PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 143, Number 3, March 2015, Pages 1315–1323 S 0002-9939(2014)12354-9 Article electronically published on October 16, 2014

PROJECTIVE AND INJECTIVE SYMMETRIC CATEGORICAL GROUPS AND

TEIMURAZ PIRASHVILI

(Communicated by Michael A. Mandell)

Abstract. We prove that the 2- of symmetric categorical groups have enough projective and injective objects.

1. Introduction Symmetric categorical groups, also known as Picard categories or abelian 2- groups, play the same role in the 2-dimensional algebra as abelian groups play in the classical algebra. In [3] D. Bourn and E. Vitale defined the notion of projective and injective symmetric categorical groups, and they asked whether there are enough projective symmetric categorical groups (see p. 104 in [3]). In this paper we show that there are enough projective and injective symmetric categorical groups, thereby giving an affirmative answer to the question of D. Bourn and E. Vitale. Moreover, we construct a projective symmetric categorical group Φ and we prove that any projective symmetric categorical group is a of copies of Φ, thus any projective symmetric categorical group is free in some sense. As an application we construct a self-duality of the 2-category of finite symmetric categorical groups. For some other applications of these facts see [5].

2. Basic notions Recall that a groupoid is a small category such that all are isomor- phisms. For a groupoid G andanobjectx ∈ G we let π0(G)andπ1(G,x)be the set of connected components of G and the group of automorphisms of x in G respectively. The notion of a symmetric categorical group is a categorification of the notion of an . More precisely, let (A, +, 0,a,l,r,c) be a symmetric monoidal category, where + : A × A → A is the composition law, 0 is the neutral element, a is the associative constraint, c is the commutativity constraint and l : Id → 0+Id and r : Id → Id + 0 are natural transformations satisfying well-known properties [6]. We will say that A is a symmetric categorical group or Picard category provided A is a groupoid and for any object x the endofunctor x+:A → A is an equivalence of categories. It follows that π (A) is an abelian group and the endofunctor x+:A → 0 ∼ A yields an isomorphism π1(A, 0) = π1(A,x) of abelian groups. In what follows we will write π1(A) instead of π1(A, 0).

Received by the editors May 23, 2013 and, in revised form, June 9, 2013. 2010 Mathematics Subject Classification. Primary 55-XX; Secondary 18-XX. This research was partially supported by the grant “DI/27/5-103/12, D-13/2 Homological and categorical methods in topology, algebra and theory of stacks”.

c 2014 American Mathematical Society Reverts to public domain 28 years from publication 1315

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For symmetric categorical groups S1 and S2 we have a groupoid (in fact a sym- metric categorical group [3]) Hom(S1, S2). Hence symmetric categorical groups form a groupoid enriched category SCG. Objects of the groupoid Hom(S1, S2)are symmetric monoidal functors, called morphisms of symmetric categorical groups, while morphisms of the groupoid Hom(S1, S2) are monoidal natural transforma- tions, called tracks.WeletHo(SCG), or simply Ho, be the additive category, with the same objects as SCG,and

HomHo(S1, S2)=π0(Hom(S1, S2)). Recall that a f : A → B of symmetric categorical groups is essen- tially surjective provided π0(A) → π0(B) is a surjective homomorphism of abelian groups. A symmetric categorical group P is projective provided for any essentially surjective morphism f : A → B of symmetric categorical groups the induced mor- phism Hom(P, A) → Hom(P, B) is essentially surjective. Equivalently, the map HomHo(P, A) → HomHo(P, B) is surjective. Dually, a morphism g : C → D of symmetric categorical groups is faithful pro- vided the induced map π1C → π1D is injective. A symmetric categorical group I is called injective provided for any faithful morphism F : C → D the induced map

