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Stream: theory: category theory

Topic: discrete fibrations are left Beck-Chevalley

James Deikun (Apr 27 2022 at 14:52):

While further contemplating the analogy between topos theory and category theory I found this alternate characterization of discrete fibrations:

Given a functor $F : \mathcal{E} \to \mathcal{B}$ consider all strict pullback squares in $\bold{Cat}$

$$\begin{CD} \mathcal{E} \times_{\mathcal{B}} \mathcal{D} @>H>> \mathcal{D} \\ @V{K}VV @VV{G}V \\ \mathcal{E} @>>F> \mathcal{B} \end{CD}$$

and the corresponding squares in $\bold{Prof}$

$$\begin{CD} \mathcal{E} \times_{\mathcal{B}} \mathcal{D} @>{\mathcal{D}(1,H)}>> \mathcal{D} \\ @A{\mathcal{E}(K,1)}AA @AA{\mathcal{B}(G,1)}A \\ \mathcal{E} @>>{\mathcal{B}(1,F)}> \mathcal{B} \end{CD}$$

There is a canonical natural transformation $\theta$ from $\mathcal{D}(1,H) \mathcal{E}(K,1)$ to $\mathcal{B}(G,1) \mathcal{B}(1,F)$. If this is an isomorphism for every $G$, then $F$ is called left Beck-Chevalley. If all pullbacks of $F$ are left Beck-Chevalley, then $F$ is called stable left Beck-Chevalley. It so happens that the left Beck-Chevalley functors are the discrete fibrations and (due to stability of discrete fibrations) also the same as stable left Beck-Chevalley functors. Is this a known characterization of discrete fibrations?

Also, if $\theta$ is epic rather than iso for every $G$, $F$ is called weak left Beck-Chevalley and there's a corresponding stable weak left Beck-Chevalley. Are these notions used for anything in category theory?

James Deikun (Apr 27 2022 at 14:58):

(BTW I found these definitions in Elephant C2.4.16; I merely generalized them to an arbitrary equipment and reapplied them to $\mathbb{P}\bold{rof}$ ...)

James Deikun (Apr 27 2022 at 16:17):

This can be described as an [[exact+square]] condition but I'm having trouble saying exactly which squares must be exact since the variance on that nLab page is making my head spin, I'm sure something is wrong with either theirs or mine ...