On Sat, 4 Mar 1995, categories wrote:
Date: Fri, 3 Mar 1995 18:50:19 -0500 From: Todd Wilson <twilson@CS.Cornell.EDU>
1. A question about notation: In accounts of internal category theory, one uses d_0, d_1 : C_1 -> C_0 as, respectively, domain and codomain, in analogy (I suppose) with simplicial objects. But then I would have expected it to be the other way around: in line with the analogy of an arrow f:A->B as an oriented one-simplex between vertices A and B , domain should rather be the face operator d_1 ("delete 1") and codomain d_0 ("delete 0"). Am I backwards?
2. The category Grp of groups has the property that every simplicial object in Grp satisfies the extension condition (i.e., is a so-called Kan complex). Is there a characterization of the categories with this property? Are there interesting uses of homotopy in these (other) categories?
--Todd Wilson
Re 1:No,You are absolutely correct if one wishes to use the usual simplicial conventions of "face opposite" in describing the simplices of t the simplicial object associated with the category : its "Nerve".which provides a most convenient numbering of the projections and compositions which occur there.However, it seems almost impossible to get people to give up what seems to them an illogical convention for arrows which must go from 0 to 1! Re 2: Barr characterised these categories as those which satisfy Malcev's condition.I have never seen his proof or know whether he ever published it.I have one of my own since it really is not too difficult. The observation was the ingenious part! Regards, Jack Duskin