HomHo(D, I) → HomHo(C, I) is surjective. We will prove that the 2-category SCG has enough injective and projective ob- jects, meaning that for any symmetric categorical group A there exists a faithful morphism A → I with I and an essentially surjective morphism P → A, with projective P. Recall also the notion of direct sum of symmetric categorical groups. Assume S ∈ S α, α A, are symmetric categorical groups. A coproduct α∈A α is a symmetric categorical group S, together with morphisms of symmetric categorical groups iα : Sα → S, such that for any symmetric categorical group A, the obvious functor  Hom(S, A) → Hom(Sα, A) α  S is an equivalence of categories. In a similar way one can define a product α∈A α of symmetric categorical groups Sα. As in the case of abelian groups, finite product and coproduct are equivalent. One easily sees that coproduct and product exist and         πi Sα = πi(Sα),πi Sα = πi(Sα),i=0, 1. α∈A α∈A α∈A α∈A Thus product and coproduct of symmetric categorical groups yield the usual prod- uct and coproduct in the homotopy category Ho.

3. The homotopy category of the 2-category of symmetric categorical groups It follows from the results of [7] that the 2-category SCG of symmetric categorical groups is 2-equivalent to the 2-category of 2-stage spectra (see also Proposition B.12 in [4]). Hence we can use the classical facts of algebraic topology to study SCG.

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Let ΓAB be the category of triples (A, B, a)whereA and B are abelian groups and a ∈ Hom(A/2A, B)=Hom(A, 2B)

where 2B = {b ∈ B | 2b =0}. A morphism (A, B, a) → (A1,B1,a1)isapair(f,g) where f : A → A1 and g : B → B1 are homomorphisms, such that a1f = ga.The functor k : Ho(SCG) → ΓAB is defined by k(S):=(π0(S),π1(S),kS)

where S is a symmetric categorical group and kS : π0(S) → 2π1(S) is the homo- morphism induced by the commutativity constraints in S: ∼ x → cx,x ∈ π1(S,x+ x) = π1(S).

Proposition 1. For any symmetric categorical groups S1 and S2, one has a short of abelian groups γ (1) 0 → Ext(π0(S1),π1(S2)) → π0(Hom(S1, S2)) −→ HomΓAB(k(S1), k(S2)) → 0 Furthermore, one has also an isomorphism of abelian groups ∼ (2) π1(Hom(S1, S2)) = Hom(π0(S1),π1(S2)). Moreover, for the k-invariant one has the equality:

(3) kHom(S1,S2) = αγ,

where α sends a pair f0 : π0(S1) → π0(S2),f1 : π0(S1) → π0(S2) to the composite

S f0 k 2 π0(S1) −→ π0(S2) −−→ 2π1(S1). Proof. The second isomorphism is obvious, while the first one is Proposition 7.1.6 in [2]. The statement on k-invariant follows from the expression of symmetric constant on Hom(S1, S2).  We see that both categories Ho and ΓAB are additive and the functor k : Ho → ΓAB preserves and products. Thus it is additive. Moreover k is a part of a linear extension of categories (see Lemma 7.2.4 and Theorem 7.2.7 in [2]). It follows from the properties of linear extensions of categories [1] that the functor k is full, reflects isomorphisms, is essentially surjective on objects and it induces a bijection on the isomorphism classes of objects. Moreover, the kernel of k (the class of morphisms which go to zero) is a square zero ideal of Ho. Hence, for a given object A of the category ΓAB we can choose a symmetric categorical group K(A) such that k(K(A)) = A. Such an object exists and is unique up to equivalence. Moreover, for any morphism f : A → B we can choose a morphism of symmetric categorical groups K(f):K(A) → K(B), such that k(K(f)) = f.Thereadermust be aware that the assignments A → K(A), f → K(f) do NOT define a functor ΓAB → Ho. Having in mind the relation with spectra, the construction K for the objects of the form (A, 0, 0) coincides with the Eilenberg-Mac Lane spectrum and in the general case is consistent with Definition 7.1.5 in [2]. We let Φ be the symmetric categorical group corresponding to the object (Z, Z/2Z,k = IdZ/2Z). We can assume that the objects of the groupoid Φ are

