From mboxrd@z Thu Jan 1 00:00:00 1970 X-Msuck: nntp://news.gmane.io/gmane.science.mathematics.categories/8315 Path: news.gmane.org!not-for-mail From: Richard Garner Newsgroups: gmane.science.mathematics.categories Subject: Re: Is the category of group actions LCCC? Date: Fri, 05 Sep 2014 11:05:12 +1000 Message-ID: References: Reply-To: Richard Garner NNTP-Posting-Host: plane.gmane.org Mime-Version: 1.0 Content-Type: text/plain; charset="utf-8" Content-Transfer-Encoding: quoted-printable X-Trace: ger.gmane.org 1409921093 17704 80.91.229.3 (5 Sep 2014 12:44:53 GMT) X-Complaints-To: usenet@ger.gmane.org NNTP-Posting-Date: Fri, 5 Sep 2014 12:44:53 +0000 (UTC) Cc: categories@mta.ca To: Ross Street , Timothy Revell Original-X-From: majordomo@mlist.mta.ca Fri Sep 05 14:44:47 2014 Return-path: Envelope-to: gsmc-categories@m.gmane.org Original-Received: from smtp3.mta.ca ([138.73.1.127]) by plane.gmane.org with esmtp (Exim 4.69) (envelope-from ) id 1XPssg-0008Fh-SM for gsmc-categories@m.gmane.org; Fri, 05 Sep 2014 14:44:47 +0200 Original-Received: from mlist.mta.ca ([138.73.1.63]:42519) by smtp3.mta.ca with esmtp (Exim 4.80) (envelope-from ) id 1XPss1-0004Dm-Pe; Fri, 05 Sep 2014 09:44:05 -0300 Original-Received: from majordomo by mlist.mta.ca with local (Exim 4.71) (envelope-from ) id 1XPss0-0005NG-6P for categories-list@mlist.mta.ca; Fri, 05 Sep 2014 09:44:04 -0300 Precedence: bulk Xref: news.gmane.org gmane.science.mathematics.categories:8315 Archived-At: It seems that the following is in fact true: Let p: E ----> B be a fibration. If B is cartesian closed, each fibre is cartesian closed with exponents stable under pullback, and Pi's exist along product projections (and satisfy BCC), then E is cartesian closed. The product of (a, phi) with (b, psi) in E is of course (a x b, pi_1^*(phi) x pi_2^*(psi)) with pi_1 : a <--- a x b ----> b : pi_2 the product projections in B. The internal hom [(b, psi), (c, gamma)] is Pi_{pi_1} [pi_2^*(psi), ev^*(theta)], where pi_1 : [b,c] <---- [b,c] x b ----> b : pi_2 and=20 ev: [b,c] x b ----> c in B. This in particular applies to Cat//=E2=80=99Set=E2=80=99 as in Ross' messag= e, seen as a fibration over Cat with reindexing along f:A--->B given by [f,1]:[B,Set]--->[A,Set]. This fibration has right adjoints to pullbacks, but they don't satisfy BCC; however, right adjoints to pullback along product projections are given just by (conical) limit functors, and these do satisfy BCC. So the preceding construction applies (and a bit of fiddling about shows that this does indeed agree with Ross' prescription). As for local cartesian closure: if B is lccc, each fibre is lccc with fibrewise Pi's stable under pullback, and E--->B has all products, then it seems that each slice fibration p/A: E/A--->B/pA will satisfy the conditions in the second paragraph, whence E is also lccc. Richard On Thu, Sep 4, 2014, at 10:19 AM, Ross Street wrote: > On 1 Sep 2014, at 7:12 pm, Timothy Revell > wrote: >=20 >> I'm wondering whether the category of ALL group actions is locally >> Cartesian closed.=20 >=20 > This is what I answered Timothy: > =3D=3D=3D=3D=3D=3D > No, it=E2=80=99s not. > Since the category has a terminal object (1,1), being a LCCC would imply > it > was cartesian closed. However, that would imply (G,X) \times =E2=80=94 pr= eserved > the initial object (1,0), which is false: (G,X)\times (1,0) =3D (G,0). > =3D=3D=3D=3D=3D=3D >=20 > But it seems there is more to the story.=20 > The thing stopping the category of actions from > being cartesian closed is that the category Gp of groups is not. However, > the category Gpd of groupoids and the category Cat of categories are. > The (2-)category Cat//=E2=80=99Set=E2=80=99 of all category actions is de= fined as > follows: > objects (A,F) are functors F : A =E2=80=94> Set and morphisms (f,t) : (A,= F) =E2=80=94>=20 > (B,G) > are functors f : A =E2=80=94> B with natural transformation t : F =3D=3D>= G f. > This (2-)category is cartesian closed: the internal hom [(B,G),(C,H)] is > ([B,C], K) where [B,C] is the functor category and K(g) =3D [B,Set](G, H > g). >=20 > However Cat//=E2=80=99Set=E2=80=99 is not locally cartesian closed basica= lly because Cat > is not. It is not even locally cartesian closed as a bicategory. > The 2-category Gpd is cartesian closed; it is not locally cartesian > closed; > it is locally cartesian closed as a bicategory.=20 >=20 > Similarly, Gpd//=E2=80=99Set=E2=80=99 is locally cartesian closed as a bi= category. > Often, in dealing with groups, we find groupoids help. > This case is a good example and I hope helps in the applications > you have in mind, Timothy. >=20 > Ross >=20 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]