application of Lemma 1.4.6, there exists an LSM functor

\[ F_{m|2n}:\mathbf{D}\to \mathsf{Rep}_k\mathrm{OSp}(m|2n),\quad U_t\mapsto k^{m|2n} \]

for every $m,n\in \mathbb {N}$ with $m-2n=t$.

2.3.3

For a commutative $k$-algebra $K$, consider the subcategory $\mathbf {C}$ of $[\mathrm {GL}_0,K]_0$ which has the same objects and for which the inclusion functor $\mathbf {C}\to [\mathrm {GL}_0,K]_0$ is full on each morphism set, except that on $\mathbf {C}({\mathbb {1}},{\mathbb {1}})$ it realises the unit morphism $k\to K$. That $\mathbf {C}$ constitutes a (monoidal) subcategory of $[\mathrm {GL}_0,K]_0$ follows from the fact that the collection of all morphisms in $[\mathrm {GL}_0,K]_0$ excluding the ones ${\mathbb {1}}\to {\mathbb {1}}$ form a (tensor) ideal.

Now the pseudo-abelian envelope $\mathbf {D}$ of $\mathbf {C}$ is a pseudo-tensor category over $k$.

Proof. Part (ii) can be derived from Theorem 2.3.1.

For part (i), we consider the sequence in Definition 2.2.2(i) for $M=N={\mathbb {1}}$ and $X=V_0$, which yields

\[ 0\to k\to K\to K \mathrm{S}_2, \]

where $\mathrm {S}_2$ is the symmetric group on two symbols, the morphism $k\to K$ is the unit morphism and the morphism $K\to K\mathrm {S}_2$ is zero. This follows either by direct computation or from the fact that $V_0$ is strongly faithful in $[\mathrm {GL}_0,K]$, which shows that the kernel of the morphism $K\to K \mathrm {S}_2$ is $K$, the endomorphism ring of ${\mathbb {1}}$ in $[\mathrm {GL}_0,K]$.

3.1.1

We consider the $k$-linear presheaf category $\mathrm {PSh}\mathbf {D}$ of $k$-linear functors $\mathbf {D}^{\mathrm {op}}\to \mathsf {Vec}_k$. Then $\mathrm {PSh}\mathbf {D}$ is symmetric closed monoidal for the Day convolution $\star$; see, for example, [Reference SchäppiSch20, § 3.2]. The tensor product of two presheaves $F,G$ is given by the co-end expression

\[ F\star G:=\int^{X,Y\in\mathbf{D}}F(X)\otimes_k G(Y)\otimes_k \mathbf{D}(-,X\otimes Y), \]

and the internal Hom is given by

\[ [G,H]= \int_{Y\in \mathbf{D}}\mathrm{Hom}_k(G(Y),H(Y\otimes{-})). \]

The Yoneda embedding $Y:\mathbf {D}\to \mathrm {PSh}\mathbf {D}$ is canonically LSM. By construction, the tensor product $-\star -$ is cocontinuous in each variable.

3.1.2

We define the full subcategory $\mathrm {Sh}\mathbf {D}$ of $F\in \mathrm {PSh}\mathbf {D}$ for which $F(D\otimes \gamma _X)$ is exact in $\mathsf {Vec}$, for every exact sequence

(4)\begin{equation} D\otimes\gamma_X:\;\;DX^\vee XX^\vee X\to DX^\vee X\to D\to 0, \end{equation}

with $X$ strongly faithful and $D$ arbitrary in $\mathbf {D}$. Since ‘limits commute’, it follows that the inclusion functor from $\mathrm {Sh}\mathbf {D}$ to $\mathrm {PSh}\mathbf {D}$ is continuous and hence (by Freyd's special adjoint functor theorem) admits a left adjoint

