Limit Categories: From Deligne to Stable Envelopes

• Limit Categories are abelian tensor categories built as the universal abelian envelopes of the Deligne category Rep(GL_t) for integral t, capturing stable supergroup representation theory.
• Their construction uses stabilization of finite tensor subcategories (Rep^k(gl(m|n))) via the Duflo–Serganova homology functor to establish equivalences in a stable range.
• The embedding of D_t into V_t ensures a universal property that classifies dualizable t-dimensional objects, distinguishing faithful from Schur-annihilated cases and recovering Deligne’s envelope.

For each integer $t$, the paper constructs a tensor category $V_t$ that plays the role of an abelian envelope for the Deligne category $D_t=\operatorname{Rep}(GL_t)$. The point of departure is that $D_t$ is the Karoubian rigid symmetric monoidal category generated by one dualizable object $X_t$ of categorical dimension $t$; when $t\notin \mathbb Z$, $D_t$ is semisimple abelian, but when $t\in\mathbb Z$, it is only Karoubian and not abelian. The category $V_t$ is constructed from the stable behavior of the categories $V_t$0 with fixed superdimension $V_t$1, and it is characterized by a universal property for dualizable $V_t$2-dimensional objects not annihilated by any Schur functor (Entova-Aizenbud et al., 2015).

1. The Deligne category at integral parameter

The Deligne category $V_t$3 is the universal Karoubian rigid symmetric monoidal category generated by a dualizable object of dimension $V_t$4. For $V_t$5, this already has the expected abelian and semisimple behavior. For $V_t$6, however, $V_t$7 is not abelian, so it cannot by itself serve as the target for the usual Tannakian-style classification of tensor-functorial realizations of a $V_t$8-dimensional object.

The category $V_t$9 is introduced precisely to repair that failure. It is an abelian tensor category equipped with a fully faithful symmetric monoidal embedding

$D_t=\operatorname{Rep}(GL_t)$0

and it is universal among faithful symmetric monoidal realizations of $D_t=\operatorname{Rep}(GL_t)$1. In this sense, $D_t=\operatorname{Rep}(GL_t)$2 is the “abelian envelope” of the Deligne category.

The paper’s conceptual claim is not merely that $D_t=\operatorname{Rep}(GL_t)$3 is an abstract enlargement of $D_t=\operatorname{Rep}(GL_t)$4. Rather, it is built so as to retain the stable tensor-combinatorial information common to the classical categories $D_t=\operatorname{Rep}(GL_t)$5 with $D_t=\operatorname{Rep}(GL_t)$6, while discarding the Schur-vanishing relations that are specific to finite superdimension.

2. Construction from stable finite tensor-degree pieces

The construction begins with the tensor categories $D_t=\operatorname{Rep}(GL_t)$7 for pairs $D_t=\operatorname{Rep}(GL_t)$8 satisfying $D_t=\operatorname{Rep}(GL_t)$9. Inside each such category, the paper considers the filtration by full abelian subcategories

$D_t$0

where $D_t$1 consists of subquotients of finite direct sums of mixed tensor powers

$D_t$2

with $D_t$3 the standard representation.

The stabilization mechanism is the Duflo–Serganova homology functor. For a rank-one odd nilpotent $D_t$4, one has

$D_t$5

sending a module $D_t$6 to $D_t$7. Theorem 7.1.1 proves that if

$D_t$8

then

$D_t$9

is an equivalence. Thus, for fixed $X_t$0, the categories $X_t$1 become canonically identified once $X_t$2 are sufficiently large with difference $X_t$3 (Entova-Aizenbud et al., 2015).

This stable range allows the definition

$X_t$4

and then

$X_t$5

The paper states explicitly that “$X_t$6 should be seen as an inverse limit of the system $X_t$7,” but the construction is carried out through the stabilized finite stages $X_t$8. The tensor structure on $X_t$9 is induced from tensor products on the finite-rank categories, and $t$0 contains a distinguished object, also denoted $t$1, obtained as the inverse limit of the standard modules $t$2, with

$t$3

3. The embedding $t$4 and the abelian-envelope theorem

Because $t$5 is generated by one dualizable object $t$6 of dimension $t$7, the distinguished object $t$8 determines a canonical symmetric monoidal functor

$t$9

Proposition 8.1.2 proves that $t\notin \mathbb Z$0 is fully faithful. This gives the basic structural relation: $t\notin \mathbb Z$1 sits inside $t\notin \mathbb Z$2 as a full rigid symmetric monoidal subcategory.

The formal abelian-envelope statement is given by Theorems 9.2.1 and 9.2.2. For any tensor category $t\notin \mathbb Z$3, composition with $t\notin \mathbb Z$4 induces an equivalence

$t\notin \mathbb Z$5

where the left-hand side is the category of exact symmetric monoidal $t\notin \mathbb Z$6-linear functors $t\notin \mathbb Z$7, and the right-hand side is the category of faithful symmetric monoidal $t\notin \mathbb Z$8-linear functors $t\notin \mathbb Z$9. Equivalently, $D_t$0 is an initial object in the relevant $D_t$1-category of faithful symmetric monoidal functors out of $D_t$2 (Entova-Aizenbud et al., 2015).

