The more I think about coequalisers the less I understand them. DOES ANYONE KNOW of a category in which REGULAR EPIS DO NOT COMPOSE? (Maybe the opposite of the category of commutative rings?) Here is a summary of the results of the previous discussion: I asked what a REGULAR EPI is. I asked this because the observation that all monos in a topos are regular is often made immediately after introducing the subobject classifier, by considering the characteristic map and constantly true. This is not a co-equivalence relation, but regular epis are usually defined in terms of equivalence relations. However if the kernel pair of a map exists, it coequalises this if it coequalises anything, so the definitions agree. You can't do much logic in a category without pullbacks, but Mike Barr and Peter Freyd gave me generalisations which are applicable without them. Categories of algebras were the original examples of REGULAR CATEGORIES. In these there is a factorisation of homomorphisms into regular epis (which are charactersed as surjections) and monos. This is stable under pullback, as is the class of regular epis, and also the class of diagrams which are kernels and coequalisers. Does a regular category have all coequalisers? The definition in the literature is ambiguous, but I am inclined to agree with that in "Categories, Allegories" by Peter Freyd and Andre Scedrov (North Holland 1992), namely that they do not. Although categories of algebras have all coequalisers, and regular epis are stable, coequaliser diagrams need not be. Here is a counterexample in the category of groups given to me by Peter Freyd. 2 --------> 2 2 >-----> 2 2 ---->> 2 ---->> 1 | | | --------> | | | | -- | | | -- | | | -- | | | | | | | V | | V 2 --------> V V 2 >-----> 2 A -------------->> 3 --------> 4 Here 2^2 is the four-element normal subgroup of the group of even permuations on four symbols. The parallel pair consists of the inclusion and the map which interchanges (12)(34) with (13)(24). The equalisers and coequalisers are illustrated and the squares are pullbacks in the obvious way. There are several reasons why I am interested in all of this. One of them is the interpretation of WHILE PROGRAMS using coequalisers of functional relations, which must be stable under pullback. I would like to be able to embed such a category in a topos. There is an obvious Grothendieck topology, and the inclusion sends the coequalising maps to epis, but why should it preserve the coequaliser DIAGRAMS? (For a draft of a paper on while programs, see /theory/papers/Taylor/while-1993.dvi at theory.doc.ic.ac.uk) The other reason is an application to SYNTHETIC DOMAIN THEORY. See my paper in LICS 1991 (also /theory/papers/Taylor/lics.dvi) for the background. Eugenio Moggi asked for a good class of monos (see /theory/papers/Moggi/ELT.dvi) and I was trying to show that what I called "extremal" monos in my LiCS paper coincided with those which arise as equalisers of parallel pairs into powers of Sigma. Somewhat to my surprise, I discovered yesterday that in any category with kernel pairs and coequalisers of them (NOT NECESSARILY STABLE), a map is mono iff it is orthogonal to all regular epis where "orthogonal" means that the universal property for factorisation systems is satisfied. Moreover if regular epis compose, (reg.epi)-(mono) is a factorisation Hence the question above. Stability of regular epis is sufficient but not necessary to make them compose. For SDT I am specifically interested in "Sigma-epis" instead of monos (maps which internally satisfy the cancellation property for parallel pairs into the object Sigma, rather than all objects). For reasons concerned with Eugenio Moggi's Evaluation Logic, we want Sigma-regular monos to be "topological subspaces", ie that maps to Sigma extend along the mono. This is also sufficient to make regular monos compose. Paul Taylor +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Thanks to Art Stone in particular and also Mike Barr, Max Kelly, Dusko Pavolovic and Walter Tholen for their helpful comments, which I still have to study. I wonder whether any specialists in group theory or commutative algebra might be able to find a single example of non-composing regular monos in the category of groups or of commutative rings. Something like: (1) a normal subgroup of a centraliser in a simple group, or (2) A chain of subrings R<S<T such that s # 1 = 1 # s in the tensor S *_R S implies s in R t # 1 = 1 # t in the tensor T *_S T implies t in S but s # 1 = 1 # s in T *_R T does not imply s # 1 = 1 # s in T *_S T This would avoid "theological" objections to discussing on-the-nose colimits in the 1-category of categories. Paul Taylor +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
Paul Taylor (pt@doc.ic.ac.uk) asks:
I wonder whether any specialists in group theory or commutative algebra might be able to find a single example of non-composing regular monos in the category of groups or of commutative rings.
Alas, every inclusion of a subgroup (monomorphism of groups) is an equalizer "for free" (a regular monomorphism), so there's no example of the desired sort in {Groups} . (The argument I know is one I learned from Eilenberg several decades ago; it uses a suitable permutation group as the target of two homomorphisms (from the larger group) whose equalizer is precisely the given subgroup. Cheers. -- Fred +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
It has been known for a long time that all monomorphisms in the category of groups are regular, therefore the regular monos compose. I do not know who proved it first. It can be shown directly by representing any inclusion map, say from H to G, as an equaliser of two arrows from G to the group of all bijections from the set of all left cosets modulo H to itself. Another way is to look at the usual descriptions of amalgamated products (i.e. poushouts of pairs of monos); then every mono m can be seen to be the equaliser ofthe two maps forming the pushout of m with itself. In the category of rings (commutative or not, unital or not), regular monos do not compose. The problem is stdied in detail in John Isbell's paper "Epimorphisms and dominions", La Jolla Proceedings, 1965. Similar arguments work for monoids or semigroups. A nice and easy, but seemingly not very well-known example is given by Gabriel and Ulmer (LNM 221): let N be the additive monoid of non-negative integers, let A be the submonoid generated by 3 and 5 and B the submonoid generated by 3, 5, and 7. Then the inclusions from A to B and from B to N can easily be see to be regular monos, but their composite is not, due to the following nice computation: assume f(3)=g(3) and f(5)=g(5) for some morphism from N to some monoid (multiplicatively written). Then f(7)=f(2)f(5)=f(2)g(5)= f(2)g(3)g(2)=f(2)f(3)g(2)=f(5)g(2)=g(5)g(2)=g(7). Gabriel and Ulmer also prove the (now?) well known fact that in any category with finite limits regular epis compose, if they are stable under pullback, and they even state their result more generally. Only recently, I realized that their result easily implies that the class of regular epimorphisms stable under pullback (i. e. of all morphisms e such that every pullback of e is a regular epi) is aways closed under compostion, even if regular epis do not compose. Greetings Reinhard Boerger +++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
participants (3)
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Fred E.J. Linton -
Paul Taylor -
Reinhard Boerger