(5)\begin{equation} S\,:\,\mathrm{PSh}\mathbf{D}\to\mathrm{Sh}\mathbf{D}, \end{equation}

the sheafification or reflection. The restriction of $S$ to $\mathrm {Sh}\mathbf {D}$ is the identity. If $F\in \mathrm {Sh}\mathbf {D}$, then clearly the functor $F(Z\otimes -)$ is also in $\mathrm {Sh}\mathbf {D}$, for each $Z\in \mathbf {D}$. It then follows as a direct application of Day's reflection theorem [Reference DayDay72, Theorem 1.2(2)] that there is a unique closed symmetric monoidal structure on $\mathrm {Sh}\mathbf {D}$ which makes $S$ LSM. We denote the tensor product on $\mathrm {Sh}\mathbf {D}$ again by $\otimes$. The Yoneda embedding $Y:\mathbf {D}\to \mathrm {PSh}\mathbf {D}$ factors through the embedding of the subcategory $\mathrm {Sh}\mathbf {D}$. We will denote the corresponding fully faithful functor by $Y_0:\mathbf {D}\to \mathrm {Sh}\mathbf {D}$. It is isomorphic to the composite $S\circ Y$, so, in particular, $Y_0$ is LSM.

3.1.3

We refer to the appendix for the notions of sieve, Grothendieck topology, the category of sheaves with respect to a topology and localisations of Grothendieck categories.

For each $D\in \mathbf {D}$, denote by $\mathcal {T}(D)$ the set of all sieves $R\subset \mathbf {D}(-,D)$ such that there exists a strongly faithful $X\in \mathbf {D}$ for which $D\otimes \mathrm {ev}_X\in R(DX^\vee X)$. Our notation $\mathrm {Sh}\mathbf {D}$ is justified by the following theorem.

Proof. Part (i) will be proved in § 3.2. We explain how (i) implies (ii) in the usual fashion. As a left adjoint, the reflection $S$ in (5) is cocontinuous. This already implies that $\mathrm {Sh}\mathbf {D}$ is cocomplete. It also follows that the coproduct in $\mathrm {Sh}\mathbf {D}$ over the set of isomorphism classes of objects in $\mathbf {D}$,

\[ G:=\bigoplus_{X\in\mathrm{Ob}\mathbf{D}/{\simeq}} X, \]

is a generator of $\mathrm {Sh}\mathbf {D}$, meaning that $\mathrm {Sh}\mathbf {D}(G,-):\mathrm {Sh}\mathbf {D}\to \mathsf {Vec}$ is faithful. Hence it suffices to show that $\mathrm {Sh}\mathbf {D}$ is abelian and that direct limits of short exact sequences are (left) exact. Both properties follow easily if $S:\mathrm {PSh}\mathbf {D}\to \mathrm {Sh}\mathbf {D}$ is (left) exact, that is, when $\mathrm {Sh}\mathbf {D}$ is a localisation of $\mathrm {PSh}\mathbf {D}$. Hence claim (ii) follows from claim (i) and Theorem A.1.4.

For part (v), we can observe that by definition and the Yoneda lemma,

\[ \mathrm{Sh}\mathbf{D}(Y_0(D\otimes\gamma_X),F)=\mathrm{PSh}\mathbf{D}(Y(D\otimes\gamma_X),F)=F(D\otimes\gamma_X), \]

for arbitrary $F\in \mathrm {Sh}\mathbf {D}$. Hence $Y_0(D\otimes \gamma _X)$ is indeed exact.

Part (iv) follows easily from the fact that in $\mathsf {Vec}$, a filtered colimit of short exact sequences is exact.

Finally, we prove part (iii). Objects in $\mathbf {D}$ are compact in $\mathrm {PSh}\mathbf {D}$, so part (iv) implies that $Y_0:\mathbf {D}\to \mathrm {Sh}\mathbf {D}$ sends every object in $\mathbf {D}$ to a compact object in $\mathrm {Sh}\mathbf {D}$. That every object is a quotient of coproduct of objects in $\mathbf {D}$ follows from the above fact that $G$ is a generator.

3.2.1

First we prove that $\mathcal {T}$ from § 3.1.3 constitutes a topology as in Definition A.1.2. Condition (T1) is immediate from Example 2.2.3(i). For condition (T2), consider $A\in \mathbf {D}$, $R\in \mathcal {T}(A)$ and a morphism $f:B\to A$ in $\mathbf {D}$. By definition, there exists a strongly faithful $X\in \mathbf {D}$ such that $A\otimes \mathrm {ev}_X$ is in $R$. It then follows that $B\otimes \mathrm {ev}_X$ is in $f^{-1}R$, so $f^{-1}R\in \mathcal {T}(B)$.

