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Fix a fibration of categories. Suppose $f:A\to B$ is an arrow in the base.

What are the relations between the following pairs?

$$f\text{ epi}\qquad f^\ast \text{ faithful}$$

$$f\text{ mono}\qquad f^\ast \text{ full}$$

$$f\text{ strong epi}\qquad f^\ast \text{ conservative}$$

I don't mind completeness assumptions. Just trying to work out whether the behavior of the codomain fibration is "general".

Is there perhaps a reference for results of this kind?

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  • $\begingroup$ Just a thought: in the case of the codomain fibration, what makes a difference is often not only the properties you mention but also whether they are stable under pullbacks. So for a general fibration maybe you'd have to ask that the fibration reflects the property of $f$, at least for cartesian arrows. $\endgroup$
    – Arnaud D.
    Commented Oct 26, 2018 at 9:44
  • $\begingroup$ @ArnaudD. I am happy to add such assumptions, but since I don't know the theory of fibrations well, I am hoping for an answer which will explain them and their consequences. $\endgroup$
    – Arrow
    Commented Oct 26, 2018 at 10:24

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In short: There is absolutely no such relations, and in general essentially no properties of arrows of the codomain (except being a split epi/split mono or an iso) have any effect on the base change functor.

Indeed, because of Grothendieck's construction, absolutely any (pseudo)-functor from $X^{op}$ to $Cat$ is obtained as the base change functor corresponding to a Fibration of codomain $X$.

So you can send you epis and mono on whatever you like, you are asking if a completely general functor send epi or mono to some specific classes of maps. The only thing that are preserved are those that are expressed using justs "equations", so: "isomorphisms", "split epis" and "split monos" and that is essentially it.

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  • $\begingroup$ Thanks for the answer! I was just poking around in Jacobs' book on categorical logic and exercise 9.5.1 says that for a locally small fibration, epis downstairs induce faithful base change functors. If you happen to have the time to explain how to prove this, and perhaps similar results for locally small fibrations, I would really appreciate it. $\endgroup$
    – Arrow
    Commented Oct 26, 2018 at 16:22
  • $\begingroup$ A Hint/ very rough sketch: The definition of "Locally small" should says that given two objects $X,Y$ in a same fiber over $Z$, you have a certain objects in the codomain "representing morphisms between $X,Y$". You need to use the condition that the map your taking base change along is an epi with respect to morphisms to this specific objects and this will give you immediately that the base change is invective on morphisms between $X$ and $Y$. $\endgroup$
    – Simon Henry
    Commented Oct 26, 2018 at 16:33

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