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integers; if n and m are integers, then there are no morphisms from n to m,if m = n, while the automorphism group of the object n is the cyclic group of order two {±1}. The monoidal structure on objects is induced by the group structure of integers and the monoidal structure on morphisms is induced by the multiplication on the cyclic group of order two. The associativity and unitality constraints are the identity morphisms, while the commutativity constraint n + m → m + n is (−1)nm. It follows from Proposition 1 that for any symmetric categorical group S the symmetric categorical groups Hom(Φ, S)andS are equivalent.

4. Projective objects in SCG In this section we prove the following theorem. Theorem 2. There are enough projective symmetric categorical groups. Moreover, any projective symmetric categorical group is equivalent to a coproduct of copies of Φ. This result is a consequence of Lemma 3 proved below. A morphism f =(f0,f1)inΓAB is essentially surjective if f0 is an of abelian groups. Moreover an object P in ΓAB is projective if for any essentially surjective morphism f : A → B in ΓAB the induced map

HomΓAB(P, A) → HomΓAB(P, B) is surjective. It is clear that a morphism F : S1 → S2 of symmetric categorical groups is essentially surjective iff k(F ):k(S1) → k(S2)issoinΓAB. For an abelian group M we introduce two objects in ΓAB:

l(M):=(M,M/2M,idM/2M ), M[0] = (M,0, 0).

Lemma 3. i) If M is an abelian group and A =(A0,A1,α) is an object in ΓAB, then one has the following functorial isomorphism of abelian groups:

HomΓAB(l(M), A)=Hom(M,A0). ii) An object P is projective in ΓAB iff it is isomorphic to the object of the form l(P ) with P . iii) Φ is a projective object in SCG and any projective object in SCG is equivalent to a coproduct of copies of Φ. iv) The 2-category SCG of symmetric categorical groups has enough projective objects.

Proof. i) Assume f =(f0,f1):l(M) → A is a morphism in ΓAB.Sof0 : M → A0 and f1 : M/2M → A1 are homomorphisms of abelian groups and the following diagram commutes: M/2M Id /M/2M

fˆ0 f1   / A0/2A0 α A1 ˆ Here f0 is induced by f0. It follows that f1 is completely determined by f0.This proves the result.

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ii) Let P be a free abelian group and let A → B be an essentially surjec- tive morphism in ΓAB.ThusA0 → B0 is an epimorphism. It follows that Hom(P, A0) → Hom(P, B0) is an epimorphism as well. Hence, by virtue of i), the map HomΓAB(l(P ), A) → HomΓAB(l(P ), B) is surjective. Thus l(P ) is projective in ΓAB. Conversely, assume P =(P0,P1,π) is a projective object in ΓAB. We claim that P0 is a free abelian group. In fact it suffices to show that it is a projective object in the category Ab of abelian groups. Take any epimorphism of abelian groups f0 : A → B and any homomorphism of abelian groups g0 : P0 → B.Wehaveto show that g0 has a lift to A.Observethatf =(f0, 0) : A[0] → B[0] is essentially surjective in ΓAB and g =(g0, 0) : P → B[0] is a well-defined morphism in ΓAB. By our assumption we can lift g to a morphismg ˜ : P → A[0]. It is clear that g˜ =(˜g0, 0) for someg ˜0 : P0 → A. Clearly, g0 = f0 ◦ g˜0. It follows that P0 is a free abelian group. Hence l(P0) is a projective object in ΓAB.Byi)theidentity → P map defines a canonical morphism i =(IdP0 ,i1):l(P0) , which is obviously essentially surjective in ΓAB.SinceP is projective, it follows that there exists P → ◦ a morphism p =(IdP0 ,p1): l(P ) such that i p = IdP.Thus,wehavea π / P0/2P0 P1