For condition (T3), consider $S\subset \mathbf {D}(-,A)$ and $R\in \mathcal {T}(A)$ as in (T3). Since there exists $f:=A\otimes \mathrm {ev}_X$ in $R(AX^\vee X)$, for some strongly faithful $X$, there must exist a strongly faithful $Y\in \mathbf {D}$ such that $AX^\vee X\mathrm {ev}_Y$ is in $f^{-1}S(AX^\vee X Y^\vee Y)$. The latter just means that $A\otimes \mathrm {ev}_X\otimes \mathrm {ev}_Y$ is in $S(AX^\vee XY^\vee Y)$, which means that $A\otimes \mathrm {ev}_{X\otimes Y}$ is also in $S(AY^\vee X^\vee XY)$. It then follows from Proposition 2.2.6(i) that $S\in \mathcal {T}(A)$.

3.2.2

We now prove the equality $\mathrm {Sh}\mathbf {D}=\mathrm {Sh}(\mathbf {D},\mathcal {T})$. Take an arbitrary presheaf $F\in \mathrm {PSh}\mathbf {D}$. By a ‘pair’ $(D,X)$ we mean an arbitrary $D\in \mathbf {D}$ and a strongly faithful $X\in \mathbf {D}$. For each pair $(D,X)$, denote by $R^D_X$ the sieve on $D$ generated by the morphism $D\otimes \mathrm {ev}_X$. This is the minimal sieve on $D$ containing $D\otimes \mathrm {ev}_X$, or equivalently the image of

\[ \mathbf{D}(-,DX^\vee X)\xrightarrow{(D \otimes\mathrm{ev}_X)\circ{-}} \mathbf{D}(-,D). \]

Since the representable functors yield a set of generators for $\mathrm {PSh}\mathbf {D}$, we can complete the epimorphism $\mathbf {D}(-, DX^\vee X)\twoheadrightarrow R^D_X$ to give an exact sequence

\[ \bigoplus_{g:B\to DX^\vee X, f\circ g=0}\mathbf{D}(-,B)\to \mathbf{D}(-, DX^\vee X)\to R^D_X\to 0 \]

in $\mathrm {PSh}\mathbf {D}$ with $f:=D\otimes \mathrm {ev}_X$. Consequently, $F(D)\to \mathrm {Nat}(R^D_X,F)$ is an isomorphism if and only if the sequence

(6)\begin{equation} 0\to F(D)\xrightarrow{F(f)} F(DX^\vee X)\to \prod_{g:B\to DX^\vee X, f\circ g=0}F(B) \end{equation}

is exact. On the other hand, by definition, $F\in \mathrm {Sh}\mathbf {D}$ if and only if

(7)\begin{equation} 0\to F(D)\xrightarrow{F(f)} F(DX^\vee X)\xrightarrow{F(D\mathcal{E}_X)}F(DX^\vee XX^\vee X) \end{equation}

is exact for every pair $(D,X)$.

Clearly, if (7) is exact, then so is (6). On the other hand, assume that (6) is exact, for a fixed $X$ but for every $D\in \mathbf {D}$. For any $g:B\to DX^\vee X$ with $f\circ g=0$, we have the following commutative diagram.

Applying $F$ yields the following commutative diagram.

The fact that the lower horizontal arrow is a monomorphism follows from our assumption that (6) with $D$ replaced by $B$ is exact. Consider $a\in F(DX^\vee X)$ such that $F(D\mathcal {E}_X)(a)=0$. By commutativity of the diagram (and using the monomorphism) we also find that $F(g)(a)=0$. Exactness of (6) thus implies that $a$ is the image of $F(f)$ and therefore (7) is exact too.

By the above two paragraphs we find that when $F\in \mathrm {Sh}(\mathbf {D},\mathcal {T})$, it follows that (6) is exact for all pairs $(D,X)$, so (7) is exact for all pairs and consequently $F\in \mathrm {Sh}\mathbf {D}$. Next, assume that $F\in \mathrm {Sh}\mathbf {D}$. Then by the reverse reasoning we find that $F(D)\to \mathrm {Nat}(R^D_X,F)$ is an isomorphism for all pairs $(D,X)$. Now take an arbitrary $R\in \mathcal {T}(D)$. It is of the form $R^D_X\subset R\subset \mathbf {D}(-,D)$ for some strongly faithful $X$. We have already found that the composite

\[ F(X)\to \mathrm{Nat}(R,F)\to \mathrm{Nat}(R^D_X,F) \]

is an isomorphism, so we only need to show that $\mathrm {Nat}(R,F)\to \mathrm {Nat}(R^D_X,F)$ is a monomorphism to be able to conclude that $F$ is a $\mathcal {T}$-sheaf. Take therefore arbitrary $A\in \mathbf {D}$ and $h\in R(A)\subset \mathbf {D}(A,X)$. In $\mathrm {PSh}\mathbf {D}$, we can expand the commutative square of morphisms $AX^\vee X\rightrightarrows D$ to give the following commutative diagram.

