NOTES ON RESTRICTED INVERSE LIMITS OF CATEGORIES

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1 NOTES ON RESTRICTED INVERSE LIMITS OF CATEGORIES INNA ENTOVA AIZENBUD Abstract. We describe the framework for the notion of a restricted inverse limit of categories, with the main motivating example being the category of polynomial representations of the group GL = n 0 GL n. This category is also known as the category of strict polynomial functors of finite degree, and it is the restricted inverse limit of the categories of polynomial representations of GL n, n 0. This note is meant to serve as a reference for future work. 1. Introduction In this note, we discuss the notion of an inverse limit of an inverse sequence of categories and functors. Given a system of categories C i (with i running through the set Z + ) and functors F i 1,i : C i C i 1 for each i 1, we define the inverse limit category lim C i to be the following category: The objects are pairs ({C i } i Z+, {φ i 1,i } i 1 ) where C i C i for each i Z + and φ i 1,i : F i 1,i (C i ) C i 1 for any i 1. A morphism f between two objects ({C i } i Z+, {φ i 1,i } i 1 ), ({D i } i Z+, {ψ i 1,i } i 1 ) is a set of arrows {f i : C i D i } i Z+ satisfying some compatability conditions. This category is an inverse limit of the system ((C i ) i Z+, (F i 1,i ) i 1 ) in the (2, 1)-category of categories with functors and natural isomorphisms. It is easily seen (see Section 3) that if the original categories C i were pre-additive (resp. additive, abelian), and the functors F i 1,i were linear (resp. additive, exact), then the inverse limit is again pre-additive (resp. additive, abelian). One can also show that if the original categories C i were monoidal (resp. symmetric monoidal, rigid symmetric monoidal) categories, and the functors F i 1,i were, monoidal (resp. symmetric monoidal functors), then the inverse limit is again a monoidal (resp. symmetric monoidal, rigid symmetric monoidal) category Motivating example: rings. We now consider the motivating example. First of all, consider the inverse system of rings of symmetric polynomials... Z[x 1,..., x n ] Sn Z[x 1,..., x n 1 ] S n 1... Z[x 1 ] Z with the homomorphisms given by p(x 1,..., x n ) p(x 1,..., x n 1, 0). We also consider the ring Λ Z of symmetric functions in infinitely many variables. This ring is defined as follows: first, consider the ring Z[x 1, x 2,...] n 0S n of all power series with integer coefficients in infinitely many indeterminates x 1, x 2,... which are invariant under any permutation of indeterminates. The ring Λ Z is defined to be the subring of all the power series such that the degrees of all its monomials are bounded. We would like to describe the ring Λ Z as an inverse limit of the former inverse system. Date: April 3,

2 1-st approach: The following construction is described in [4, Chapter I]. Take the inverse limit lim Z[x 1,..., x n ] Sn (this is, of course, a ring, isomorphic to Z[x 1, x 2,...] n 0S n ), n 0 and consider only those elements (p n ) n 0 for which deg(p n ) is a bounded sequence. These elements form a subring of lim Z[x 1,..., x n ] Sn which is isomorphic to the n 0 ring of symmetric functions in infinitely many variables. 2-nd approach: Note that the notion of degree gives a Z + -grading on each ring Z[x 1,..., x n ] Sn, and on the ring Λ Z. The morphisms Z[x 1,..., x n ] Sn Z[x 1,..., x n 1 ] S n 1 respect this grading; furthermore, they do not send to zero any polynomial of degree n 1 or less, so they define an isomorphism between the i-th grades of Z[x 1,..., x n ] Sn and Z[x 1,..., x n 1 ] S n 1 for any i < n. One can then see that Λ Z is an inverse limit of the rings Z[x 1,..., x n ] Sn in the category of Z + -graded rings, and its n-th grade is isomorphic to the n-th grade of Z[x 1,..., x n ] Sn Motivating example: categories. We now move on to the categorical version of the same result. Let GL n (C) (denoted by GL n for short) be the general linear group over C. We have an inclusion GL n GL n+1 with the matrix A GL n corresponding to a block matrix A GL n+1 which has A as the upper left n n-block, and 1 in the lower right corner (the rest of the entries are zero). One can consider a similar inclusion of Lie algebras gl n gl n+1. Next, we consider the polynomial representations of the algebraic group GL n (alternatively, the Lie algebra gl n ): these are the representations ρ : GL n Aut(V ) which can be extended to an algebraic map Mat n n (C) End(V ). These representations are direct summands of finite sums of tensor powers of the tautological representation C n of GL n. The category of polynomial representations of GL n, denoted by Rep(gl n ) poly, is a semisimple symmetric monoidal category, with simple objects indexed by integer partitions with at most n parts. The Grothendieck ring of this category is isomorphic to Z[x 1,..., x n ] Sn. We also have functors Res n 1,n = ( ) En,n : Rep(gl n ) poly Rep(gl n 1 ) poly On the Grothendieck rings, these functors induce the homomorphisms Z[x 1,..., x n ] Sn Z[x 1,..., x n 1 ] S n 1 p(x 1,..., x n ) p(x 1,..., x n 1, 0) discussed above. Finally, we consider the infinite-dimensional group GL = n 0 GL n, and its Lie algebra gl = n 0 gl n. The category of polynomial representations of this group (resp. Lie algebra) is denoted by Rep(gl ) poly, and it is the free Karoubian symmetric monoidal category generated by one object (the tautological representation C of GL ). It is also known that this category is equivalent to the category of strict polynomial functors of finite degree (c.f. [3]), it is semisimple, and its Grothendieck ring is isomorphic to the ring Λ Z. The category Rep(gl ) poly possesses symmetric monoidal functors Γ n : Rep(gl ) poly Rep(gl n ) poly with the tautological representation of gl being sent to tautological representation of gl n. These functors are compatible with the functors Res n 1,n (i.e. Γ n 1 = Resn 1,n Γ n ), and the functor Γ n induces the homomorphism Λ Z Z[x 1,..., x n ] Sn p(x 1,..., x n, x n+1,...) p(x 1,..., x n, 0, 0,...) 2

