Incompressible categories, uniqueness and reconstruction

Written by Claude Opus 5.5 (Anthropic), October 2026. Self-checked by the writing AI. Public domain (CC0).

Lesson 8 proved that every finite symmetric tensor category \(\mathcal C\) admits a fibre functor to some higher Verlinde category \(\operatorname{Ver}_{p^n}\). Combined with what is known about the higher Verlinde categories themselves, this has three consequences, proved in this lesson. First, the finite symmetric tensor categories that cannot be compressed, in the sense that every exact symmetric tensor functor out of them is fully faithful, are exactly the higher Verlinde categories and their even parts (Corollary 2.2). Second, in characteristic two the fibre functor to the union of the tower is unique (Corollary 3.1). Third, every finite symmetric tensor category is the category of representations of an affine group scheme internal to a higher Verlinde category, generalizing the reconstruction theorem of the lesson Tannakian categories (Corollary 4.1).

Prerequisites. Descent to a Verlinde category, Theorem 8.1; Finite symmetric tensor categories and fibre functors, Sections 3 and 4; Tannakian categories, Section 1.

1. Results from the literature

All functors in this lesson are \(k\)-linear. A tensor subcategory of a finite symmetric tensor category is a full subcategory containing \(\mathbf 1\) and closed under tensor products, duals, finite direct sums and subquotients. For \(p^m>2\), the category \(\operatorname{Ver}_{p^m}^+\) is the tensor subcategory of \(\operatorname{Ver}_{p^m}\) generated by the images of the tilting modules \(T(a)\) with \(a\) even [BEO, Section 4.1]. For odd \(p\) put \(\operatorname{Ver}_1=\operatorname{sVec}_k\) and \(\operatorname{Ver}_1^+=\operatorname{Vec}_k\), the tensor subcategories of \(\operatorname{Ver}_p\) generated by \(L_1,L_{p-1}\) and by \(L_1\) [CEOi, Section 1.3]. The low levels are \(\operatorname{Ver}_2\simeq\operatorname{Vec}_k\), \(\operatorname{Ver}_3\simeq\operatorname{sVec}_k\) and \(\operatorname{Ver}_3^+\simeq\operatorname{Vec}_k\) [BEO, Theorem 4.5, Corollary 4.11, Example 4.52].

Theorem 1.1 (named results N1 and N2, continued).

  1. The tensor subcategories of \(\operatorname{Ver}_{p^n}\) are the categories \(\operatorname{Ver}_{p^m}\) and \(\operatorname{Ver}_{p^m}^+\) with \(m\le n\), including \(m=0\) for odd \(p\) [BEO, Corollary 4.61; CEOi, Section 1.3].
  2. Every exact symmetric tensor functor from \(\operatorname{Ver}_{p^n}\), or from \(\operatorname{Ver}_{p^n}^+\) when \(p^n>2\), to a symmetric tensor category is fully faithful [BEO, Theorem 4.71 and Remark 4.72(iii); BE, Theorem 4.4].
  3. Let \(F\colon\mathcal C\to\mathcal D\) be an exact functor between finite abelian categories. The full subcategory of \(\mathcal D\) of subquotients of objects \(F(X)\) is a finite abelian category [EGNO, Definition 1.8.18 and Proposition 1.8.19]; if \(F\) is a tensor functor between finite symmetric tensor categories, it is a tensor subcategory, since a subquotient of \(F(X)\otimes F(Y)\cong F(X\otimes Y)\) or of \(F(X)^*\cong F(X^*)\) is again a subquotient of an object in the image.
  4. Let \(F\colon\mathcal C\to\mathcal D\) be an exact tensor functor between finite tensor categories, and \(\operatorname{FPdim}(\mathcal C)=\sum_Sd(S)\,d(P_S)\), the sum over simple objects of the products of the dimensions of \(S\) and its projective cover. If \(F\) is fully faithful, then \(\operatorname{FPdim}(\mathcal C)\le\operatorname{FPdim}(\mathcal D)\); if every object of \(\mathcal D\) is a subquotient of some \(F(X)\), then \(\operatorname{FPdim}(\mathcal C)\ge\operatorname{FPdim}(\mathcal D)\); in either case equality holds only if \(F\) is an equivalence [EGNO, Propositions 6.3.3 and 6.3.4].