Id p1   Id / P0/2P0 P0/2P0

Id i1   π / P0/2P0 P1

with i1p1 = IdP1 . It follows that p1 and i1 are mutually inverse isomorphisms of abelian groups. Hence p : P → l(P )andl : l(P ) → P are mutually inverse isomorphisms in ΓAB. iii) First of all observe that k preserves coproduct and k(Φ) = l(Z). Hence our assertion is equivalent to the following one: For any free abelian group P the symmetric categorical group K(l(P )) is a projective symmetric categorical group and conversely, if S is a projective symmetric categorical group, then π0(S)isafree abelian group and S is equivalent to K(l(π0(S))). To prove the last assertion, let F : S1 → S2 be an essentially surjective morphism of symmetric categorical groups and let G : K(l(P )) → S2 be a morphism of symmetric categorical groups. Apply the functor k to get morphisms k(F ):k(S1) → k(S2)andk(G):l(P ) → k(S2)inΓAB.Sinceπ0(F ):π0(S1) → π0(S2)isan epimorphism of abelian groups, it follows that k(F ):k(S1) → k(S2) is an essentially surjective morphism in ΓAB.SinceP is a free abelian group, l(P ) is projective in ΓAB by ii). Thus we can lift k(F ) to get a morphismg ˆ : l(P ) → k(S1) such that k(F ) ◦ gˆ = k(G)holdsinΓAB.SinceP = π0(K(l(P )) is a free abelian group, the Ext-term in the exact sequence (1) disappears and we get the isomorphism ∼ (4) π0(Hom(K(l(P )), Si)) = HomΓAB(l(P ), k(Si)),i=0, 1.

Take a morphism L : K(l(P )) → S1 of symmetric categorical groups which cor- responds to the morphismg ˆ : l(P ) → k(S1). By our construction one has an equality k(FL)=k(G)inHomΓAB(l(P ), k(S2)) = π0(Hom(H(l(P )), S2)). Thus

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the classes of FL and of G in π0(Hom(K(l(P )), S1)) are the same. Hence there exists a track from FL to G. This shows that K(l(P )) is a projective symmetric categorical group. Conversely assume S is a projective symmetric categorical group. Since S and K(k(S)) are equivalent, it follows that K(k(S)) is also projective. We claim that k(S)isprojectiveinΓAB. In fact, take any essentially surjective morphism f = (f0,f1):A → B and any morphism g : k(S) → B in ΓAB.ThenK(f):K(A) → K(B) is essentially surjective in SCG. Hence for K(g):K(k(S)) → K(B)we have a morphism G˜ : K(k(S)) → K(A)andatrackK(f) ◦ G˜ → K(g). Thus K(f) ◦ G˜ = K(g)inπ0(Hom(K(k(S)),K(B))). Now apply the functor k to get the equality f ◦ k(G˜)=g, showing that k(S)isprojectiveinΓAB. Hence k(S)is isomorphic to l(P ) for a free abelian group P .ThusS and K(l(P )) are equivalent. iv) Let S be a symmetric categorical group. Choose a free abelian group P and an epimorphism of abelian groups f0 : P → π0(S). By part i) of Lemma 3, f0 has a unique extension to a morphism f =(f0,f1):l(P ) → k(S) which is essentially surjective. Since P is a free abelian group, we have the isomorphism (4), which shows that there exists a morphism of symmetric categorical groups K(l(P )) → S which realizes f0 on the level of π0. Clearly this morphism does the job. 

5. Injective objects In this section we prove the following result. Theorem 4. The groupoid enriched category SCG has enough injective objects. This is just part iv) of Lemma 5 proved below. A morphism f =(f0,f1)inΓAB is faithful provided f1 is injective and an object I =(I0,I1,ι)ofΓAB is injective if for any faithful morphism f : A → B in ΓAB the induced map

HomΓAB(B, I) → HomΓAB(A, I)

is surjective. It is clear that a morphism F : S1 → S2 of symmetric categorical groups is faithful iff k(F ):k(S1) → k(S2)isfaithfulinΓAB. For an abelian group M we introduce two objects in ΓAB:

r(M)=(2M,M,id2M ),

M[1] = (0,M,0).