3 This gives us a fully faithful functor Γ lim : Rep(gl ) poly lim n 0 Rep(gl n ) poly. Finding a description of the image of the functor Γ lim inspires the following two frameworks for special inverse limits, which turn out to be useful in other cases as well Restricted inverse limit of categories. To define the restricted inverse limit, we work with categories C i which are finite-length categories; namely, abelian categories where each object has a (finite) Jordan-Holder filtration. We require that the functors F i 1,i be shortening : this means that these are exact functors such that given an object C C i, we have l Ci 1 (F i 1,i (C)) l Ci (C) In that case, it makes sense to consider the full subcategory of lim C i whose objects are of the form ({C i } i Z+, {φ i 1,i } i 1 ), with {l Cn (C n )} n 0 being a bounded sequence (the condition on the functors implies that this sequence is weakly increasing). This subcategory will be called the restricted inverse limit of categories C i and will be denoted by lim C, restr i. It is the inverse limit of the categories C i in the (2, 1)-category of finite-length categories and shortening functors. Considering the restricted inverse limit of the categories Rep(gl n ) poly, we obtain a functor Γ lim : Rep(gl ) poly lim n 0, restr Rep(gl n ) poly It is easy to see that Γ lim is an equivalence. Note that in terms of Grothendieck rings, this construction corresponds to the first approach described in Subsection Inverse limit of categories with filtrations. Another construction of the inverse limit is as follows: let K be a filtered poset, and assume that our categories C i have a K-filtration on objects; that is, we assume that for each k K, there is a full subcategory F il k (C i ), and the functors F i 1,i respect this filtration (note that if we consider abelian categories and exact functors, we should require that the subcategories be Serre subcategories). We can then define a full subcategory lim C,K filtr i of lim C i whose objects are of the form ({C i } i Z+, {φ i 1,i } i 1 ) such that there exists k K for which C i F il k (C i ) for any i 0. The category lim C,K filtr i is automatically a category with a K-filtration on objects. It is the inverse limit of the categories C i in the (2, 1)-category of categories with K- filtrations on objects, and functors respecting these filtrations. Remark A more general way to describe this setting would be the following. Assume that for each i, the category C i is a direct limit of a system ( ( )) (Ci k ) k Z+, G k 1,k i : C k 1 i Ci k Furthermore, assume that the functors F i 1,i induce functors Fi 1,i k : Ci 1 k Ci k for any k Z +, and that the latter are compatible with the functors G k 1,k i. One can then define the category lim lim C i k k K i Z + which will be the directed inverse limit of the system. When Ci k := F il k (C i ) and G k 1,k i are inclusion functors, the directed inverse limit coincides with lim C,K filtr i. All the statements in this note concerning inverse limits of categories with filtrations can be translated to the language of directed inverse limits. 3

4 Considering appropriate Z + -filtrations on the objects of the categories Rep(gl n ) poly, we obtain a functor Γ lim : Rep(gl ) poly lim Rep(gl n ) poly n 0,Z + filtr One can show that this is an equivalence. Note that in terms of Grothendieck rings, this construction corresponds to the second approach described in Subsection 1.1 (in fact, in this particular case one can use a grading instead of a filtration; however, this is not the case in [2]). These two special inverse limits may coincide, as it happens in the case of the categories Rep(gl n ) poly, and in [2]. We give a sufficient condition for this to happen. In such case, each approach has its own advantages. The restricted inverse limit approach does not involve defining additional structures on the categories, and shows that the constructed inverse limit category does not depend on the choice of filtration, as long as the filtration satisfies some relatively mild conditions. Yet the object-filtered inverse limit approach is sometimes more convenient to work with, as it happens in [2]. 2. Conventions Let C be an abelian category, and C be an object of C. A Jordan-Holder filtration for C is a finite sequence of subobjects of C 0 = C 0 C 1... C n = C such that each subquotient C i+1 /C i is simple. The Jordan-Holder filtration might not be unique, but the simple factors C i+1 /C i are unique (up to re-ordering and isomorphisms). Consider the multiset of the simple factors: each simple factor is considered as an isomorphism class of simple objects, and its multiplicity is the multiplicity of its isomorphism class in the Jordan-Holder filtration of C. This multiset is denoted by JH(C), and its elements are called the Jordan-Holder components of C. The length of the object C, denoted by l C (C), is defined to be the size of the finite multiset JH(C). Definition An abelian category C is called a finite-length category if every object admits a Jordan-Holder filtration. 3. Inverse limit of categories In this section we discuss the notion of an inverse limit of categories, based on [8, Definition 1], [6, Section 5]. This is the inverse limit in the (2, 1)-category of categories with functors and natural isomorphisms Inverse limit of categories. Consider the partially ordered set (Z +, ). We consider the following data ( system ): (1) Categories C i for each i Z +. (2) Functors F i 1,i : C i C i 1 for each i 1. Definition Given the above data, we define the inverse limit category lim C i to be the following category: The objects are pairs ({C i } i Z+, {φ i 1,i } i 1 ) where C i C i for each i Z + and φ i 1,i : F i 1,i (C i ) C i 1 for any i 1. 4

5 A morphism f between two objects ({C i } i Z+, {φ i 1,i } i 1 ), ({D i } i Z+, {ψ i 1,i } i 1 ) is a set of arrows {f i : C i D i } i Z+ such that for any i 1, the following diagram is commutative: F i 1,i (C i ) F i 1,i (f i ) F i 1,i (D i ) φ i 1,i Ci 1 f i 1 ψ i 1,i Di 1 Composition of morphisms is component-wise. The definition of lim C i implies that for each i Z +, we can define functors Pr i : lim i Z + C i C i C = ({C i } i Z+, {φ i 1,i } i 1 )) C i f = {f i : C i D i } i Z+ f i which satisfy the following property (this property follows directly from the definition of lim i Z + C i ): Lemma For any i 1, F i 1,i Pr i = Pri 1, with a natural isomorphism given by: (here C = ({C i } i Z+, {φ i 1,i } i 1 ))). (F i 1,i Pr i )(C) φ i 1,i Pr i 1 (C) Let A be a category, together with a set of functors G i : A C i which satisfy: for any i 1, there exists a natural isomorphism η i 1,i : F i 1,i G i G i 1 Then lim C i is universal among such categories; that is, we have a functor G : A lim C i i Z + A ({G i (A)} i Z+, {η i 1,i } i 1 ) (f : A 1 A 2 ) {f i := G i (f)} i Z+ and G i = Pri G for every i Z +. Finally, we give the following simple lemma: Lemma Let N Z +, and assume that for any i N, F i 1,i is an equivalence. Then Pr i : lim j Z+ C j C i is an equivalence for any i N. Proof. Set F ij := F i,i+1... F j 1,j for any i j (in particular, F ii := Id Ci ). Fix i N. Let j i; then F ij is an equivalence, i.e. we can find a functor G j : C i C j such that F ij G j = IdCi, and G j F ij = IdCj (for j := i, we put G i := Id Ci ). For any j > i, fix natural transformations η j 1,j : F j 1,j G j For any j i, put G j := F ji, and η j 1,j := Id. 5 G j 1

