CONGRUENCES OF MORITA EQUIVALENT SMALL CATEGORIES

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1 Theory and pplications of Categories, Vol. 26, No. 12, 2012, pp CONGRUENCES OF MORIT EQUIVLENT SMLL CTEGORIES VLDIS LN bstract. Two categories are called Morita equivalent if the categories of functors from these categories to the category of sets are equivalent. We prove that congruence lattices of Morita equivalent small categories are isomorphic. 1. Preliminaries Categories and are called Morita equivalent if the functor categories Fun, Set and Fun, Set are equivalent. The basic theory of Morita equivalent categories has been developed already in 1960s [rtin, Grothendieck and Verdier, 1972]. One natural question to ask in Morita theory is that which properties are shared by Morita equivalent structures. For example, it is well known see Proposition in [nderson and Fuller, 1974] that ideal lattices of Morita equivalent rings are isomorphic. We shall prove an analogue of this result: Morita equivalent small categories have isomorphic congruence lattices. lthough this result can be proved easily by topos-theoretic methods, we give an elementary proof using Cauchy completions. In [Elkins and Zilber, 1976], a construction of a Cauchy completion of a small category is given as follows: objects of are idempotents e : of i.e. endomorphisms with e 2 = e; morphism sets are where e : and e : ; composition is given by The following result is well known. e, e = {e, a, e a,, e ae = a}, e, b, e e, a, e = e, ba, e. This research was supported by the Estonian Science Foundation grant no and Estonian Targeted Financing Project SF s08. Received by the editors and, in revised form, Transmitted by Robert Rosebrugh. Published on Mathematics Subject Classification: Key words and phrases: category, Morita equivalence, congruence. c Valdis Laan, Permission to copy for private use granted. 331

2 332 VLDIS LN 1.1. Theorem. Two small categories are Morita equivalent if and only if their Cauchy completions are equivalent. We write 0 1 for the class of objects morphisms of a category Definition. [See [Mac Lane, 1998], p. 52] congruence on a category is a family ρ = ρ,, 2 0 of equivalence relations ρ, on morphism sets, that are compatible with the composition of morphisms. If ρ is a congruence on a category then one can form a quotient category /ρ, where /ρ 0 = 0, /ρ, =, /ρ, and [g][f] = [gf] for every f,, g, C. We say that a congruence ρ is contained in a congruence σ and denote ρ σ if ρ, σ, for all, 0. This is obviously an order relation, and it is not difficult to see that the set Con of congruences on a small category is a lattice. 2. Congruence lattices of equivalent small categories 2.1. Proposition. If and are equivalent small categories then there is an isomorphism Γ : Con Con between their congruence lattices. Moreover, if ρ Con then /ρ is equivalent to /Γρ. Proof. Let F G be equivalence functors and let η : 1 GF, ε : F G 1 be natural isomorphisms. We define mappings Con Γ Con by Γρ, := { ε F fε, ε } F gε f, g ρg,g, σ, := { η Gkη, η Glη k, l σf,f }, ρ Con, σ Con,, 0,, 0. It is easy to see that Γρ, is an equivalence relation. To prove its compatibility, take f, g ρ G,G and b,. Then ε bε = F a for some a : G G and af, ag ρ G,G. Consequently, bε F fε, bε F gε = ε ε bε F fε, ε ε bε F gε = ε F afε, ε F agε Γρ,. b ε F f F G F G F G F g ε F a ε

3 CONGRUENCES OF MORIT EQUIVLENT SMLL CTEGORIES 333 Hence Γρ and similarly σ is a congruence. Clearly the mappings Γ and preserve order. Note that ρ GF,GF = { η uη, η vη u, v ρ, } for every, 0. Using this and properties of equivalence functors we obtain Γρ, = Γρ, = { } η G k η, η G l η k, l ΓρF,F { } = η G ε F F fε F η, η G ε F F gε F η f, g ρ GF,GF { } = η G ε F F η uη ε F η, η G ε F F η vη ε F η u, v ρ, = { } η GF uη, η GF vη u, v ρ, = { η η u, η η v u, v ρ, } = ρ,. Hence Γ = 1 Con and, analogously, Γ = 1 Con. To prove the second assertion, take ρ Con, denote σ := Γρ and define a functor F ρ : /ρ /Γρ by F ρ := F, F ρ [f] ρ, := [F f] σf,f,, 0, f,. First we show that, for every f, g,, η fη, η gη ρgf,gf ε F F η fη ε F, ε F F η gη ε F σ F,F. Suppose that the last holds. Then ε F F η fη ε F, ε F F η gη ε F = ε F F f ε F, ε F F g ε F for some f, g ρ GF,GF. This implies F η fη = F f, F η gη = F g, and hence η fη, η gη = f, g ρ GF,GF. The converse is obvious. Using the proved fact together with f, g ρ, η fη, η gη ρgf,gf and we conclude that ε F F η fη ε F, ε F F η gη ε F = F f, F g f, g ρ, F f, F g σ F,F.