Theorem 1.2 (named result N10 [CEOi]).

  1. (Subterminality.) Let \(p=2\) and let \(\operatorname{Ver}_{2^\infty}=\bigcup_n\operatorname{Ver}_{2^n}\) along the inclusions of the tower. For every symmetric tensor category \(\mathcal T\) over \(k\) there is, up to monoidal natural isomorphism, at most one exact symmetric tensor functor \(\mathcal T\to\operatorname{Ver}_{2^\infty}\) [CEOi, Theorem 9.4.1].
  2. (Reconstruction.) Let \(\mathcal V\) be a finite symmetric tensor category and \(F\colon\mathcal C\to\mathcal V\) an exact symmetric tensor functor from a finite symmetric tensor category. Let \(G=\underline{\operatorname{Aut}}^{\otimes}(F)\), an affine group scheme in \(\mathcal V\), and \(\phi\colon\pi(\mathcal V)\to G\) the canonical homomorphism from the fundamental group \(\pi(\mathcal V)=\underline{\operatorname{Aut}}^{\otimes}(\mathrm{Id}_{\mathcal V})\). Then \((G,\phi)\) is a \(\mathcal V\)-group, and there is a symmetric tensor equivalence \(E\colon\mathcal C\to\operatorname{Rep}(G,\phi)\) with a monoidal natural isomorphism \(\omega\circ E\cong F\), where \(\omega\) forgets to \(\mathcal V\) [CEOi, Sections 4.1.1 to 4.1.4 and Theorem 4.2.1].

Here an affine group scheme in \(\mathcal V\) is a commutative Hopf algebra \(\mathcal O(G)\) in the category \(\operatorname{Ind}(\mathcal V)\) of ind-objects. The group \(\pi(\mathcal V)\) acts canonically on every object of \(\operatorname{Ind}(\mathcal V)\). A \(\mathcal V\)-group is a pair \((G,\phi)\) with \(\phi\colon\pi(\mathcal V)\to G\) a homomorphism such that the coaction of \(\pi(\mathcal V)\) on \(\mathcal O(G)\) by conjugation through \(\phi\) is the canonical one, and \(\operatorname{Rep}(G,\phi)\) is the category of representations of \(G\) in \(\mathcal V\) whose restriction along \(\phi\) is the canonical action. For \(\mathcal V=\operatorname{Vec}_k\) the group \(\pi(\mathcal V)\) is trivial and Theorem 1.2(2) is the reconstruction theorem of the lesson Tannakian categories.

2. Incompressible categories

Definition 2.1. A finite symmetric tensor category \(\mathcal C\) is incompressible if every exact symmetric tensor functor from \(\mathcal C\) to a symmetric tensor category over \(k\) is fully faithful [CEOi, Definition 3.2.1].

Corollary 2.2 ([OpenAI-208, Corollary 8.1]). A finite symmetric tensor category over \(k\) is incompressible if and only if it is equivalent, as a symmetric tensor category, to one of the following:

Proof. Let \(\mathcal C\) be incompressible, and choose a fibre functor \(F\colon\mathcal C\to\operatorname{Ver}_{p^n}\) by Theorem 8.1 of Lesson 8. Let \(\mathcal I\) be the full subcategory of subquotients of the objects \(F(X)\). By Theorem 1.1(3) it is a finite tensor subcategory, and the corestriction \(F\colon\mathcal C\to\mathcal I\) has the property that every object of \(\mathcal I\) is a subquotient of an object in its image. By incompressibility the corestriction is fully faithful. Theorem 1.1(4) then gives both \(\operatorname{FPdim}(\mathcal C)\le\operatorname{FPdim}(\mathcal I)\) and \(\operatorname{FPdim}(\mathcal C)\ge\operatorname{FPdim}(\mathcal I)\), so equality holds and the corestriction is an equivalence. By Theorem 1.1(1), \(\mathcal I\) is \(\operatorname{Ver}_{p^m}\) or \(\operatorname{Ver}_{p^m}^+\) for some \(m\le n\). With the low levels listed in Section 1, these are exactly the displayed categories: for \(p=2\), \(\operatorname{Ver}_2=\operatorname{Vec}_k\) and the even part exists only for \(m\ge2\); for \(p=3\), the case \(m=0\) repeats \(\operatorname{Ver}_3=\operatorname{sVec}_k\) and \(\operatorname{Ver}_3^+=\operatorname{Vec}_k\); for \(p\ge5\), the case \(m=0\) gives \(\operatorname{sVec}_k\) and \(\operatorname{Vec}_k\).

Conversely, \(\operatorname{Ver}_{p^m}\) and \(\operatorname{Ver}_{p^m}^+\) are incompressible by Theorem 1.1(2). For \(\operatorname{Vec}_k\), an exact symmetric tensor functor sends \(k\) to \(\mathbf 1\) and is bijective on \(\operatorname{Hom}(k,k)=k\to\operatorname{End}(\mathbf 1)=k\), hence on morphisms between finite direct sums of \(k\). For \(\operatorname{sVec}_k\) with \(p\) odd, a symmetric tensor functor sends the odd line \(\Pi\) to an invertible object whose self-symmetry is \(-1\). It is not isomorphic to \(\mathbf 1\), whose self-symmetry is \(1\), so the two simple objects go to non-isomorphic simple objects and the functor is fully faithful on all finite direct sums of them. \(\square\)

Every finite symmetric tensor category has moderate growth: the lengths of the tensor powers of each object grow at most exponentially, by Lemma 3.1 of Lesson 6. So Corollary 2.2 is the finite case of the conjecture of Coulembier, Etingof and Ostrik that every incompressible category of moderate growth is a tensor subcategory of \(\operatorname{Ver}_{p^\infty}\) [CEOi, Conjecture B].

3. Uniqueness in characteristic two

Corollary 3.1 ([OpenAI-208, Corollary 8.2]). Let \(p=2\) and let \(\mathcal C\) be a finite symmetric tensor category.

  1. There is, up to monoidal natural isomorphism, exactly one exact symmetric tensor functor \(\mathcal C\to\operatorname{Ver}_{2^\infty}\), and it factors through \(\operatorname{Ver}_{2^n}\) for some finite \(n\).
  2. For each \(n\ge1\) there is at most one fibre functor \(\mathcal C\to\operatorname{Ver}_{2^n}\), up to monoidal natural isomorphism.

Proof. (1) Theorem 8.1 of Lesson 8 gives a fibre functor to some \(\operatorname{Ver}_{2^n}\), and composing with the inclusion of the tower gives a functor to the union. Uniqueness is Theorem 1.2(1) applied to \(\mathcal C\).

(2) Let \(F_1,F_2\colon\mathcal C\to\operatorname{Ver}_{2^n}\) be fibre functors and \(\iota\) the inclusion of \(\operatorname{Ver}_{2^n}\) into the union, which is fully faithful. By (1) there is a monoidal natural isomorphism \(\iota F_1\cong\iota F_2\). Its components are morphisms between objects of \(\operatorname{Ver}_{2^n}\), so by fullness they are morphisms of \(\operatorname{Ver}_{2^n}\), and naturality and compatibility with the tensor structures hold there because \(\iota\) is faithful. \(\square\)

Theorem 8.1 provides some level \(n\), depending on \(\mathcal C\); it does not provide a fibre functor at every prescribed level.