6 Then the universal property of lim j Z+ C j implies that there exists a functor G : C i lim C j j Z + such that Pr j G = G j for any j. The functor G is given by G : C i lim C j j Z + C ({G j (C)} j Z+, {η j 1,j } j 1 ) f : C C {f j := G j (f)} j Z+ In particular, we have: Pr i G = Id Ci. It remains to show that G Pr i = Idlim j Z + C j, and this will prove that Pr i is an equivalence of categories. For any C lim j Z+ C j, C := ({C j } j Z+, {φ j 1,j } j 1 ), and for any l j we define isomorphisms φ lj : F lj (C j ) C l given by φ lj := φ l,l+1 F l,l+1 (φ l+1,l+2 F l+1,l+2 (φ l+2,l+3... F j 2,j 1 (φ j 1,j )...)) Define θ(c) := {θ(c) j : C j Pr j (G(C i )) = G j (C i )} j I by setting { φ 1 ji if j i θ(c) j := G j (φ ij ) if j > i Now, let C := ({C j } j Z+, {φ j 1,j } j 1 ), D := ({D j } j Z+, {ψ j 1,j } j 1 ) be objects in lim C j, together with a morphism f : C D, f := {f j : C j D j } j Z+. j Z + Then the diagram f C D θ(c) is commutative, since for j i, the diagrams φ 1 ji (G Pr i )(C) (G Pr i )(f) θ(d) (G Pr i )(D) C j Pr j (G(C i )) = G j (C i ) f j G j (f i ) D j ψ 1 ji Pr j (G(D i )) = G j (D i ) are commutative, and for j > i, the diagrams C j G j (φ ij ) Pr j (G(C i )) = G j (C i ) f j G j (f i ) D j G j (ψ ij ) Pr j (G(D i )) = G j (D i ) are commutative. 6

7 3.2. Inverse limit of pre-additive, additive and abelian categories. In this subsection, we give some more or less trivial properties of the inverse limit corresponding to the system ((C i ) i Z+, (F i 1,i ) i 1 ) depending on the properties of the categories C i and the functors F i 1,i. Lemma Assume the categories C i are C-linear pre-additive categories (i.e. the Hom-spaces in each C i are complex vector spaces), and the functors F i 1,i are C-linear. Then the category lim C i is automatically a C-linear pre-additive category: given f, g : C D in lim C i, where C = ({C i } i Z+, {φ i 1,i } i 1 ), D = ({D i } i Z+, {ψ i 1,i } i 1 ), f = {f i : C i D i } i Z+, g = {g i : C i D i } i Z+, we have: where α, β C. The functors Pr i are then C-linear. αf + βg := {(αf i + βg i ) : C i D i } i Z+ Lemma Assume the categories C i are additive categories (i.e. each C i is preadditive and has biproducts), and the functors F i 1,i are additive. Then the category lim C i is automatically a additive category: i Z + The zero object in lim C i is ({0 Ci } i Z+, {0} i 1 ). Given C, D in lim C i, where C = ({C i } i Z+, {φ i 1,i } i 1 ), D = ({D i } i Z+, {ψ i 1,i } i 1 ), we have: C D := ({(C i D i } i Z+, {φ i 1,i ψ i 1,i } i 1 ) with obvious inclusion and projection maps. The functors Pr i are then additive. Proof. Let X, Y lim C i, X = ({X i } i Z+, {µ i 1,i } i 1 ), Y = ({Y i } i Z+, {ρ i 1,i } i 1 ), and let f C : X C, f D : X D, g C : C Y, g D : D Y (we denote the components of the map f C by f Ci, of the map f D by f Di, etc.). Denote by ι Ci, ι Di, π Ci, π Di the inclusion and projection maps between C i, D i and C i D i. By definition, ι C := {ι Ci } i Z+, ι D := {ι Di } i Z+, π C := {π Ci } i Z+, π D := {π Di } i Z+ are the inclusion and projection maps between C, D and C D. For each i, there exists a unique map f i : X i C i D i and a unique map g i : C i D i Y i such that π Ci f i = f Ci, π Di f i = f Di, g i ι Ci = g Ci, g i ι Di = g Di for any i Z +. This means that we have a unique map f : X C D and a unique map g : C D Y such that π C f = f C, π D f = f D, g ι C = g C, g ι D = g D (these are the maps f = {f i } i, g = {g i } i ). Lemma Let f : C D in lim C i, where C = ({C i } i Z+, {φ i 1,i } i 1 ), D = ({D i } i Z+, {ψ i 1,i } i 1 ), f = {f i : C i D i } i Z+. Assume f i are isomorphisms for each i. Then f is an isomorphism. 7

