Let $Grpd$ be the category of groupoids and $p:E\rightarrow B$ a fibration in the standard model structure on $Grpd$ (ie an isofibration). How do you prove that the pullback functor $p^{\star}:Grpd/B \rightarrow Grpd/E$ has a right adjoint ?
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$\begingroup$ Do you happen to know that it has a right adjoint? $\endgroup$– Fernando MuroCommented Dec 20, 2013 at 17:52
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$\begingroup$ It seems it's true but I was not able to find a demonstration. Especially it would be nice to have an explicit and elementary construction of this right adjoint even if I would be also grateful for a theorem that solves the question. $\endgroup$– user2664Commented Dec 20, 2013 at 18:03
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$\begingroup$ So is the existence of a right adjoint part of the question? If so, it would be nice if you could edit your question. $\endgroup$– Fernando MuroCommented Dec 20, 2013 at 18:04
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1$\begingroup$ It's kind of strange. If you know it's true, then you must know some proof, but you still ask about it, why is it so? $\endgroup$– Fernando MuroCommented Dec 20, 2013 at 20:34
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3$\begingroup$ If you read it somewhere you should include that reference as part of the question. That way others would know where to look and what to reference when formulating an answer $\endgroup$– David WhiteCommented Dec 20, 2013 at 21:00
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1 Answer
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Here are some references: Theorem 4.4 on p.40 of:
Giraud, Jean Méthode de la descente. Bull. Soc. Math. France Mém. 2 1964
which is available from Numdam. The result was rediscovered by F. Conduché,
F. Conduché, Au sujet de l'existence d'adjoints à droite aux foncteurs "image réciproque" dans la catégorie des catégories, C. R. Acad. Sci. Paris 275 (1972), A891–894.
and developed for crossed complexes in
Howie, James Pullback functors and crossed complexes. Cahiers Topologie Géom. Différentielle 20 (1979), no. 3, 281–296.
See also
Bunge, Marta; Niefield, Susan Exponentiability and single universes J. Pure Appl. Algebra 148 (2000), no. 3, 217–250.
Update: I'd like to add an amusing application of the result on fibrations of groupoids and pullbacks. An epimorphism of groups, say $p: E \to B$, is a special case of a fibration of groupoids. So the pullback functor $p^*$ from groupoids over $B$ to groupoids over $E$ preserves colimits. Now the inclusion of categories $\mathbf{ Groups} \to \mathbf{ Groupoids} \;\;$ preserves colimits of connected diagrams. It follows that the pullback functor $p^*: \mathbf{ Groups}/B \to \mathbf{ Groups}/E\;\; $ preserves colimits of connected diagrams. In particular, it preserves pushouts. (This was published with P.R. Heath in "Lifting amalgamated sums and other colimits of groups and topological groups'', Math. Proc. Camb. Phil. Soc. 102 (1987) 273-280.)
answered Dec 21, 2013 at 15:33
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$\begingroup$ Thanks Ronnie for these helpful references, unfortunetly as I said in my previous comments I know this result and in particular this reference but my problem is to understand it. I must confess that I'm a bit lazzy and afraid by Giraud's paper which uses old notations. $\endgroup$– user2664Commented Jan 8, 2014 at 13:35
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$\begingroup$ @Asymrotik: I have a more down-to-earth paper by Heath and Kamps, which is a republication of a previous article of theirs in 1988 conference on Categorical Topology in Prague Czechoslovakia. I can email it to you if you send me an email. Susan Niefeld also has some sophisticated and more general treatments, and she told me of the Giraud reference. $\endgroup$ Commented Jan 9, 2014 at 22:22
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$\begingroup$ @Asymptotik: See my previous comment, misadressed. $\endgroup$ Commented Jan 10, 2014 at 22:51
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$\begingroup$ Here is the Heath-Kamps paper: pages.bangor.ac.uk/~mas010/pdffiles/heath-kamps-LiftCol.pdf Hope it is of use. $\endgroup$ Commented Jan 12, 2014 at 10:30