HSP SUBCATEGORIES OF EILENBERG-MOORE ALGEBRAS

HSP SUBCATEGORIES OF EILENBERG-MOORE ALGEBRAS MICHAEL BARR Abstract. Given a triple T on a complete category C and a actorization system E /M on the category o algebras, we show there is a 1-1 correspondence between ull subcategories o the category o algebras that are closed under U-split epimorphisms, products, and M -subobjects and triple morphisms T S or which the induced natural transormation between ree unctors belongs to E. 1. Introduction A celebrated theorem o Birkho s says that a subcategory o the category o algebras or a signature is given by a set o equations (which is to say, an equivalence relation on the ree algebras) i and only i it is closed under homomorphic images, subobjects, and products (HSP). It is easy to generalize this to any equational category, so that a subcategory o an equational category is given by urther equations i and only i it is closed under HSP. When one tries to extend this theorem to arbitrary categories, one immediately runs into a number o diiculties. What is a homomorphic image, what is a subobject and can the theorem be proved. As we will see, the homomorphic image seems to be a bit misleading. Even Birkho s theorem seems to be true only modulo the axiom o choice, which is obviously not available. Then the proo itsel seems to depend on the act that the product o surjections is a surjection, which is also a orm o AC. At any rate, I present here a theorem that holds in some generality and does reduce to Birkho s theorem over sets. It is based on the choice o a actorization system in the category o algebras, which may or may not be induced by one in the base category. As a matter o act, even over sets, there are examples that are not examples o Birkho s theorem. It would be nice i the theorem could be stated in terms o a actorization system in the base category. But over sets, or example, there is no actorization system that corresponds in any way to epimorphisms in the categories o monoids or o rings. When I irst announced these results, I received a number o comments rom people that the class E did not have to consist entirely o epimorphisms and that completeness was unnecessary. Some o them also suggested that the triple would have to preserve the class E (although with the class E being in the category o algebras, perhaps what they meant was the cotriple preserved it). As will be seen below, the hypothesis that E is included in the epics is crucial to going rom triples to HSP subcategories and completeness This research has been supported by the NSERC o Canada 2000 Mathematics Subject Classiication: 18C05, 18A20, 18A40. Key words and phrases: Birkho subcategories, actorizations, relective subcategories. c Michael Barr, 2002. Permission to copy or private use granted. 1

is crucial to going the other way. So the theorem could be stated with two dierent sets o hypotheses or the two directions. But this seems to miss the main the point o Birkho s theorem which is the 1-1 correspondence between HSP subcategories and certain kinds o morphisms o triples. Moreover, the hypothesis that every map in E is epic is much less exigent than that the cotriple preserve the class. The ormer hypothesis is normally true, while the latter is not, except in the category o sets and even there it requires either the axiom o choice or a initary theory. 2 2. Factorization systems We collect here a number o acts about actorization systems that we will need. They are all well-known. This is not the most general kind o actorization and some o the hypotheses could perhaps be relaxed a bit, but at the price o a considerable complication. We assume a complete category C and two classes o morphisms E and M such that FS1. FS2. FS3. E M consists o the isomorphisms; E and M are each closed under composition; Every map in E is an epimorphism and every map in M is a monomorphism; FS4. Every map in C actors = m e with m M and e E ; FS5. Given any commutative diagram e m with e E and m M there is a unique arrow cod(e) triangles commute. This is called the diagonal ill-in. dom(m) making both FS6. For any object C o C, there is, up to isomorphism, only a set o arrows o E whose domain is C. An epimorphism in a category is called extremal i it does not actor through any proper subobject o its codomain and dually, a monomorphism is called extremal i it does not actor through any proper epimorphism rom its codomain. 2.1. Proposition. The ollowing acts about actorization systems are well-known and we omit the proos. 1. Every extremal epimorphism is in E and dually, every extremal monomorphism is in M.