8 Proof. Let g i := f 1 i for each i Z + (this morphism exists since f i is an isomorphism, and is unique). All we need is to show that g := {g i : D i C i } i is a morphism from D to C in lim C i, i.e. that the following diagram is commutative for any i 1: F i 1,i (C i ) F i 1,i (g i ) φ i 1,i Ci 1 g i 1 F i 1,i (D i ) ψ i 1,i Di 1 The morphism g i 1 ψ i 1,i is inverse to ψ 1 i 1,i f i 1, and φ i 1,i F i 1,i (g j i) is inverse to F i 1,i (f i ) φ 1 i 1,i. But ψ 1 i 1,i f i 1 from C to D in lim = F i 1,i (f i ) φ 1 i 1,i, since f = {f i : C i D i } i Z+ is a morphism C i. The uniqueness of the inverse morphism then implies that g i 1 ψ i 1,i = φ i 1,i F i 1,i (g i ), and we are done. Proposition Assume the categories C i are abelian, and the functors F i 1,i are exact. Then the category lim C i is automatically abelian: Given f : C D in lim C i, where C = ({C i } i Z+, {φ i 1,i } i 1 ), D = ({D i } i Z+, {ψ i 1,i } i 1 ), f = {f i : C i D i } i Z+, f has a kernel and a cokernel: Ker(f) := ({Ker(f i )} i Z+, {ρ i 1,i } i 1 ), Coker(f) := ({Coker(f i )} i Z+, {µ i 1,i } i 1 ) where ρ i 1,i, µ i 1,i are the unique maps making the following diagram commutative: Ker(F i 1,i (f i )) = F i 1,i (Ker(f i )) ρ i 1,i Ker(fi 1 ) F i 1,i (C i ) F ij (f i ) F i 1,i (D i ) φ i 1,i ψ i 1,i Ci 1 f i 1 Di 1 Coker(F i 1,i (f i )) = F i 1,i (Coker(f i )) µ i 1,i Coker(fi 1 ) Given f : C D in lim C i, we have: Im(f) := Ker(Coker(f)) = Coker(Ker(f)) =: Coim(f). Proof. The universal properties of Ker(f), Coker(f) hold automatically, as a consequence of the universal properties of Ker(f i ), Coker(f i ). Now, let f : C D in lim C i, where C = ({C i } i Z+, {φ i 1,i } i 1 ), D = ({D i } i Z+, {ψ i 1,i } i 1 ), f = {f i : C i D i } i Z+. Consider the objects Im(f) := Ker(Coker(f)), Coim(f) := Coker(Ker(f)) in lim C i. We have a canonical map f : Coim(f) Im(f), such that f : C D i Z + is the composition f C Coim(f) Im(f) D Consider the maps f i for each i Z +, where f i is the canonical map such that f i : C i D i is the composition C i Coim(f i ) f i Im(fi ) D i 8

9 One then immediately sees that f = { f i : Coim(f i ) Im(f i )} i. Since the category C i is abelian for each i Z +, the map f i is an isomorphism. Lemma then implies that f is an isomorphism as well. The following is a trivial corollary of the previous proposition: Corollary The functors Pr i are exact. This corollary, in turn, immediately implies the following statement: Corollary Let (C i, F ij ) be a system of pre-additive (respectively, additive, abelian) categories, and linear (respectively, additive, exact) functors. Let A be a pre-additive (respectively, additive, abelian) category, together with a set of linear (respectively, additive, exact) functors G i : A C i which satisfy: for any i 1, there exists a natural isomorphism η i 1,i : F i 1,i G i G i 1 Then lim C i is universal among such categories; that is, we have a linear (respectively, additive, exact) functor and G i = Pri G for every i Z +. G : A lim C i i Z + A ({G i (A)} i Z+, {η i 1,i } i Z+ ) f : A 1 A 2 {f i := G i (f)} i Z+ 4. Restricted inverse limit of finite-length categories 4.1. We consider the case when the categories C i are finite-length. We would like to give a notion of an inverse limit of the system ((C i ) i Z+, (F i 1,i ) i 1 ) which would be a finitelength category as well. In order to do this, we will define the notion of a shortening functor, and define a stable inverse limit of a system of finite-length categories and shortening functors. Definition Let A 1, A 2 be finite-length categories. An exact functor F : A 1 A 2 will be called shortening if for any object A A 1, we have: l A1 (A) l A2 (F(A)) Since F is exact, this is equivalent to requiring that for any simple object L A 1, the object F(L) is either simple or zero. Definition Let ((C i ) i Z+, (F i 1,i ) i 1 ) be a system of finite-length categories and shortening functors. We will denote by lim C, restr i the full subcategory of lim C i whose objects C := ({C j } j Z+, {φ j 1,j } j 1 ) satisfy: the integer sequence {l Ci (C i )} i 0 stabilizes. Note that the since the functors F i 1,i are shortening, the sequence {l Ci (C i )} i 0 is weakly increasing. Therefor, this sequence stabilizes iff it is bounded from above. We now show that lim C, restr i is a finite-length category. Lemma The category C := lim C, restr i is a Serre subcategory of lim C i, and its objects have finite length. Moreover, given an object C := ({C i } i Z+, {φ i 1,i } i 1 ) in C, we have: l C (C) max{l Ci (C i ) i 0} 9

10 Proof. Let C := ({C j } j Z+, {φ j 1,j } j 1 ), C := ({C j} j Z+, {φ j 1,j} j 1 ), C := ({C j } j Z+, {φ j 1,j} j 1 ) be objects in lim C i, together with morphisms f : C C, g : C C such that the sequence 0 C f C g C 0 is exact. If C lies in the subcategory C, then the sequence {l Ci (C i )} i 0 is bounded from above, and stabilizes. Denote its maximum by N. For each i, the sequence 0 C i f i Ci g C i 0 is exact. Therefore, l Ci (C i), l Ci (C i ) N for each i, and thus C, C lie in C as well. Vice versa, assuming C, C lie in C, denote by N, N the maximums of the sequences {l Ci (C i)} i, {l Ci (C i )} i respectively. Then l Ci (C i ) N + N for any i 0, and so C lies in the subcategory C as well. Thus C is a Serre subcategory of lim C i. Next, let C lie in C. We would like to say that C has finite length. Denote by N the maximum of the sequence {l Ci (C i )} i 0. It is easy to see that C has length at most N; indeed, if {C, C,..., C (n) } is a subset of JH C (C), then for some i >> 0, we have: Pr i (C (k) ) 0 for any k = 1, 2,..., n. Pr i (C (k) ) are distinct Jordan Holder components of C i, so n l Ci (C i ) N. In particular, we see that l C (C) N = max{l Ci (C i ) i 0} Notation Denote by Irr(C i ) the set of isomorphism classes of irreducible objects in C i, and define the pointed set Irr (C i ) := Irr(C i ) {0} The shortening functors F i 1,i then define maps of pointed sets f i 1,i : Irr (C i ) Irr (C i 1 ) ( ) Similarly, we define Irr lim C to be the set of isomorphism classes of irreducible objects in C, and define the pointed i Z +, restr set Irr (C) := Irr(C) {0} Let C := ({C j } j Z+, {φ j 1,j } j 1 ) be an object in C. We denote by JH(C j ) the multiset of the Jordan-Holder components of C j, and let JH (C j ) := JH(C j ) {0} The corresponding set lies in Irr (C j ), and we have maps of (pointed) multisets f j 1,j : JH (C j ) JH (C j 1 ) Denote by lim Irr (C i ) the inverse limit of the system ({Irr (C i } i 0, {f i 1,i } i 1 ). We will also denote by pr j : lim Irr (C i ) Irr (C j ) the projection maps. The elements of the set lim Irr (C i ) are just sequences (L i ) i 0 such that L i Irr (C i ), and f i 1,i (L i ) = L i 1. The following lemma describes the simple objects in the category C := lim, restr C i. 10