Evaluation of $\mathrm {Nat}(-,F)$ yields the commutative diagram

where the left vertical arrow is a monomorphism by exactness in (7) for the pair $(A,X)$. A natural transformation $\eta :R\Rightarrow F$ which is sent to zero in $\mathrm {Nat}(R_X^D,F)$ therefore satisfies $\eta _A(h)=0$ for all $A\in \mathbf {D}$ and $h\in R(A)$, or in other words $\eta =0$. This proves the required monomorphism and thus concludes the proof of the claim $\mathrm {Sh}\mathbf {D}=\mathrm {Sh}(\mathbf {D},\mathcal {T})$.

Analogy 3.2.3. Consider a category $\mathbf {B}$ with finite products, with terminal object $\ast$. By [Reference Demazure and GrothendieckSGA3, IV.1.3], the morphism $U\to \ast$ is a ‘universal effective epimorphism’ if the induced

(8)\begin{equation} V\times U\times U\rightrightarrows V\times U\to V \end{equation}

is a coequaliser for every $V\in \mathbf {B}$. By [Reference Demazure and GrothendieckSGA3, IV.1.8], if $U\to \ast$ and $U'\to \ast$ are universal effective epimorphisms, the same is true for $U\times U'\to \ast$. Take $C\subset \mathrm {Ob}\mathbf {B}$, containing $\ast$ and closed under products, such that $U\to \ast$ is a universal effective epimorphism for every $U\in C$. The corresponding collection of coverings $V\times U\to V$ forms a classical Grothendieck pretopology. The sheaves are the presheaves $F:\mathbf {B}^{\mathrm {op}}\to \mathsf {Set}$ which send the diagrams (8) to equalisers.

Remark 3.2.4 Denote by $\Sigma \subset \Xi (\mathbf {D})$ the class of all exact sequences

\[ D\otimes\gamma_X\quad\text{and}\quad D\otimes (X^\vee\otimes X\xrightarrow{\mathrm{ev}_X} {\mathbb{1}}\to 0\to 0), \]

for arbitrary $D\in \mathbf {D}$ and strongly faithful $X\in \mathbf {D}$. It follows easily, and from similar arguments to those used in § 3.2.2, that $\Sigma$ constitutes an ‘ind-class’, as in Definition A.2.1 in the appendix. We can therefore also prove Theorem 3.1.4(i) by applying Propositions A.2.2 and A.2.3.

4.2.4

In [Reference DeligneDel07, § 2], a pseudo-tensor category $[\mathrm {S}_t,k]$ is defined for every $t\in k$, which is a semisimple tensor category when $t\not \in \mathbb {N}$ by [Reference DeligneDel07, Théorème 2.18]. In [Reference DeligneDel07, Proposition 8.18] it is shown that, for $n\in \mathbb {N}$, the pseudo-tensor category $[\mathrm {S}_n,k]$ admits a fully faithful tensor $F$ functor into a tensor category $\mathbf {T}_n$. By Corollary 2.2.7, every non-zero object in $[\mathrm {S}_n,k]$ is strongly faithful. Furthermore, it is proved in [Reference Comes and OstrikCO14, Lemma 3.11] that there exists a non-zero object which splits every morphism in $[\mathrm {S}_n,k]$. Theorem 4.1.1 therefore demonstrates that $[\mathrm {S}_n,k]$ admits an abelian envelope. This recovers one of the main results in [Reference Comes and OstrikCO14].

It seems worthwhile to point out the following observations (although the equivalent properties are of course known to be true by [Reference Comes and OstrikCO14]), which do not rely on [Reference Comes and OstrikCO14, Lemma 3.11]. The latter lemma is one of the cornerstones in both the original and above proof that $[\mathrm {S}_n,k]$ admits an abelian envelope, but has a rather intricate proof.

