regular monos, epis and categories
I am a bit confused about what is or should be the correct definitions of regular monos, epis and categories. In the traditional examples from algebra there are plenty of limits and colimits around, so it doesn't much matter whether you say they're the (co)equalisers of arbitrary pairs or their (co)kernel pairs. Has anyone made use of these concepts in categories which have only the bare minimum (whatever that may be) of limits and colimits? Paul Taylor. =========================================================================
Here is what I think ought to be the definition in a general category. Say that f:A --> B is a regular epi if whenever g:A --> C is an arrow with the property that [(forall parallel pairs u,v with codmain A) (fu = fv ==> gu = gv)] ==> (exists! h:B --> C) (hf = g). If there is a kernel pair, this is equivalent to the more common definition. Obviously, the dual definition should be used for regular monics. Michael =========================================================================
Paul Taylor asks for the "correct" definition of regular monos and epis. I think a regular epi has always been an epi that appears as a coequalizer. The lemma is that if an epi has a pullback with itself then it is regular iff it is the coequalizer of the relevant pair of maps from that pullback (aka the "kernel pair", the "level", the "congruence"). But I can't imagine anyone insisting that split epis are regular only when they have kernel pairs. =========================================================================
Much as I try to get out of the habit of finding counterexamples to things, here, for what it's worth, is a category with a map satisfying Mike Barr's condition but which is not a coequaliser. * ====> * <==== * | \ / | \ / | | \ / f \ / | | X | X | | / \ | / \ | V V V V V V V * * * or, more prettily, $$\begin{diagram} \bullet&\pile{\rArr\\\rArr}&A&\pile{\rArr\\\rArr}&\bullet\ \dArr&\SE\SW&\dArr~f&\SE\SW&\dArr\ \bullet&&B&&\bullet \end{diagram}$$ ("=" above means a parallel pair. Believe it or not, I typed that without an editor!) A mono satisfying Mike's condition is invertible. Can't see how to relate it to orthogonality (in the sense of factorisation systems) with monos. What I had in mind was that someone may have used coequalisers in distributive categories to model "while". Only those coequalisers which, as relations and hence directed graphs (the node set being the target of the parallel pair), are directed arise in this way. Paul Taylor =========================================================================
To make my remark, ("I think a regular epi has always been an epi that appears as a coequalizer") compatible with Mike Barr's proposed definition just understand the word "coequalizer" to allow the case of a (joint) coequalizer of a family of pairs of maps. =========================================================================
What Peter says about coequalizers of a family of pairs is consistent with what I said so long as you allow a family to be large. As I wrote to PT earlier, these epics are strict, orthogonal to all monics and are coequalizers of their kernel pairs, if such exist. --Michael =========================================================================
My message related to Paul Taylor's question seems not to have been received since it has not yet appeared on the Bulletin Board: I recommend the following two papers in answer to this question of Paul Taylor: G.M. Kelly, Monomorphisms, epimorphisms, and pullbacks, J. Australian Math Soc Volume IX (1969) 124-142 P. Freyd and G.M. Kelly, Categories of continuous functors I, J. Pure Appl Algebra Volume 2 #3 (1972) 169-191 Regards, --Ross Street In any case, these references are also consistent with the comments of Mike Barr and Peter Freyd. --Ross =========================================================================
participants (4)
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barr＠fermat.Math.McGill.CA -
Paul Taylor -
pjf＠saul.cis.upenn.edu -
street＠macadam.mpce.mq.edu.au