11 Lemma Let C := ({C j } j Z+, {φ j 1,j } j 1 ) be an object in C := lim, restr C i. Then C Irr (C) Pr j (C) = C j Irr (C j ) j In other words, C is a simple object (that is, C has exactly two distinct subobjects: zero and itself) iff C 0, and for any j 0, the component C j is either a simple object in C j, or zero. Proof. The direction is obvious, so we will only prove the direction. Let n 0 be a position in which the maximum of the weakly-increasing integer sequence {l Ci (C i )} i 0 is obtained. By definition of n 0, for j > n 0, the functors F j 1,j do not kill any Jordan-Holder components of C j. Now, consider the socles of the objects C j for j n 0. For any j > 0, we have: and thus for j > n 0, we have Thus the sequence F j 1,j ( socle(c j ) ) φ j 1,j socle(c j 1 ) l Cj ( socle(c j ) ) = l Cj 1 (F j 1,j ( socle(c j ) )) l Cj 1 ( socle(c j 1 ) ) {l Cj ( socle(c j ) )} j n0 is a weakly decreasing sequence, and stabilizes. Denote its stable value by N. We conclude that there exists n 1 n 0 so that F j 1,j ( socle(c j ) ) φ j 1,j socle(c j 1 ) is an isomorphism for every j > n 1. Now, denote: { F j,n1 ( socle(c n1 ) ) if j < n 1 D j := socle(c j ) if j n 1 and we put: D := ((D j ) j 0, (φ j 1,j ) j 1 ) (this is a subobject of C in the category lim C i ). Of course, l Cj (D j ) N for any j, so D is an object in the full subcategory C of lim C i. i Z + Furthermore, since C 0, we have: for j >> 0, socle(c j ) 0, and thus 0 D C. D is a semisimple object C, with simple summands corresponding to the elements of the inverse limit of the multisets lim JH (D j ). j Z+ We conclude that D = C, and that socle(c j ) = C j has length at most one for any j 0. Remark Note that the latter multiset is equivalent to the inverse limit of multisets JH ( socle(c j ) ), so D is, in fact, the socle of C. Corollary The set of isomorphism classes of simple objects in lim, restr C i is in bijection with the set lim Irr (C i ) \ {0}. That is, we have a natural bijection Irr (C) = lim i Z + Irr (C i ) 11

12 In particular, given an object C := ({C j } j Z+, {φ j 1,j } j 1 ) in lim C, restr i, we have: JH (C) = lim JH (C i ) (an inverse limit of the system of multisets JH (C j ) and maps f j 1,j ). It is now obvious that the projection functors Pr i are shortening as well: Corollary The projection functors Pr i are shortening, and define the maps Lemma and Corollary give us: pr i : Irr (C) Irr (C i ) Corollary Given an object C := ({C i } i Z+, {φ i 1,i } i 1 ) in C, we have: l C (C) = max{l Ci (C i ) i 0} It is now easy to see that the restricted inverse limit has the following universal property: Proposition Let A be a finite-length category, together with a set of shortening functors G i : A C i which satisfy: for any i 1, there exists a natural isomorphism η i 1,i : F i 1,i G i G i 1 Then lim, restr C i is universal among such categories; that is, we have a shortening functor and G i = Pri G for every i Z +. G : A lim C i i Z +, restr A ({G i (A)} i Z+, {η i 1,i } i 1 ) f : A 1 A 2 {f i := G i (f)} i Z+ Proof. Consider the functor G : A lim C i induced by the functors G i. We would like to say that for any A A, the object G(A) lies in the subcategory lim C, restr i, i.e. that the sequence {l Ci (G i (A))} i is bounded from above. Indeed, since G i are shortening functors, we have: l Ci (G i (A)) l A (A). Thus the sequence {l Ci (G i (A))} i is bounded from above by l A (A). Now, using Corollary 4.1.9, we obtain: l C (G(A)) = max i 0 {l C i (G i (A))} l A (A) and we conclude that G is a shortening functor. 5. Inverse limit of categories with a filtration 5.1. We now consider the case when the categories C i have a filtration on the objects (we will call these filtered categories ), and the functors F i 1,i respect this filtration. We will then define a subcategory of the category lim C i which will be denoted by lim C i Z +,K filtr i and will be called the inverse limit of filtered categories C i. Fix a directed partially ordered set (K, ) ( directed, means that for any k 1, k 2 K, there exists k K such that k 1, k 2 k). Definition (K-filtered categories). We say that a category A is a K-filtered category if for each k K we have a full subcategory A k of A, and these subcategories satisfy the following conditions: (1) A k A l whenever k l. 12

