From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/7006 Path: news.gmane.org!not-for-mail From: "F. William Lawvere" Newsgroups: gmane.science.mathematics.categories Subject: Re: Simplicial versus (cubical with connections) Date: Wed, 26 Oct 2011 17:27:49 -0400 Message-ID: References: Reply-To: "F. William Lawvere" NNTP-Posting-Host: lo.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="iso-8859-1" Content-Transfer-Encoding: quoted-printable X-Trace: dough.gmane.org 1319716934 6508 80.91.229.12 (27 Oct 2011 12:02:14 GMT) X-Complaints-To: usenet@dough.gmane.org NNTP-Posting-Date: Thu, 27 Oct 2011 12:02:14 +0000 (UTC) Cc: categories To: , , Original-X-From: majordomo@mlist.mta.ca Thu Oct 27 14:02:09 2011 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtpx.mta.ca ([138.73.1.4]) by lo.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1RJOev-0007Ti-9r for gsmc-categories@m.gmane.org; Thu, 27 Oct 2011 14:02:09 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:33065) by smtpx.mta.ca with esmtp (Exim 4.76) (envelope-from ) id 1RJOd9-0004Cd-OO; Thu, 27 Oct 2011 09:00:19 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1RJOd8-0002LT-1u for categories-list@mlist.mta.ca; Thu, 27 Oct 2011 09:00:18 -0300 Importance: Normal In-Reply-To: Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:7006 Archived-At: Trivial objects should NOT be admitted to categories A whose set valued fun= ctors should form a combinatorial topos intended as a surrogate for contin= uous spaces in some sense.That is to say=2C there is a reason why the delta= of simplicial sets does not have a unit object (unlike the delta that as a= strict monoid in CAThas precisely all monads as its actions). Similarly th= e basic cubical sets are functors on the part A of the algebraic category o= f two nullary operations which consist of finitely presented STRICT algebra= s=3B thus this is the classifying topos for bipointed objects (in arbitrary= toposes) toposes that satisfy the non-equational entailment csub0 =3Dcsub= 1 entails false.The reason is this: if a site C=3DAop has an initial object= =2C then the leftadjoint to the inclusion of constant functors =2C which s= hould model the notion of connected components=2C is representable =2C hen= ce preserves equalizers=3B but the basic intuitive examples of spaces withn= on-trivial higher connectivity are constructed as equalizers betweenconnect= ed spaces ! ( Note that restrictions (on the structures classified) involving falsity= =2C disjunction=2C or existential quantification typically give sheaf topos= esbut exceptionally may just give smaller presheaf categories =3B another e= xample is in algebraic geometry where classifying the algebrassubject to th= e disjunctive conditionx^2=3Dx entails x=3D0 or x=3D1merely involves the to= pos of all presheaves on those fp algebras satisfying the same condition.) According to the paradigm set by Milnor=2C the relation between continuous = and combinatorial is a pair of adjoint functors called traditionally singul= ar and realization . ("Singular"=2C as emphasized by Eilenberg=2C means tha= t the figures on which the combinatorial structure of a space lives should = not be required to be monomorphisms=2Cin order that in order that that stru= cture should be functorial wrt all continuous changes of space =3B "realiza= tion" refers to a process analogous to to the passage from blueprints to a= ctual buildings of beton and steel).As emphasized by Gabriel and Zisman=2C = the exactness of realizationforces us to refine the default notion of space= itself=2C in the directionproposed by Hurewicz in the late 40s and well-de= scribed by J L Kelley in 1955. Further refinements suggest that the notion = of continuous could well be taken as a topos=2C of a cohesive (or gros) kin= d.The exactness of realization is a example of the striving to make the sur= rogate combinatorial topos (=3D having a site with finite homs ???) describ= e the continuous category as closely as possible. For example the finite pr= oducts of combinatorial intervals might be required to admit the diagonal m= aps that their realizations have. There is one point however where perfect agreement cannot be achieved ( Is = this a theorem?) : the contrast between continuous and combinatorialforced = Whitehead to introduce a specific notion he called weak equivalence=2C as e= xplained by Gabriel-Zisman=2C in order to extract the correct homotopy cate= gory. The contrast can readily be read off of my list of axioms for Cohesio= n (TAC) : the reasonable combinatorial toposes satisfy all but one of the a= xioms=2Cbut only the continuous examples satisfy it. That Continuity axiom = (preservationof infinite products by pizero) was introduced in order to obt= ain homotopytypes that are "qualities" in an intuitive sense (as they shou= ld be automaticallyin the continuous case).=20 > Date: Sat=2C 22 Oct 2011 09:07:59 -0400 > From: trimble1@optonline.net > Subject: categories: Re: Simplicial versus (cubical with connections) > To: ross.street@mq.edu.au=3B pratt@cs.stanford.edu > CC: categories@mta.ca >=20 > My impression is that there are at least two distinct notions of > cubical set which have entered this discussion. One version > describes cubical sets as presheaves on the Lawvere theory > generated by two constants or 0-ary operations=3B this is close > to what Vaughan described. More precisely=2C instead of taking > the category whose objects are finite sets equipped with two > distinct points (which is opposite to the Lawvere theory)=2C he > adds in a terminal object (where the two constants are forced > to coincide)=2C giving a category C. Anyway=2C whether one takes > the Lawvere theory or C^{op}=2C the result is a category with > finite cartesian products and an interval object=2C and one notion > of cubical set is that of presheaf on this category. >=20 > Whereas cubical sets in the sense described by Ross are > different: they are presheaves on the free *monoidal* category > with an interval object. This category does not include diagonal > maps. I expect this is the notion of cubical set that Dmitry and > Urs were actually concerned with=2C but in any event=2C both the > cartesian version and the monoidal version of the cubical site > appear in the literature=2C and it is important to clarify which > notion is meant. >=20 > Todd Trimble >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]