4.3.1

We work in the tensor category $\mathsf {Rep}\, {\rm SL}_2$ of finite-dimensional algebraic representations of the algebraic group ${\rm SL}_2/k$. We have the pseudo-tensor subcategory $\mathbf {D}:=\mathsf {Tilt}\, {\rm SL}_2$ of tilting modules; see [Reference JantzenJan03, §II.E]. We denote the simple module and the indecomposable tilting module with highest weight $i\omega$ (with $\omega$ the fundamental weight) by $L_i$ and $T_i$, for $i\in \mathbb {N}$. The Steinberg modules (see [Reference JantzenJan03, II.3.18]) are

\[ \mathrm{St}_j=L_{p^j-1}= T_{p^j-1},\quad\text{for } j\in\mathbb{N}. \]

For $r\in \mathbb {Z}_{>0}$, we consider the tensor ideal $\mathcal {J}_r$ in $\mathsf {Tilt}\, {\rm SL}_2$ of morphisms which factor through a direct sum of objects $T_i$, with $i\ge (p^r-1)$. This gives a complete and irredundant list of the non-trivial tensor ideals in $\mathsf {Tilt}\, {\rm SL}_2$; see [Reference CoulembierCou18, § 5.3]. Consequently, $\mathcal {J}_r$ is generated by $\mathrm {id}_{\mathrm {St}_{r}}$.

The condition $p>2$ is not required and only reflects the limitations of the proof of Lemma 4.3.5 below. Indeed, the equivalent of Theorem 4.3.2 for $p=2$ is already known by [Reference Benson and EtingofBE19]. We start the proof with the following lemma.

Proof. By the Steinberg tensor product theorem [Reference JantzenJan03, II.3.17], for $i< p^r$ we have

\[ L_i\simeq \bigotimes_{a=0}^{r-1}L_{p^ai_a},\quad \text{with}\ i=\sum_{a=0}^{r-1} p^a i_a\ \text{and}\ 0\le i_a< p. \]

By Lemma 1.3.8, it therefore suffices to prove that $L_{p^ab}\otimes \mathrm {St}_{r-1}$ is a tilting module for $a< r$ and $b< p$. We prove the more general claim that $L_m\otimes \mathrm {St}_{r-1}$ is a tilting module for $m\le p^r-p^{r-1}.$ By [Reference JantzenJan03, Proposition E.1], it then suffices to prove that

(12)\begin{equation} \mathrm{Ext}^1(\Delta_n,L_m\otimes\mathrm{St}_{r-1} )=0,\quad\text{for } n\in\mathbb{N} \text{ and } m\le p^r-p^{r-1}, \end{equation}

where $\Delta _n$ is the Weyl module with top $L_n$.

We divide (12) into two cases. First assume that $n\ge p^r-1$. Then $n\ge m+p^{r-1}-1$, so $L_m\otimes \mathrm {St}_{r-1}$ belongs to the Serre subcategory ${\mathsf {Rep}\, {\rm SL}_2}^{\le n}$ generated by simples $L_j$ with $j\le n$ in which $\Delta _n$ is projective. Hence (12) is satisfied. Now assume that $n< p^r-1$. The left-hand side of (12) can be rewritten as $\mathrm {Ext}^1(\Delta _n\otimes L_m,\mathrm {St}_{r-1} )$, and by our assumption

\[ n+m<2p^r-p^{r-1}-1. \]

By Lemma 4.3.3, this means that the direct summand of $\Delta _n\otimes L_m$ in the block of $\mathrm {St}_{r-1}$ is a direct sum of copies of $\mathrm {St}_{r-1}$, so the extension vanishes and (12) is again satisfied.

We set $\mathbf {D}=\mathsf {Tilt}\, {\rm SL}_2$ and $\mathbf {C}=\mathbf {D}/\mathcal {J}_r$.

Proof. By definition, we need to prove that the sequence

(13)\begin{equation} \mathbf{C}(\gamma_{\mathrm{St}_{r-1}}, T_i):\ 0\to \mathbf{C}({\mathbb{1}}, T_i)\to \mathbf{C}(\otimes^2\mathrm{St}_{r-1}, T_i)\to \mathbf{C}(\otimes^4\mathrm{St}_{r-1}, T_i) \end{equation}

is exact, for each $0\le i< p^r-1$.

