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
I have no opinion on the notational question. I suppose the standard notation is backward, but I am not going to change now. I use d^0 and d^1, BTW, reserving the lower index for the dimension, which in this case is 1. If an equational category has a Mal'cev operator (a ternary operation t with t(x,x,y) = t(y,x,x) = y), then every simplicial object is Kan. This is essentially clear from John Moore's proof of the group case, in which forms xy^{-1}z appear repeatedly. For an equational category, the converse is also true, since a special case of every simplicial object being Kan is that every reflexive relation is an equivalence relation, a well-known characterization of Mal'cev. Most familiar Mal'cev categories have a group structure, so the Kan condition follows from the case for groups. Just about the only familiar Mal'cev category that I can think of that lacks a group op is Heyting algebras and I don't know any application of simplicial Heyting algebras. --Michael Barr
Todd Wilson asked on 3 Mar: 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? An answer may be found in A. Carboni, G.M. Kelly, and M.C. Pedicchio, Some remarks on Maltsev and Goursat categories, Applied Categorical Structures 1 (1993), 385-421. The slogan is "Kan = Maltsev". Max Kelly.
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
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
participants (5)
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James Stasheff -
Max Kelly -
Michael Barr -
MTHDUSKN＠ubvms.cc.buffalo.edu -
Todd Wilson