Lemma 5. i) If M is an abelian group and A =(A0,A1,α) is an object in ΓAB, then one has the following functorial isomorphism of abelian groups

HomΓAB(A,r(M)) = Hom(A1,M). ii) An object D is an injective object in ΓAB iff it is isomorphic to the object of the form r(D), with divisible abelian group D. iii) For any divisible abelian group D the symmetric categorical group K(r(D)) is injective. Conversely, if S is an injective categorical group, then π1(S) is a divisible abelian group and S is equivalent to K(r(π1(S))). iv) The 2-category SCG of symmetric categorical groups has enough injective objects.

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Proof. i) Assume g =(g0,g1):A → r(M) is a morphism in ΓAB.Sog0 : A0 → 2M and g1 : A1 → M are homomorphisms of abelian groups and we have a commutative diagram α / i / A0 2A1 A1

g0 g¯1 g1    Id / j / 2M 2M M

whereg ¯1 is induced by g1 and i, j are inclusions. It follows that g0 is completely determined by g1 and the result follows. ii) Let D be a divisible abelian group and let A → B be a faithful morphism in ΓAB.ThusA1 → B1 is a monomorphism. Since D is an injective object in Ab it follows that Hom(B1,D) → Hom(A1,D) is an epimorphism of abelian groups. So by i) the map HomΓAB(B,r(D)) → HomΓAB(A,r(D)) is surjective. Thus r(D)is injective in ΓAB. Conversely, assume D =(D0,D1,χ) is injective in ΓAB. We claim that D1 is a divisible abelian group. In fact, it suffices to show that it is an injective object in the category Ab. Take any monomorphism of abelian groups f1 : A → B and any homomorphism of abelian groups g1 : A1 → D0. We have to show that g1 hasalifttoB1.Observethatf =(0,f1):A[1] → B[1]isfaithfulinΓAB and g =(0,g1):A[1] → D is a well-defined morphism in ΓAB. By assumption there exists a morphismg ˜ : B[1] → D.Itisclearthat˜g =(0, g˜1)forsome˜g1 : B → D1. Thus D1 is a divisible abelian group and r(D1) is an injective object in ΓAB. D → By i) the identity map defines a canonical morphism i =(i0, IdD1 ): r(D1), which is obviously faithful in ΓAB.SinceD is injective, it follows that there exists → D ◦ a morphism q =(q0, IdD1 ):r(D1) such that q i = IdQ. Thuswehavea commutative diagram χ / D0 2D1

i0 Id   Id / 2D1 2D1

q0 Id   χ / D0 2D1 D → with q0i0 = IdD0 . It follows that i0 is an isomorphism. Hence i : r(D1)isan isomorphism and we are done. iii) Let F : S1 → S2 be a faithful morphism of symmetric categorical groups and let G : S1 → K(r(D)) be a morphism of symmetric categorical groups. Apply the functor k to get morphisms k(F ):k(S1) → k(S2)andk(G):k(S1) → r(D) in ΓAB.Sinceπ1(F ):π1(S1) → π1(S2) is a monomorphism of abelian groups, it follows that k(F ):k(S1) → k(S2) is a faithful morphism in ΓAB.SinceD is a divisible abelian group, r(D)isinjectiveinΓAB by ii) and we can extend k(F ) to get a morphismg ˆ : k(S2) → r(D). Thus we have the equalityg ˆ ◦ k(F )=k(G) in HomΓAB(k(S1),r(Q)). Since D = π1(K(r(D))) is a divisible abelian group, the Ext-term in the exact sequence (1) disappears and we get the isomorphism ∼ (5) π0(Hom(Si,K(r(D)))) = HomΓAB(k(Si),r(D)),i=0, 1.