4. Reconstruction over a higher Verlinde category

Corollary 4.1 ([OpenAI-208, Corollary 8.3]). Let \(\mathcal C\) be a finite symmetric tensor category, and \(F\colon\mathcal C\to\mathcal V=\operatorname{Ver}_{p^n}(k)\) a fibre functor given by Theorem 8.1 of Lesson 8. With \(G=\underline{\operatorname{Aut}}^{\otimes}(F)\) and \(\phi\colon\pi(\mathcal V)\to G\) the canonical homomorphism, \((G,\phi)\) is a \(\mathcal V\)-group, and there is a symmetric tensor equivalence \[ E\colon\mathcal C\xrightarrow{\ \sim\ }\operatorname{Rep}(G,\phi),\qquad\omega\circ E\cong F . \]

Proof. The categories \(\mathcal C\) and \(\mathcal V\) are finite symmetric tensor categories, with simple units by Lemma 1.3 of Lesson 1, and \(F\) is an exact symmetric tensor functor. Theorem 1.2(2) applies. \(\square\)

The group scheme \(G\) is affine; nothing is asserted about it being of finite type.

5. Exercises

5.1. Show that \(\operatorname{Rep}_k(\mathbb Z/p)\) is not incompressible, by exhibiting an exact symmetric tensor functor out of it that is not fully faithful.

5.2. Let \(p\ge5\). Show that \(\operatorname{Vec}_k\), \(\operatorname{sVec}_k\), \(\operatorname{Ver}_p^+\) and \(\operatorname{Ver}_p\) are pairwise non-equivalent, using the numbers of simple objects and Frobenius–Perron dimensions.

5.3. Let \(\mathcal C=\operatorname{Rep}_k(G)\) for a finite group \(G\) and \(F\) the forgetful functor to \(\operatorname{Vec}_k\). Identify the \(\operatorname{Vec}_k\)-group of Corollary 4.1.

5.4. In the proof of Corollary 3.1(2), explain why the components of a natural isomorphism \(\iota F_1\cong\iota F_2\) are isomorphisms in \(\operatorname{Ver}_{2^n}\) and not merely in the union.

6. Solutions

5.1. The forgetful functor \(\operatorname{Rep}_k(\mathbb Z/p)\to\operatorname{Vec}_k\) is an exact symmetric tensor functor. It is not full: \(\operatorname{Hom}_{\mathbb Z/p}(k,J_2)\) is one-dimensional, while the linear maps \(k\to k^2\) form a two-dimensional space.

5.2. The numbers of simple objects are \(1\), \(2\), \((p-1)/2\) and \(p-1\). For \(p\ge7\) these are four different numbers. For \(p=5\) the categories \(\operatorname{sVec}_k\) and \(\operatorname{Ver}_5^+\) both have two simple objects, but \(\operatorname{Ver}_5^+\) contains \(L_3\) of dimension \(\sin(3\pi/5)/\sin(\pi/5)=(1+\sqrt5)/2\), while every object of \(\operatorname{sVec}_k\) has integer dimension; an equivalence preserves Frobenius–Perron dimensions (Lemma 3.2(3) of Lesson 1).

5.3. Here \(\pi(\operatorname{Vec}_k)\) is trivial and \(\underline{\operatorname{Aut}}^{\otimes}(F)\) is the constant group scheme \(G\): the tensor automorphisms of the forgetful functor are given by the elements of \(G\), by the reconstruction theorem of the lesson Tannakian categories. The equivalence \(E\) is the identity of \(\operatorname{Rep}_k(G)\).

5.4. The inclusion \(\iota\) is fully faithful. A component \(\eta_X\colon\iota F_1(X)\to\iota F_2(X)\) is a morphism between objects in the image of \(\iota\), so it is \(\iota\) of a unique morphism of \(\operatorname{Ver}_{2^n}\); the inverse of \(\eta_X\) likewise, and the two composites are identities because \(\iota\) is faithful.

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