4 334 VLDIS LN Thus F ρ is well defined and faithful. It clearly is dense. To prove that F ρ is full, take a morphism k F, F. Then [η F ρ Gkη = ]ρ [ F η, F GkF η ] [ ] = F η ρ, F Gkε F ρ, [ ] = ε F ε F k = [k]. ρ, ρ, 3. Congruence lattices of Morita equivalent small categories Our main result is the following Theorem. If and are Morita equivalent small categories then there is an isomorphism Π : Con Con between their congruence lattices. Moreover, if ρ Con then /ρ is Morita equivalent to /Πρ. It follows from Theorem 1.1, Proposition 2.1 and the next proposition Proposition. If is a small category then there is an isomorphism Π : Con Con between congruence lattices. Moreover, if ρ Con then /ρ is Morita equivalent to /Πρ. Proof. Let E be the set of idempotents of. Define a mapping Π : Con Con by Πρ e,e := {e, e se, e, e, e te, e s, t ρ, }, where ρ Con, e, e E, e : and e :. It is not difficult to see that Πρ e,e is a reflexive and symmetric relation on e, e. To prove transitivity of Πρ e,e, take p, q, q, r Πρ e,e. Then there exist s, t, u, v ρ, such that p, q = e, e se, e, e, e te, e and q, r = e, e ue, e, e, e ve, e, in particular e te = e ue. Since ρ is a congruence, we have e se ρ, e te = e ue ρ, e ve, and hence e se, e ve ρ,. Consequently, p, r = e, e se, e, e, e ve, e = e, e e see, e, e, e e vee, e Πρ e,e. If s, t ρ, and e, a, e : e e in, where e :, then a : in, hence as, at ρ, and e, a, e e, e se, e, e, a, e e, e te, e = e, ae se, e, e, ae te, e = e, e ase, e, e, e ate, e Πρ e,e. Similarly Πρ is compatible with precomposition and thus Πρ Con. We also define a mapping Ω : Con Con by Ωτ, := {a, b 1, a, 1, 1, b, 1 τ 1,1 },

5 CONGRUENCES OF MORIT EQUIVLENT SMLL CTEGORIES 335 where τ Con,, 0. Clearly Ωτ, is an equivalence relation. Let us show that Ωτ is compatible with precomposition. Take a, b Ωτ, and c C,. Note that 1, ac, 1 C, 1, bc, 1 C = 1, a, 1 1, c, 1 C, 1, b, 1 1, c, 1 C τ 1C,1, because τ Con. Consequently, ac, bc Ωτ C,. The proof that Ωτ is compatible with postcomposition is symmetric. Hence Ωτ = Ωτ,, 2 0 Con. Clearly Π and Ω preserve order. Now, for ρ Con and,, ΩΠρ, = ΩΠρ, = {a, b 1, a, 1, 1, b, 1 Πρ 1,1 } = {a, b a, b ρ, } = ρ,. On the other hand, if τ Con and e, e E, e :, e :, then ΠΩτ e,e = ΠΩτ e,e = {e, e se, e, e, e te, e s, t Ωτ, }. Let us prove the inclusion ΠΩτ e,e τ e,e. Note that s, t Ωτ, if and only if 1, s, 1, 1, t, 1 τ 1,1. Composing with e, e, 1 and 1, e, e we obtain e, e se, e, e, e te, e τ e,e. To prove the converse, let e, u, e, e, v, e τ e,e. Then 1, u, 1, 1, v, 1 τ 1,1 implies u, v Ωτ,. Thus e, u, e, e, v, e = e, e ue, e, e, e ve, e ΠΩτ e,e, and we have proven the equality ΠΩτ = τ. Let us now prove the second assertion. We denote à := /ρ, à := /Πρ and τ := Πρ. Thus Ã, =, /ρ, and Ãe, e = e, e /τ e,e,, 0, e, e E. We shall show that the Cauchy completions à and à constructed as in Section 1 are equivalent. Observe that the the objects of à are the idempotents of Ã, that is, the equivalence classes of endomorphisms e : in such that e 2 ρ, e. The morphism sets are Ã[e] ρ,, [e ] ρ, = {[e ] ρ,, [a] ρ,, [e] ρ, e, e, a 1, e 2 ρ e, e 2 ρ e, e ae ρ a}. The objects of à are the idempotents of and the morphisms are the equivalence classes [e, a, e] τe,e of morphisms e, a, e in, where e 2 = e, e 2 = e and e ae = a in. Hence the idempotents of à i.e. the objects of à are the classes [e, a, e] τ e,e such that e 2 = e, eae = a in and e, a 2, e τ e,e e, a, e. If [e, a, e] τe,e and [e, a, e ] τe,e are two objects of à then the morphism set à [e, a, e] τe,e, [e, a, e ] τe,e is {[e, a, e ] τe,e, [e, x, e] τe,e, [e, a, e] τe,e } x 1, e xe = x, e, a xa, e τ e,e e, x, e.

