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Let $F:C \to D$ be a functor. Then, for all $c \in C_0$ we have an induced functor $F/c:C/c \to D/Fc$. Suppose that for each $c$, $F/c$ has a right adjoint. There's surely a name for such a functor. What is it and where can I read about them?

Note: I am not assuming that $C$ has a terminal object- in fact, the example I have in mind does NOT have one.

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    $\begingroup$ Locally a left-adjoint? $\endgroup$
    – Harry Gindi
    Commented Oct 17, 2010 at 23:38
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    $\begingroup$ Saying that $F$ has locally a right adjoint seems a reasonnable terminology to me. You might be interested by the paper of Maltsiniotis: "Structures d'asphéricité, foncteurs lisses, et fibrations", Ann. Math. Blaise Pascal 12 (2005), 1-39 (an English translation is available as arXiv:0912.2432). You will see that what you are looking at are the locally aspheric functors with respect to the minimal asphericity structure on the category of small categories (see Examples 1.2 and 1.7 as well as 1.17 and Corollary 3.11 in loc. cit.). $\endgroup$
    – D.-C. Cisinski
    Commented Oct 18, 2010 at 0:39
  • $\begingroup$ I guess you mean $Fc$, not $Fd$. What example do you have in mind? $\endgroup$
    – Todd Trimble
    Commented Oct 18, 2010 at 0:52
  • $\begingroup$ @Todd: Thanks, I'll fix that. And, the example I am looking at is $C_0$ is (a Grothendieck universe of) topological spaces and $C_1$ is local homeomorphisms, $D$ is $Sh(Top)$, and $F$ is "Yoneda". Note, that $C/X\cong Sh(X)$ and $Sh(Top)/X \cong Sh(Top/X)$, and under these equivalences, $F/X \cong j_!$, where $j:O(X) \to Top/X$ is the functor that sends $U \subset X$ to $U \to X$. $\endgroup$
    – David Carchedi
    Commented Oct 18, 2010 at 1:14
  • $\begingroup$ The usage of local here is standard, by the way (c.f. SGA 4 or M. Hakim's thesis). $\endgroup$
    – Harry Gindi
    Commented Oct 18, 2010 at 1:42

2 Answers 2

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I would call a functor $F : C \to D$ locally a left-adjoint if for every functor $G : I \to C$ ($I$ small), the induced functor $F : (C \downarrow G) \to (D \downarrow FG)$ is a left-adjoint. This is basically the assertion that every $(C \downarrow c) \to (D \downarrow Fc)$ is a left-adjoint plus a natural compatibility condition on the adjoints, which is probably given automatically in most examples.

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If the right adjoints are also a right inverses, than such functors are called (Grothendieck) fibrations. If the right adjoint are full and faithful, than such functors are called Street fibrations.

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  • $\begingroup$ What if instead, the each $F/c$ is fully faithful? $\endgroup$
    – David Carchedi
    Commented Oct 18, 2010 at 9:23

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