Date: Sat, 4 Mar 1995 00:49:32 -0400 (AST) From: categories To: categories Subject: A couple of simplicial questions Date: Fri, 3 Mar 1995 18:50:19 -0500 From: Todd Wilson 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 Date: Sun, 5 Mar 1995 23:57:00 -0400 (AST) From: categories To: categories Subject: Re: A couple of simplicial questions Date: Sat, 4 Mar 95 14:44:07 EST From: Michael Barr 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 Date: Thu, 9 Mar 1995 01:27:15 -0400 (AST) From: categories To: categories Subject: Re: A couple of simplicial questions Date: Thu, 9 Mar 95 14:47:51 +1000 From: Max Kelly 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. Date: Fri, 10 Mar 1995 02:48:15 -0400 (AST) From: categories To: categories Subject: some simplicial questions Date: Thu, 09 Mar 1995 12:09:04 +0000 From: Prof R. Brown The nerve of a groupoid (not just a group) is also a Kan complex in the strong sense of a simplicial T-complex (Keith Dakin's definition, 1975). A simplicial T-complex (K,T) has sets T_n in K_n of elements called thin with the property that: 1) any horn has a unique thin filler 2) degenerate implies thin 3) if all faces but one of a thin element are thin, so is the last face. This is of rank less than or equal to n if all simplices in dimensions greater than n are thin. Simplicial T-complexes of rank 1 are equivalent to groupoids (Dakin). Simplicial T-complexes in general are equivalent to crossed complexes (Ashley, see Diss Math 265 (1988) Simplicial T-complexes and crossed complexes; Nan Tie, JPAA 56 (1989) 195-209), to infinity-groupoids, and to other things. The notion of cubical T-complex is crucial in the proof by Brown-Higgins of an n-dimensional Van Kampen Theorem (JPAA 22 (1981) 11-41), because of the technical point that it easily enables the handling of multiple compositions of homotopy addition lemmas (the boundary of a simplex or cube is the "sum" of its faces) without writing down formulae. For polyhedral versions, see D W Jones, Diss Math 266 (1988) A general theory of polyhedral sets and the corresponding T-complexes.) Part of the point is that the singular complex SX of a space X is a Kan complex but the fillers come from the models, so ought, morally, to be canonical and to satisfy relations. But these relations are up to homotopy, it seems. So it is difficult to be more precise. The T-complex condition is very strong, but is nice in that it is easy to see how to weaken it. These weakenings need lots more investigation. A further point is that a filler of a horn of a triangle determines a product, as if it were a `computation'. Analogous notions for categories are studied by Ross Street and by Dominic Verity. Ronnie Brown Prof R. Brown Tel: (direct) +44 248 382474 School of Mathematics (office) +44 248 382475 Dean St Fax: +44 248 355881 University of Wales email: mas010@bangor.ac.uk Bangor wwweb for maths: http: //www.bangor.ac.uk/ma Gwynedd LL57 1UT UK Date: Sat, 11 Mar 1995 01:20:40 -0400 (AST) From: categories To: categories Subject: Re: A couple of simplicial questions Date: Fri, 10 Mar 1995 17:02:09 -0500 (EST) From: MTHDUSKN@ubvms.cc.buffalo.edu On Sat, 4 Mar 1995, categories wrote: > Date: Fri, 3 Mar 1995 18:50:19 -0500 > From: Todd Wilson > > 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 Date: Tue, 14 Mar 1995 22:28:05 -0400 (AST) From: categories To: categories Subject: Re: A couple of simplicial questions Date: Mon, 13 Mar 1995 09:27:43 -0500 (EST) From: James Stasheff 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 > > 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 > > Date: Tue, 14 Mar 1995 22:41:06 -0400 (AST) From: categories To: categories Subject: Kan and Maltsev Date: Tue, 14 Mar 95 14:31:15 +1000 From: Max Kelly Subject: Kan and Maltsev Recently, Todd Wilson asked for a characterization of those categories wherein every simplicial object is Kan; and Michael Barr replied that this is so for a variety if and only if it admits a Maltsev operation, whereupon it suffices to imitate John Moore's original proof for the category of groups. Jack Duskin also recalls this unpublished observation of Michael's, remarking that he has his own proof. I noted that a complete proof has been published in [A.Carboni, G.M.Kelly, and M.C.Pedicchio, Some remarks on Maltsev and Goursat categories, Applied Categorical Structures 1(1993), 385-421]. I mention this again, since I failed to point out that our paper shows the equivalence Kan = Maltsev for any REGULAR category, where "Maltsev" now means that equivalence relations commute. The proof is pretty; it does follow the lines of John Moore's, but replaces his calculations with elements by reasonings about equivalence relations. Max Kelly.