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Take a morphism L : S2 → K(r(D)) of symmetric categorical groups which cor- responds to the morphismg ˆ : k(S2) → r(D). By our construction, one has the equality k(LF )=k(G). This is an equality in π0(Hom(S1,K(r(D)))), which im- plies that the classes of LF and of G in π0(Hom(S1,K(r(D)))) are the same. Thus there exists a track from LF to G. This shows that K(r(D)) is an injec- tive symmetric categorical group. Conversely, assume S is an injective symmetric categorical group. Since S and K(k(S)) are equivalent, it follows that K(k(S)) is also projective. We claim that k(S) is injective in ΓAB. In fact, take any faithful morphism f =(f0,f1):A → B in ΓAB and any morphism g : A → k(S)inΓAB. Then K(f):K(A) → K(B)isfaithfulinSCG. Hence for K(g):K(A) → K(k(S)) we have a morphism G˜ : K(B) → K(k(S)) and a track G˜ ◦ K(f) → K(g). Thus G˜ ◦ K(f)=K(g)inπ0(Hom(K(k(S)),K(B))). Now apply the functor k to get the equality k(G˜) ◦ f = g, showing that k(S)isinjectiveinΓAB. Hence k(S) is isomorphic to r(D) for a divisible abelian group D.ThusS and K(r(D)) are equivalent. iv) Let S be a symmetric categorical group. Choose a divisible abelian group D and a monomorphism of abelian groups f1 : π1(S) → D. By Lemma 5 f1 has a unique extension to a morphism f =(f0,f1):k(S) → r(D) which is essentially surjective. Since D is a divisible abelian group, we have the isomorphism (5), which shows that there exists a morphism of symmetric categorical groups S → K(r(D)) which realizes f1 on the level of π1 andwegettheresult.  6. Duality For an abelian group A we set d(A)=Hom(A, Q/Z), where Hom is taken in the category of abelian groups. It is well known that d yields a duality of the category of finite abelain groups. We will prove that there is a similar duality of the 2-category of finite symmetric categorical groups. Here, a symmetric categorical group S is called finite provided πi(S) is finite, i =0, 1. To describe the new duality we put J = K(r(Q/Z)). By Lemma 5 we know that J is an injective object of the 2-category SCG. For a symmetric categorical group S we set D(S)=Hom(S, J). Lemma 6. i) For any symmetric categorical group S one has

π0(D(S)) = d(π1(S)),π1(D(S)) = d(π0(S)) and kD(S) = d(kS ). ii) If S is finite, then D(S) is also finite.

Proof. ii) follows immediately from i). Since π1(J)=Q/Z it follows from the exact sequence (1) that π0(D(S)) = HomΓAB(k(S), k(J)) = d(π1(S)). The last equality follows from part i) of Lemma 5. The isomorphism π1(D(S)) = d(π0(S)) follows directly from the isomorphism (2). To prove the equation π1(D(S)) = d(π0(S)), letusrecallthatkS is a homomorphism π0(S)/2π0(S) → π1(S), hence d(kS)isthe homomorphism d(π1(S)) → d(π0(S)/2π0(S)) = 2d(π0(S)). The fact on k-invariants follows from the equality (3).  Theorem 7. The 2-functor D : SCGop → SCG yields duality on finite symmetric categorical groups.

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Proof. For any symmetric categorical group S the obvious evaluation map yields a morphism of symmetric categorical groups S → D(D(S)). We claim that this map is an equivalence, provided S is finite. For this it suffices to show that it induces an isomorphism on π0 and π1. But this directly follows from Lemma 6. In fact the map π0(S) → π0(DD(S)) equals to π0(S) → d(π1(D(S))), which is the same as π0(S) → d(d(π0(S))), which is an isomorphism, by the classical duality of finite abelian groups. Same for π1.  References

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Department of Mathematics, University of Leicester, University Road, Leicester LE1 7RH, United Kingdom E-mail address: tp59-at-le.ac.uk

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