The structure of tensor ideals recalled in § 4.3.1 implies that, for $i\ge p^{r-1}-1$, the module $T_i$ is a direct summand of an object $T\otimes \mathrm {St}_{r-1}$. That (13) is exact for $i\ge p^{r-1}-1$ is thus an example of Lemma 2.2.9.

Next, we consider $T_i$ with $i< p^{r-1}-1$. We claim that

\[ \mathcal{J}_r({\mathbb{1}},T_i)=0=\mathcal{J}_r(\otimes^2\mathrm{St}_{r-1}, T_i)=\mathcal{J}_r(\otimes^4\mathrm{St}_{r-1}, T_i). \]

That the leftmost space is zero follows immediately from the description of the ideals $\mathcal {J}_l$ in [Reference CoulembierCou18, § 3.2]. We now prove the claim for the rightmost space; the proof for the middle space is similar but easier. By adjunction, we can equivalently prove

\[ \mathcal{J}_r(\mathrm{St}_{r-1}, \otimes^3\mathrm{St}_{r-1}\otimes T_i)=0. \]

By definition of $\mathcal {J}_r$ and Lemma 4.3.3, the contrary would necessarily imply that

\[ [\otimes^3\mathrm{St}_{r-1}\otimes T_i:L_a]\not=0,\quad\text{for some}\ a\ge 2p^r-p^{r-1}-1. \]

However, since we have

\[ i+3(p^{r-1}-1)< 4p^{r-1} -4 < 2p^r-p^{r-1}-1, \]

under the assumption $p> 2$, this non-vanishing multiplicity is impossible. It follows that for $i< p^{r-1}-1$ we have the following commutative diagram, with the second row given by (13).

The first row is exact by Corollary 2.2.7 and the inclusion $\mathbf {D}\subset \mathsf {Rep}\, {\rm SL}_2$. Hence the second row is exact. This concludes the proof.

4.4.1

A thick tensor ideal in $\mathbf {D}$ is a full Karoubi subcategory $J$ of $\mathbf {D}$ such that $X\in J$ implies that $Y\otimes X\in J$ for all $Y\in \mathbf {D}$. The decategorification map (see [Reference CoulembierCou18, § 4.1]) sends a tensor ideal $\mathcal {J}$ in $\mathbf {D}$ to the thick tensor ideal of objects $X$ with $\mathrm {id}_X\in \mathcal {J}$. By [Reference CoulembierCou18, Theorem 4.1.2], this map is always surjective.

Proof. By Corollary 4.1.3, it suffices to prove that any epimorphism in $\mathbf {V}$ is split by a non-zero object in $\mathbf {D}$. We do this in three steps.

(1) Consider a non-zero morphism $f:D\to {\mathbb {1}}$ in $\mathbf {D}$. This is automatically an epimorphism in $\mathbf {V}$. By [Reference CoulembierCou18, Proposition 4.2.2], $f$ is unique up to composition with endomorphisms of $D$. Furthermore, [Reference CoulembierCou18, Lemma 4.2.4] then implies that there exists $X\in \mathbf {D}$ such that $\mathrm {ev}_X$ is given by a composition

\[ \mathrm{ev}_X:\;X^\vee\otimes X\to D\; \xrightarrow{f}\;{\mathbb{1}}. \]

By Lemma 2.1.2(i) the morphism $X\otimes \mathrm {ev}_X$, and hence also $X\otimes f$, is split.

(2) Consider an epimorphism $g:M\twoheadrightarrow {\mathbb {1}}$ in $\mathbf {V}$. By assumption, there exists $D\in \mathbf {D}$ such that we have an epimorphism $\pi :D\twoheadrightarrow M$. By step 1, $X\otimes (g\circ \pi )$ is split for some non-zero $X\in \mathbf {D}$ from which it follows that also $X\otimes g$ is split.

(3) Finally, we consider an arbitrary epimorphism $h:M\twoheadrightarrow N$ in $\mathbf {V}$. Tensoring with $N^\vee$ and taking a pullback yields the following commutative diagram.

By step 2, there exists a non-zero $X\in \mathbf {D}$ which splits the epimorphism on the lower line. After applying $X\otimes -$, the diagram thus admits a diagonal morphism $X\to XMN^\vee$ which makes the upper triangle commute. It then follows that the associated morphism $XN\to XM$ ensures that $X\otimes h$ is split.