3 2. I g is a pullback and g M then M ; dually i the square is a pushout and E, then g E. 3. The actorization in FS 4 is unique up to isomorphism. 4. I m is an arrow o C that has the diagonal ill-in property with respect to every e E then M ; dually i e has the diagonal ill-in property with respect to every m M, then e E. 3. Morphism o triples Let C be a category and T = (T, η, µ) be a triple on C. I it is necessary to distinguish η and µ rom the unit and multiplication o another triple, we will write η T and µ T. I S = (S, η S, µ S ) is another triple on C, a morphism α: T S is a natural transormation o T S such that the diagrams T η T Id η S S α T 2 T α T S αs S 2 µ T T α S µ S commute. 3.1. Proposition. Suppose that α: T S is a morphism o triples. Then α induces a unctor C α : C S C T that takes the S-algebra (, ) to the T-algebra (, x = ). Proo. On objects, this can be read rom the diagrams T 2 µ T T T S αs S 2 T S T S T η T η S S T µ S S id

4 I : (, ) (Y, ŷ) is a morphism o S-algebras, then the diagram T S T S T Y αy SY ŷ Y commutes so that is also a morphism o T-algebras. Let F T : C C T and F S : C C S denote the ree algebra unctors given by F T = (T, µ T ) and F S = (S, µ S ). It ollows rom the preceding proposition that (F S, µ S αs) is a T-algebra or any Ob(C ). Since : (S, µ S ) (, ) is a morphism o S-algebras, it also ollows that : (S, µ S αs) (, ) is a morphism o T-algebras. 3.2. Proposition. The arrow : (T, µ T ) (S, µ S αs) is a morphism o T-algebras or any object Ob(C ). Proo. This is immediate rom the commutation o T 2 µ T T T α α which is part o the deinition o α. µ S αs T S S 2 S 3.3. Proposition. Suppose that α: F T F S is an epimorphism in C T or every Ob(C ). Then C α is ull and aithul. Proo. I (, ) and (Y, ŷ) are S-algebras and : (, x = ) (Y, y = ŷ αy ) is a morphism o T-algebras, then the outer and let hand squares o F T F T F T Y αy F S (, x) F S F S Y (Y, y) ŷ

Since is an epimorphism o T-algebras, so does the right hand square. Applying the underlying unctor, we conclude that 5 T T T Y ŷ Y commutes. The converse o this proposition is alse. For example, the category o Q-modules (rational vector spaces) is a ull subcategory o Ab, but the map Z Q between the ree objects on one generator is certainly not epic. The ollowing theorem identiies those S-algebras that are T-algebras. It is interesting on its own and is also used in the proo o the main theorem. Since the category o S-algebras is a ull subcategory o the category o T-algebras, it makes sense to ask i a T-algebra is an S-algebra. 3.4. Theorem. Suppose that α: T S is a morphism o triples such that the induced α: F T F S is an epimorphism in the category o T-algebras. Then the T- algebra (, x) is an S-algebra i and only i x can be actored in the category o T-algebras as. Proo. The crucial thing here is that x: F T (, x) is a morphism o T-algebras, in contrast with the situation with n-ary operations which are not, in most cases homomorphisms o algebras. The hypothesis is that there is a morphism : F S (, x) in the category o T-algebras such that = x. This means that the outer square o T S αs S 2 µ S S T S T S commutes. This square underlies the square in C T F T S αs F S S µ S F S F T F T F S F S (, x)

which implies, given that αs is epic in C T, that the right hand square commutes. Apply the underlying unctor it ollows that the right hand square o the irst diagram also commutes. This shows that commutes properly with µ S. The unitary law ollows immediately rom the commutativity o T η T η S S 6 id Notice, incidentally, that i αs should happen to be epic in C, then the actorization o x need only take place in C. This happens over sets when E is the class o regular epimorphisms. O course, the existence o such a actorization o x in the underlying category will orce the existence o a actorization (usually dierent) in C T. 4. The main theorem 4.1. Theorem. Suppose that T is a triple on the complete category C and E /M a actorization system on C T as described in Section 2. Then there is a 1-1 correspondence between morphisms α: T S o triples or which every instance o the induced α: F T F S lies in E and ull subcategories B C T that are closed under U-split quotients, M -subobjects, and products. Proo. Suppose that α: T S is such that every instance lies in E. Suppose that (, ) is an S-algebra and that : (, x = αx) (Y, y) is a U-split epimorphism T-algebras. Let u: Y be a right inverse o in C. Then Su = F S u and T u = F T u are maps in C T. Deine ŷ = Su: SY Y. Then ŷ αy = Su αy = T u = x T u = y T T u = y and it ollows rom Theorem 3.4 that (Y, ŷ) is an S-algebra. Suppose that (, ) is an S-algebra and that (Y, y) is an M -subalgebra o (, x = ). The diagram F T Y αy F S Y commutes because y F S m (Y, y) (, x) m m y = x F T m = F T m = F S m αy

