Date: Wed, 9 Nov 1994 15:55:52 +0400 (GMT+4:00) From: categories Subject: Fibrations<-->Adjunctions???? Date: Wed, 9 Nov 1994 19:10:37 +0100 From: Maura Cerioli Hi, I need to use the following result about the equivalence between the existence of a fibration and the existence of a right adjoint to an opportune functor between comma categories: ----------------- Let A and B be categories and let U:A-->B be a functor. Let C denote the comma category A arrow (whose objects are the arrows of A) and let D denote the comma category B over U (whose objects are arrows in B from a generic object b into U(a), together with a). Then U induces a functor V:C-->D, that basically is U (as both objects and arrows in C are arrows in A too) and is defined as follows: * V(f:a-->a')=(U(f):U(a)-->U(a'),a') for each object f:a-->a' in A; * V(g,g')=(U(g),g') for each arrow (g,g'):(f1:a1-->a'1)----->(f2:a2-->a'2) in A. Then U is a fibration iff V has a right adjoint G:D-->C s.t. G;V is the identity on D and the counit of the adjunction is the identity natural transformation. ----------------- Now, I know (or at least I believe) that the result holds, as I have a proof for it; but a collegue said to me that this is a known result and suggested to look at the works by Street in the Sydney Category Seminar. Unfortunately, being quite ignorant about 2categories, I find very hard even to understand in which sense the result I want is there. Does anybody know an easier reference, possibly not involving 2categories? Thanks Maura Maura Cerioli DISI - Dipartimento di Informatica e Scienze dell'Informazione Viale Benedetto XV, 3 16132 Genova ITALY Date: Thu, 10 Nov 1994 18:53:32 +0400 (GMT+4:00) From: categories Subject: Re: Fibrations<-->Adjunctions???? From: Jim Otto To: categories@mta.ca (the message being replied to is appended.) dear Maura Cerioli yes, the Street paper is hard to read. but he refers to a paper of Gray's which has the result you mention. in fact Gray credits it to a manuscript of Chevalley's which was lost in the mail. while i didn't read your description of the result, the way i think of it is as follows. you have a functor p: E --> B this induces a functor (with comma category B/p relative to id: B --> B <-- E :p and with 2 the ordinal) p.: E^2 --> B/p which takes `paths in E' to `paths in B with the to end in E remembered' then (modulo the axiom of choice) p is a fibration iff p. has a right adjoint c. with identity couint. c. is then a cleavage. this unfolds (viewing c. as terminal objects in comma categories) to the usual elementary definition in terms of cartesian maps. Gray says p. is rari, i.e. has a right adjoint right inverse. one can show that this isn't weaker. as i recall, Street's definition is weaker in order to be more invariant. (cod: B^2 --> B is an important example. then c. is pull-back.) actually i need fibrations in 2-categories of monoidal categories. note that, in cat the 2-category of categories, E^2 is the comma category E/E (relative to id: E --> E <-- E :id). further, still in cat, E^2 is the cotensor 2 -o E. but in any 2-category, finite weighted limits, including comma objects, can be built up from cotensors with 2 and (2-) terminal objects and pull-backs. to help you read Street, i now define cotensor with 2. 2-categories have hom categories rather than hom sets. 2 -o E is defined by isomorphisms hom(W, 2 -o E) iso 2 -o hom(W, E) (2-) natural in W. (sometimes such isomorphisms don't exist, but such equivalences do.) bon jour, Jim Otto >Date: Wed, 9 Nov 1994 19:10:37 +0100 >From: Maura Cerioli > >Hi, >I need to use the following result about the equivalence between the >existence of a fibration and the existence of a right adjoint to >an opportune functor between comma categories: >----------------- >Let A and B be categories and let U:A-->B be a >functor. >Let C denote the comma category A arrow (whose objects are the arrows >of A) >and >let D denote the comma category B over U (whose objects are arrows in B >from a generic object b into U(a), together with a). >Then U induces a functor V:C-->D, that basically is U (as both objects >and arrows in C are arrows in A too) >and is defined as follows: >* V(f:a-->a')=(U(f):U(a)-->U(a'),a') for each object > f:a-->a' in A; >* V(g,g')=(U(g),g') for each arrow > (g,g'):(f1:a1-->a'1)----->(f2:a2-->a'2) in A. > >Then U is a fibration iff V has a right adjoint G:D-->C s.t. >G;V is the identity on D and the counit of the adjunction is >the identity natural transformation. >----------------- >Now, I know (or at least I believe) that the result holds, as >I have a proof for it; but a collegue said to me that this is >a known result and suggested to look at the works by Street >in the Sydney Category Seminar. >Unfortunately, being quite ignorant about 2categories, I find very >hard even to understand in which sense the result I want is there. > >Does anybody know an easier reference, possibly not involving >2categories? Date: Thu, 10 Nov 1994 12:58:07 +0400 (GMT+4:00) From: categories Subject: Re: Fibrations<-->Adjunctions???? Date: Wed, 9 Nov 1994 21:22:16 +0000 (GMT) From: Dusko Pavlovic The result goes back to Claude Chevalley's seminar. (One direction does require the Axiom of Choice.) I think a simple formulation is in J.Gray's paper on fibrations in the proceedings of La Yolla conference, Springer 1965. -- Dusko Pavlovic