13 (2) A is the union of A k, k K: that is, for any A A, there exists k K such that A A k. A functor F : A 1 A 2 between K-filtered categories A 1, A 2 is called a K-filtered functor if for any k K, F(A k 1) is a subcategory of A k 2. Remark Let F : A 1 A 2 be a K-filtered functor between K-filtered categories A 1, A 2. Assume the restriction of F to each filtration component k is an equivalence of categories A k 1 A k 2. Then F is obviously an equivalence of (K-filtered) categories. Remark The definition of a K-filtration on the objects of a category A clearly makes A a direct limit of the subcategories A k. Definition We say that the system ((C i ) i Z+, (F i 1,i ) i 1 ) is K-filtered if for each i Z +, C i is a category with a K-filtration, and the functors F i 1,i are K-filtered functors. Definition Let ((C i ) i Z+, (F i 1,i ) i 1 ) be a K-filtered system. We define the inverse limit of this Z + -filtered system (denoted by lim C,K filtr i) to be the full subcategory of lim C i whose objects C satisfy: there exists k C K such that Pr i (C) C k C i for any i Z +. The following lemma is obvious: Lemma The category lim C,K filtr i is automatically K-filtered: the filtration component F il k (lim C,K filtr i) can be defined to be the full subcategory of lim C i Z +,K filtr i of objects C such that Pr i (C) Ci k for any i Z +. This also makes the functors Pr i : lim C,K filtr i C i K-filtered functors. Remark Note that by definition, for any k K F il k ( lim i Z +,K filtr ) C i = lim C i k i Z + where the inverse limit is taken over the system ((C k i ) i Z+, (F i 1,i C k i ) i 1 ). Thus lim i Z +,K filtr C i := lim k K lim C i k i Z + Lemma Let ((C i ) i Z+, (F i 1,i ) i 1 ) be a K-filtered system. (1) Assume the categories C i are additive, the functors F i 1,i are additive, and for any k K, Ci k is an additive subcategory of C i. Then the category lim C,K filtr i is an additive subcategory of lim C i, and all its filtration components are additive subcategories. (2) Assume the categories C i are abelian, the functors F i 1,i are exact, and for any k K, C k i is a Serre subcategory of C i. Then the category lim,k filtr C i is abelian (and a Serre subcategory of lim C i ), and all its filtration components are Serre subcategories. i Z + Proof. To prove the first part of the statement, we only need to check that F il k (lim C,K filtr i) is an additive subcategory of lim C i. This follows directly from the construction of direct sums in lim C i : let C, D F il k (lim C,K filtr i) lim C i. Then Pr i (C) C k i Z + i, Pr i (D) Ci k for any i Z +. Since Ci k is an additive 13

14 subcategory of C i, we get: Pr i (C D) Ci k taken in lim C i ). Thus lim,k filtr C i for any i Z + (the direct sum C D is is an additive subcategory of lim C i, and all its filtration components are additive subcategories as well. To prove the second part of the statement, it is again enough to check that F il k (lim C,K filtr i) is a Serre subcategory of lim C i. Indeed, let 0 C C C 0 be a short exact sequence in lim C i. We want to show that C F il k (lim C,K filtr i) iff C, C F il k (lim C,K filtr i). The functors Pr i are exact, so the sequence 0 C i C i C i 0 is exact for any i Z +. Since Ci k is a Serre subcategory of C i, we have: C i Ci k iff C i, C i Ci k, and we are done. We now have the following universal property, whose proof is straight-forward: Proposition Let ((C i ) i Z+, (F i 1,i ) i 1 ) be a K-filtered system, and let A be a category with a K-filtration, together with a set of K-filtered functors G i : A C i which satisfy: for any i 1, there exists a natural isomorphism η i 1,i : F i 1,i G i G i 1 Then lim,k filtr C i is universal among such categories; that is, we have a functor G : A lim C i i Z +,K filtr A ({G i (A)} i Z+, {η i 1,i } i 1 ) f : A 1 A 2 {f i := G i (f)} i Z+ which is obviously K-filtered, and satisfies: G i = Pri G for every i Z +. Next, consider the case when A, {G i } i Z+ satisfy the following stabilization condition: Condition For every k K, there exists i k Z + such that G j : A k Cj k is an equivalence of categories for any j i k. In this setting, the following proposition holds: Proposition The functor G : A lim,k filtr C i is an equivalence of (Kfiltered) categories. Proof. To prove that G is an equivalence of (K-filtered) categories, we neeed to show that ( ) G : A k F il k lim C i i Z +,K filtr is an equivalence of categories for any k K. Recall that ( ) F il k lim C i = lim C i k i Z +,K filtr i Z + 14

15 By Condition , for any i > i k we have a commutative diagram where all arrows are equivalences: A k G i Ci k G ik C k i 1 F i 1,i By Lemma 3.1.3, we then have: Pr i : lim C k i Ci k is an equivalence of categories for ) any i > i k, and thus G : A k F il k (lim C i Z +,K filtr i is an equivalence of categories. 6. Restricted inverse limit and inverse limit of categories with a K-filtration 6.1. Let ((C i ) i Z+, (F i 1,i ) i 1 ) be a system of finite-length categories with K-filtrations and shortening K-filtered functors, whose the filtration components are Serre subcategories. We would like to give a sufficient condition on the K-filtration for the inverse limit of K-filtered categories to coincide with the restricted inverse limit of these categories. Recall that since the functors F i 1,i are shortening, we have maps f i 1,i : Irr (C i ) Irr (C i 1 ) and we can consider the inverse limit lim Irr (C i ) of the sequence of sets Irr (C i ) and maps f i 1,i ; we will denote by pr j : lim Irr (C i ) Irr (C j ) the projection maps. Notice that the sets Irr (C i ) have a natural K-filtration, and the maps f i 1,i are K- filtered maps. Proposition Assume the following conditions hold: (1) There exists a K-filtration on the set lim Irr (C i ). That is, we require: For each L in lim Irr (C i ), there exists k K so that pr i (L) F il k (Irr (C i )) for any i 0. We would then say that such an object L belongs in the k-th filtration component of lim Irr (C i ). (2) Stabilization condition : for any k K, there exists N k 0 such that the map f i 1,i : F il k (Irr (C i )) F il k (Irr (C i 1 )) be an injection for any i N k. That is, for any k K there exists N k Z + such that the (exact) functor F i 1,i is faithful for any i N k. Then the two full subcategories lim C, restr i, lim C,K filtr i of lim C i coincide. Proof. Let C := ({C j } j Z+, {φ j 1,j } j 1 ) be an object in lim C, restr i. As before, we denote by JH(C j ) the multiset of Jordan-Holder components of C j, and let JH (C j ) := JH(C j ) {0}. The first condition is natural: giving a K-filtration on the objects of lim C, restr i is equivalent to giving a K-filtration on the simple objects of lim C, restr i, i.e. on the set lim Irr (C i ). i Z + Assume C lim C, restr i. Let n 0 0 be such that l Cj (C j ) is constant for j n 0. Recall that we have (Corollary 4.1.7): JH (C) = lim i Z + JH (C j ) 15