and hence the actorization system gives a map ŷ: F S Y Y or which ŷ αy = y. It ollows rom Theorem 3.4 that (Y, Ŷ ) is an S-algebra. Finally, i ( i, i ) is a collection o S-algebras, then it is evident that ( i, i ) is also an S-algebra since the underlying unctor creates limits. This proves the only i part o the statement. Conversely, suppose that I: B C T is a ull inclusion and that B is closed in C T under U-split epics, M -subobjects, and products. Given a T-algebra (, x), any morphism : (, x) (Y, y) with codomain in B has an E /M actorization (, x) (Z, z) (Y, y) and the hypothesis implies that (Z, z) Ob(B). Let e i : (, x) (Y i, y i ) range over a complete set o representatives or morphisms in E, whose codomains are objects o B. Their product lies in B and so does the M -subobject gotten rom the E /M -actorization o the induced map e i : (, x) (Yi, y i ). It is clear that this gives a let adjoint Î or I and then Î F is a let adjoint to U I. It is also obvious that every instance o the adjunction Id I Î belongs to E. The only thing let is to show that U I is tripleable. Since I is ull and aithul and U relects isomorphisms, so does U I. So suppose that B B 7 is a U I-split ork. Then with U tripleable, there is a morphism IB that IB IB C C in C T so is a U-split coequalizer diagram. But the hypothesis that B be closed under U-split epics implies that C = IB and it is easy to see that B B B is a coequalizer in B. 5. Examples The most obvious examples are over sets, where, by taking E as the class o regular epics, which are the same as surjections, we recover Birkho s original theorem. But even over the category o sets, there are other possibilities. For example, i we take the epic/regular monic actorization in the category o monoids, it is easy to see that the ull subcategory o groups is closed under surjective image, regular subobjects, and products and it ollows that the map rom a monoid to its associated group is always epic. A similar example is that o strongly regular rings in the category o rings. A ring R is strongly regular i or all a R there is a b R with a 2 b = a. It can be shown that this implies that there is a unique b such that a 2 b = aba = ba 2 = a and b 2 a = bab = ab 2 = b and this unique choice is preserved by ring homomorphisms. The ull subcategory is closed under U-split coequalizers, extremal subobjects, and products. Thus it is an epi-relective subcategory.

It is well-known that the category o (small) categories is tripleable over the category o graphs. The ull subcategory o pre-orders is closed under surjections, arbitrary subobjects, and products and quotients or which the underlying graph morphism splits. It is not suicient that the set o nodes and the set o arrows split as sets; the splitting is required to preserve domain and codomain. Here is a relevant example: the regular epimorphism in the category o categories gotten by identiying cod with cod g in the pre-order g produces the category g which is not a pre-order because there are two maps between the two ends. Yet in the category o graphs, this regular epimorphism is surjective on both objects and arrows and those both are split epimorphisms in the category o sets, but not simultaneously in the category o graphs. Let C be the category o completely regular topological spaces. The category o topological groups is easily shown to be tripleable over C. There is a actorization system in which E consists o the maps with dense image and M o closed inclusions. Compact groups are clearly closed under U-split epics, M -subobjects, and products and hence the adjoint is E -relective. Another actorization system on topological groups consists o surjections and injections. The commutative topological groups are closed under U-split surjections, injective homomorphisms and products. Hence the ull subcategory o topological abelian groups is a surjective-relective subcategory o topological groups. 8 Department o Mathematics and Statistics, McGill University 805 Sherbrooke St. W., Montreal, QC, H3A 2K6 Email: barr@barrs.org