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Suppose $X:D \to C$ is a fibered category (I do not assume the fibers to be groupoids). Suppose that $X$ is actually left adjoint to a fully faithful embedding $C \hookrightarrow D$. Is there a special name for such a fibration $X$?

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I don't know any specific name for this. However, it seems that such a property can be reformulated as follows: for each object $c$ of $C$, the fiber $X^{-1}(c)=D_c$ has a terminal object, and the collection of these terminal objects defines a cartesian section of $X$. So, for $C$ the terminal category, you are asking for a special name for "category with a terminal object". So you are talking somehow of "relative terminal object". Another suggestion: just say that $X$ has a cartesian right adjoint (i.e. a right adjoint as a $1$-cell of the bicategory of fibred categories over $C$). – Denis-Charles Cisinski Sep 23 '10 at 20:05
@Denis: Thanks for the insight. Indeed, what you say holds for the example I have (though I secretly am looking at a fibered bicategory). It's seems like an interesting property anyway- the category D, as a fibered category over C, should (and obviously can since it's the Grothendieck construction of something) be thought of as objects of C with extra structure (or data), so to have the fibration admit a fully-faithful right-adjoint is saying that you can localize (or reflect) this extra data away. – David Carchedi Sep 23 '10 at 21:45
Perhaps there is a term specific to fibrations but, in general, left adjoints to embeddings are reflection functors or "reflectors." – Jeremy Brazas Sep 24 '10 at 11:40
@Jeremy: Thanks. In fact I am aware that this is an example of a reflective subcategory (see my above comment). I'm just wondering whether such reflectors which are also fibrations have been studied, since I have stumbled upon an off example. – David Carchedi Sep 24 '10 at 13:08
(I meant "odd", not "off") – David Carchedi Sep 24 '10 at 16:53

There is a useful simple result related to this question:

Suppose $X: C \to D$ is a fibred category, such that $X$ has a right adjoint, and $C$ is cocomplete. Then for each object $J$ of $D$ the inclusion $C_J \to C$ of the fibre over $J$ into $C$ preserves colimits of connected diagrams. See the proof of Theorem B.1.7 on p. 579 of Nonabelian algebraic topology EMS Tracts in Mathematics Vol 15 (pdf downloadable from my web page). Actually the conclusion is true without the assumptions, but this useful case has a short proof, given there.

This usefully applies to pushout diagrams.

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Ronnie, if you register your account, your various MO questions and answers will be compiled under that account and reflect the positive contributions you have made here at MO. See also comment 148 in this discussion: – Todd Trimble Dec 16 '11 at 12:15

Isn't the answer simply that the fibration has fibred terminal objects?

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Yes, I think you're right! – Mike Shulman May 6 '12 at 20:45

I don't know how widespread this terminology is, but at least some people call this a fibration "with codiscrete objects". See this blog post for instance. An interesting fact is that if $C$ has and $X$ preserves finite limits, then $X$ is a (weak) fibration if and only if it has a fully faithful right adjoint.

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