16 Choose k such that all the elements of JH (C) lie in the k-th filtration component of lim Irr (C i ). This is possible due to the first condition. i Z + Then for any L j JH(C j ), we have: L j = pr j (L) for some L JH (C), and thus L j F il k (Irr (C j )). We conclude that C F il k (lim C,K filtr i). Thus we proved that the first condition of the Theorem holds iff lim C, restr i is a full subcategory of lim C,K filtr i. Now, let C lim C,K filtr i, and let k K be such that C F il k (lim C,K filtr i). We would like to show that l Ci (C i ) is constant starting from some i. Indeed, the second condition of the Theorem tells us that there exists N k 0 such that the map f i 1,i : F il k (Irr (C i )) F il k (Irr (C i 1 )) is an injection for any i N k. We claim that for i N k, l Ci (C i ) is constant. Indeed, if it weren t, then there would be some i N k + 1 and some L i JH(C i ) such that f i 1,i (L i ) = 0. But this is impossible, due to the requirement above. 7. gl and the restricted inverse limit of representations of gl n In this section, we give a nice example of a restricted inverse limit of categories; namely, we will show that the category of polynomial representations of the Lie algebra gl is a restricted inverse limit of the categories of polynomial representations of gl n for n 0. The representations of the Lie algebra gl (or the group GL ) are discussed in detail in [5], [1], as well as [7, Section 3] The Lie algebra gl. Let C be a complex vector space with a countable basis {e 1, e 2, e 3,...}. Consider the Lie algebra gl of infinite matrices A = (a ij ) i,j 1 with finitely many non-zero entries. We have a natural action of gl on C, with gl = C C. Here C = span C (e 1, e 2, e 3,...), where e i is the linear functional dual to e i : e i (e j ) = δ ij. We now insert more notation. Let N Z + { }, and let m 1. We will consider the Lie subalgebra gl m gl N consisting of matrices A = (a ij ) 1 i,j N for which a ij = 0 whenever i > m or j > m. We will also denote by gl m the Lie subalgebra of gl N consisting of matrices A = (a ij ) 1 i,j N for which a ij = 0 whenever i m or j m. Remark Note that gl n = gl N m for any N, m Categories of polynomial representations. In this subsection, N Z + { }. We will consider the symmetric monoidal category Rep(gl N ) poly of polynomial representations of gl N. As a tensor category, it is generated by the tautological representation C N of gl N. Namely, this is the category of gl N -modules which are direct summands in finite direct sums of tensor powers of C N, and gl N -equivariant morphisms between them. This category is discussed in detail in [7, Section 2.2]. It is easy to see that this is a semisimple abelian category, whose simple objects are parametrized (up to isomorphism) by all Young diagrams of arbitrary sizes: the simple object corresponding to λ is L N λ = Sλ C N. 16

17 Remark Note that Rep(gl ) poly is the free abelian symmetric monoidal category generated by one object (c.f. [7, (2.2.11)]). It has a equivalent definition as the category of polynomial functors of bounded degree, which can be found in [3], [4, Chapter I], [7]. Remark For N Z +, one can describe these representations as finite-dimensional representations ρ : GL N Aut(W ) which can be extended to an algebraic map End(GL N ) End(W ) Specialization functors. We now define specialization functors from the category of representations of gl to the categories of representations of gl n (c.f. [7, Section 3]): Definition Γ n : Rep(gl ) poly Rep(gl n ) poly, Γ n := ( ) gl n Lemma The functor Γ n is well-defined. Proof. First of all, notice that the subalgebras gl n, gl n gl commute, and therefore the subspace of gl n -invariants of a gl -module automatically carries an action of gl n. We need to check that given a polynomial gl -representation M of gl n, the gl n - invariants of M form a polynomial respresentation of gl n. It is enough to check that this is true when M = (C ) r. The latter statement is checked explicitly on basis elements of the form e i1 e i2... e ir. The subspace of gl n -invariants is spanned by the basis elements e i1 e i2... e ir for which i 1,..., i r n. Thus the gl n -invariants of (C ) r form the gl n -representation (C n ) r. In particular, one proves in the same way that the gl n -invariants of (C ) r form the gl n -representation (C n ) r. The following Lemmas are proved in [5], [7, Section 3]: Lemma The functors Γ n are symmetric monoidal functors. The functors Γ n : Rep(gl ) poly Rep(gl n ) poly are additive functors between semisimple categories, and their effect on simple objects is given by the following Lemma (a direct consequence of Lemma 7.3.3): Lemma For any Young diagram λ, Γ n (S λ C ) = S λ C n Restriction functors. Definition Let n 1. We define the functor Res n 1,n : Rep(gl n ) poly Rep(gl n 1 ) poly, Res n 1,n := ( ) gl n 1 The proof that this functor is well-defined is exactly the same as that of Lemma Remark Here is an alternative definition of the functors Res n 1,n. We say that a gl n -module M is of degree d if Id C n gl n acts by d Id M on M. Also, given any gl n -module M, we may consider the maximal submodule of M of degree d, and denote it by deg d (M). This defines an endo-functor deg d of Rep(gl n ) poly. Note that a simple module S λ C n is of degree λ. The notion of degree gives a decomposition Rep(gl n ) poly = d Z + Rep(gl n ) poly,d 17