6 336 VLDIS LN Note that if e : and e : in then e, u, e τ e,e e, v, e = u ρ, v. 1 Indeed, e, u, e τ e,e e, v, e if and only if there exists s, t ρ, and e te = v. Due to compatibility, also u, v = e se, e te ρ,. Define a functor F : à à by the assignment such that e se = u [e, a, e] τe,e [a] ρ, [e,a,e ] τ,[e,x,e] τ,[e,a,e] τ [e, a, e ] τe,e [a ] ρ, [a ] ρ,[x] ρ,[a] ρ where e : and e : in. If [e, a, e] τe,e is an object of the category then e, a 2, e τ e,e e, a, e, hence a 2 ρ, a by 1, and so [a] ρ, is an object of Ã. If e, a, e τ e,e e, a 1, e then again a ρ, a 1, and therefore F is well defined on objects. similar argument shows that the definition of F on morphisms does not depend on the choice of the representatives of τ-classes. Moreover, if [e, a, e ] τe,e, [e, x, e] τe,e, [e, a, e] τe,e is a morphism in à then e, a xa, e τ e,e e, x, e, so a xa ρ, x and [a ] ρ,, [x] ρ,, [a] ρ, is indeed a morphism in Ã. Straightforward calculations show that F is a functor. If [a] ρ, is an object in à then a2 ρ, a, hence 1, a 2, 1 τ 1,1 1, a, 1. Consequently, [1, a, 1 ] τ1,1 is an object of à which maps to [a] ρ,. So F is dense. To prove that F is full, let [e, a, e] τe,e, [e, a, e ] τe,e be two objects in à and consider a morphism [a ] ρ,, [x] ρ,, [a] ρ, in Ã. Then a xa ρ, x, e 2 = e, e 2 = e, eae = a, e a e = a, e, a 2, e τ e,e e, a, e and e, a 2, e τ e,e e, a, e. We have e e xee = e xe and e, e a xae, e = e, a e xea, e τ e,e e, e xe, e, because a xa ρ, x implies e, e a xae, e τ e,e e, e xe, e. lso, e xe ρ, e a xae = a xa ρ, x. Thus we have shown that [e, a, e ] τe,e, [e, x, e] τe,e, [e, a, e] τe,e is a morphism in à which maps to [a ] ρ,, [e xe] ρ,, [a] ρ, = [a ] ρ,, [x] ρ,, [a] ρ,. To show that F is faithful, suppose that [e, a, e ] τe,e, [e, x, e] τe,e, [e, a, e] τe,e and [e, a, e ] τe,e, [e, x, e] τe,e, [e, a, e] τe,e are two morphisms in à that map to the same morphism [a ] ρ,, [x] ρ,, [a] ρ, = [a ] ρ,, [x ] ρ,, [a] ρ, in Ã. Then x ρ, x and hence e, e xe, e τ e,e e, e x e, e. ut e xe = x and e x e = x, so [e, x, e] τe,e = [e, x, e] τe,e.

7 cknowledgement CONGRUENCES OF MORIT EQUIVLENT SMLL CTEGORIES 337 I am grateful to Professor Peter Johnstone and the referee for useful suggestions. References F. nderson, K. Fuller, Rings and categories of modules. Springer-Verlag, M. rtin,. Grothendieck and J. L. Verdier, Théorie des Topos et Cohomologie Étale des Schémas. Lecture Notes in Mathematics 269, Springer-Verlag, erlin-new York, Elkins and J.. Zilber 1976, Categories of actions and Morita equivalence, Rocky Mountain J. Math 6, S. Mac Lane, Categories for the Working Mathematician. Springer-Verlag, Institute of Mathematics, University of Tartu Ülikooli 18, Tartu, Estonia valdis.laan@ut.ee This article may be accessed at or by anonymous ftp at ftp://ftp.tac.mta.ca/pub/tac/html/volumes/26/12/26-12.{dvi,ps,pdf}

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