category theorists are well known to be of the opposite handedness/orientation Jim Stasheff jds@math.unc.edu Math-UNC (919)-962-9607 Chapel Hill NC FAX:(919)-962-2568 27599-3250 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