18 where Rep(gl n ) poly,d is the full subcategory of Rep(gl n ) poly consisting of all polynomial gl n -modules of degree d. Then Res n 1,n = d Z+ Res d,n 1,n : Rep(gl n ) poly Rep(gl n 1 ) poly where Res d,n 1,n : Rep(gl n ) poly,d Rep(gl n 1 ) poly,d, Res d,n 1,n := deg d Res gl n gl n 1 where Res gl n gl n 1 is the usual restriction functor for the pair gl n 1 gl n. Again, Res n 1,n are additive functors between semisimple categories, so we are interested in checking the effect of these functors on simple modules: Lemma Res n 1,n (S λ C n ) = S λ C n 1 for any Young diagram λ. Proof. This is a simple corollary of the branching rues for gl n, gl n 1. Next, we notice that these functors are compatible with the functors Γ n defined before. Lemma For any n 1, we have a commutative diagram: Γ n Rep(gl ) poly Rep(gl n ) poly Γ n 1 Res n 1,n Rep(gl n 1 ) poly That is, there is a natural isomorphism Γ n 1 = Resn 1,n Γ n. Proof. By definition of the functors Γ n 1, Res n 1,n, Γ n, we have a natural transformation θ : Γ n 1 Res n 1,n Γ n which is given by the injection θ M : Γ n 1 (M) (Res n 1,n Γ n ) (M) for any M Rep(gl ) poly. We would like to say that θ M are isomorphisms. The categories in question are semisimple, so it is enough to check what happens to the simple objects. Lemmas and then tell us that θ S λ C is an isomorphism for any Young diagram λ, and we are done. Lemma The functors Res n 1,n : Rep(gl n ) poly Rep(gl n 1 ) poly are symmetric monoidal functors. Proof. The functor Γ n is full and essentially surjective, as well as a tensor functor. The natural isomorphism from Lemma then provides a monoidal structure on the functor Res n 1,n, and we can immediately see that it is symmetric as well The restricted inverse limit of categories Rep(gl n ) poly. This subsection describes the category Rep(gl ) poly as a stable inverse limit of categories Rep(gl n ) poly. We now define a Z + -filtration on Rep(gl n ) poly for each n Z +. Notation For each k Z +, let Rep(gl n ) poly, length k be the full additive subcategory of Rep(gl n ) poly generated by S λ C n such that l(λ) k. Clearly the subcategories Rep(gl n ) poly, length k give us a Z + -filtration of the category Rep(gl n ) poly, and by Lemma 7.4.3, the functors Res n 1,n are Z + -filtered functors (see Section 5). This allows us to consider the inverse limit lim n Z +,Z + filtr Rep(gl n ) poly 18

19 of Z + -filtered categories Rep(gl n ) poly. This inverse limit is an abelian category with a Z + -filtration (by Lemma 5.1.8). Note that by Lemma 7.4.3, the functors Res n 1,n are shortening functors (see Definition 4.1.1); futhermore, the system ((Rep(gl n ) poly ) n Z+, (Res n 1,n ) n 1 ) satisfies the conditions in Proposition 6.1.1, and therefore the inverse limit of this Z + -filtered system is also its restricted inverse limit (see Section 4). Of course, since the functors Res n 1,n are symmetric monoidal functors, the above restricted inverse limit is a symmetric monoidal category. Proposition We have an equivalence of symmetric monoidal abelian categories Γ lim : Rep(gl ) poly induced by the symmetric monoidal functors lim Rep(gl n ) poly n Z +, restr Γ n = ( ) gl n : Rep(gl ) poly Rep(gl n ) poly Proof. Define a Z + -filtration on the semisimple category Rep(gl ) poly by requiring the simple object S λ C to lie in filtra l(λ). Lemma then tells us that for any k Z + and any n k, the functor Γ n : F il k (Rep(gl ) poly ) F il k (Rep(gl n ) poly ) := Rep(gl n ) poly, length k is an equivalence. Proposition completes the proof. Remark The same result has been proved in [3]; the approach used there is equivalent to that of inverse limits of Z + -filtered categories - namely, the authors give a Z + -grading on the objects of each category Rep(gl n ) poly, with S λ C n lying in grade λ. The stable inverse limit of these graded categories, as defined in [3], is just the inverse limit of the Z + -filtered categories Rep(gl n ) poly with the appropriate filtrations. Note that by Proposition 6.1.1, this construction is equivalent to our construction of a lim Rep(gl n Z +, restr n) poly. In this case, this is also equivalent to taking the compact subobjects inside lim Rep(gl n Z + n ) poly. Remark The adjoint (on both sides) to functor Γ lim is the functor Γ lim : lim Rep(gl n ) poly Rep(gl ) poly n Z +, restr defined below. For any object ((M n ) n 0, (φ n 1,n ) n 1 ) of lim Rep(gl n Z+, restr n) poly, the gl n 1 -module M n 1 is isomorphic (via φ n 1,n ) to a gl n 1 -submodule of M n. This allows us to cosider a vector space M which is the direct limit of the vector spaces M n and the inclusions φ n 1,n. On this vector space M we have a natural action of gl : given A gl n gl and m M, we have m M N for N >> 0. In particular, we can choose N n, and then A acts on m through its action on M N. We can easily check that the gl -module M is polynomial: indeed, due to the equivalence in Proposition 7.5.2, there exists a polynomial gl -module M such that M n = Γn (M ) for every n, and φ n 1,n are induced by the inclusions Γ n 1 (M ) Γ n (M ). By definition of M, we have a gl - equivariant map M M, and it is easy to check that it is an isomorphism. 19

20 We put Γ lim ((M n) n 0, (φ n 1,n ) n 1 ) := M, and require that the functor Γ lim act on morphisms accordingly. The above construction then gives us a natural isomorphism Γ lim Γ lim Id Rep(gl ) poly. References [1] E. Dan-Cohen, I. Penkov, V. Serganova, A Koszul category of representations of finitary Lie algebras; arxiv: [math.rt]. [2] I. Entova-Aizenbud, Schur-Weyl duality in complex rank: the limit case, in preparation. [3] J. Hong, O. Yacobi, Polynomial Representations and Categorifications of Fock Space, Algebras and Representation Theory, Volume 16, Issue 5, Springer Netherlands (2012), pp ; arxiv: [math.rt]. [4] I. G. Macdonald, Symmetric Functions and Hall polynomials, Oxford Mathematical Monographs, [5] I. Penkov, K. Styrkas, Tensor representations of classical locally finite Lie algebras, Developments and trends in infinite-dimensional Lie theory, Progr. Math. 288, Birkhauser Boston, (2011), pp ; arxiv: [6] O. M. Schnrer, Equivariant sheaves on flag varieties, Math. Zeitschrift 267 (2011), no. 1-2, 27-80; arxiv: [math.rt]. [7] S. V. Sam, A. Snowden, Stability patterns in representation theory; arxiv: [math.rt]. [8] B. Webster, G. Williamson, The bounded below equivariant derived category, btw4e/equivariant.pdf. Inna Entova Aizenbud, Massachusetts Institute of Technology, Department of Mathematics, Cambridge, MA USA. address: inna.entova@gmail.com 20