The proof is categorical but rests on two representation-theoretic inputs extracted from the construction. Proposition 8.4.1 shows that every object $D_t$3 admits a presentation

$D_t$4

with $D_t$5 in the image of $D_t$6. Proposition 8.6.1 states that for any epimorphism

$D_t$7

in $D_t$8, there exists a nonzero $D_t$9 such that

$t\in\mathbb Z$0

splits; a similar statement holds for monomorphisms. Together with the full faithfulness of $t\in\mathbb Z$1, these properties give the formal hypotheses needed for the universality theorem.

4. Schur-generic objects and the dichotomy with $t\in\mathbb Z$2

The paper’s classification theorem is formulated in terms of Schur functors. If $t\in\mathbb Z$3 is a dualizable object of integral dimension $t\in\mathbb Z$4 in a tensor category $t\in\mathbb Z$5, there is a canonical symmetric monoidal functor

$t\in\mathbb Z$6

Lemma 11.1.1 proves that $t\in\mathbb Z$7 is faithful if and only if $t\in\mathbb Z$8 is not annihilated by any Schur functor; explicitly, for every partition $t\in\mathbb Z$9, one must have

$V_t$0

This yields the first branch of the classification: if $V_t$1 is not annihilated by any Schur functor, then $V_t$2 uniquely factors through

$V_t$3

and extends to an exact symmetric monoidal functor

$V_t$4

The second branch is the supergroup case. Theorem 11.1.2 states that if $V_t$5 is annihilated by some Schur functor, then there exists a unique pair $V_t$6 with $V_t$7 such that $V_t$8 factors through

$V_t$9

and gives an exact symmetric monoidal functor

$V_t$00

sending the standard representation $V_t$01 (Entova-Aizenbud et al., 2015).

Accordingly, every symmetric monoidal realization of the Deligne generator of integral dimension $V_t$02 lands either in the stable envelope $V_t$03 or in one of the finite supergroup representation categories $V_t$04 with $V_t$05. The paper describes this as the “prime spectrum” of $V_t$06: $V_t$07 corresponds to the faithful realization, while the $V_t$08 yield an infinite descending chain of proper kernels.

5. Internal representation theory of $V_t$09

The category $V_t$10 is not only universal; it also has a detailed stable highest-weight structure. Simple objects are indexed by all bipartitions $V_t$11, extending the stable parametrization visible in large $V_t$12. For each $V_t$13, the finite stage $V_t$14 is a highest weight category with duality whose standard objects are $V_t$15 with $V_t$16. Projective objects exist in each $V_t$17, although $V_t$18 itself has no projectives or injectives globally.

Block decomposition is described using infinite weight diagrams $V_t$19. Lemma 8.3.3 gives

$V_t$20

with

$V_t$21

This is the stable analogue of block theory for the finite supergroup categories.

The paper also constructs translation functors

$V_t$22

and proves compatibility with specialization: $V_t$23 This is one of the main ways the paper shows that $V_t$24 captures stable block-theoretic and highest-weight behavior.

A related ingredient is the exact symmetric monoidal functor

$V_t$25

from the infinite-rank category for $V_t$26. Lemma 8.1.4 states that $V_t$27 sends the simple objects $V_t$28 of $V_t$29 to the standard objects $V_t$30 of $V_t$31. The specialization functors

$V_t$32

satisfy

$V_t$33

and Proposition 8.4.1 implies that these $V_t$34 are full (Entova-Aizenbud et al., 2015).

6. Deligne’s proposed envelope and the limit interpretation

The paper also identifies $V_t$35 with the categories proposed by Deligne as candidate abelian envelopes. If

$V_t$36

and

$V_t$37

then Corollary 11.3.2 gives a canonical equivalence

$V_t$38

carrying $V_t$39 to $V_t$40. Thus the category obtained by stabilization of $V_t$41 agrees with Deligne’s abstractly proposed envelope.

From the perspective of stable representation theory, $V_t$42 is the canonical abelian tensor category interpolating the representation categories $V_t$43 with fixed $V_t$44. The paper is explicit that it is not a naive inverse limit of the whole categories, because the homology functors are not exact globally. Rather, it is built from the stable inverse limits of the bounded tensor-degree subcategories $V_t$45, where exactness and equivalence do hold.

This construction solves the precise defect of $V_t$46 at integral parameter. The Deligne category remains the universal rigid Karoubian category generated by a $V_t$47-dimensional dualizable object, but $V_t$48 is the universal abelian tensor category that retains the stable information common to sufficiently large $V_t$49 while classifying exactly the Schur-generic realizations of dimension $V_t$50 (Entova-Aizenbud et al., 2015).