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Let $\mathcal{C}$ a category with pullbacks. Does $\mathsf{cod}: \mathcal{C}^{\to}\to\mathcal{C}$ have any kind of universal property in the category of (co)fibrations over $\mathcal{C}$? I'd want it to be "terminal" in some way, or to play a role in factorizing certain (all?) (co)fibrations.

Note: I'm aware that $\mathsf{cod}$ can be easily interpreted via comma categories, but I'm not very familiar with them, so any pointer in that direction would be helpful, too.

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First a note about terminology: older literature sometimes uses the term "cofibration" for a functor $p:E\to B$ such that $p^{\rm op} : E^{\rm op} \to B^{\rm op}$ is a fibration, but it's more common nowadays to call this an "opfibration", with "cofibration" used instead for a functor whose corresponding morphism $B\to E$ in ${\rm Cat}^{\rm op}$ is a fibration. One reason for this is that opfibrations, like fibrations, are characterized by a lifting property, like fibrations in homotopy theory, whereas cofibrations (in the modern sense) are instead characterized by an extension property, like cofibrations in homotopy theory.

With that out of the way, the most basic universal property of $\rm cod$ is that it is the free opfibration over $C$ generated by ${\rm id} : C\to C$. In other words, if $p:E\to C$ is an opfibration equipped with a section $s:C\to E$, such that $p s = 1_C$, then there is an essentially unique map of opfibrations $\tilde{s} : C^{\to} \to E$. More generally, for any functor $f:A\to C$, the opfibration $A\times_C C^{\to}$ is the free opfibration generated by $f$. This doesn't even need the fact that $C$ has pullbacks.

A fancier universal property, which does use finite limits in $C$ and the consequent fact that $\rm cod$ is also a fibration, is that in the "universe" of fibrations over $C$, the codomain fibration is the "free coproduct completion of a point", in the same way that $\rm Set$ is the free coproduct completion of a point in the universe of ordinary categories. Here "coproduct complete" means having "indexed coproducts", meaning that a fibration is also an opfibration and satisfies the Beck-Chevalley condition. Thus, for any fibration $p:E\to C$ with this property, and any object $X\in p^{-1}(1)$, there is an essentially unique $g:C^{\to}\to E$ that is a morphism of both fibrations and opfibrations and maps $1_1$ to $X$.

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  • $\begingroup$ Thank you! This second property was exactly what I was looking for. Where can I read more about all of this? $\endgroup$
    – eta
    Commented Apr 20, 2021 at 9:56
  • $\begingroup$ I was afraid you were going to ask that. (-: I can't think of a good reference that proves that universal property in that particular case. There's a lot of general theory of fibrations (or equivalently indexed categories) in Part B of Sketches of an Elephant, including the notion of indexed coproducts, but I don't see this particular result there. It's closely related to Diaconescu's theorem over an arbitrary base (B3.3 of the Elephant), but with left-exactness omitted on both sides (and specialized to presheaves on the terminal internal category). $\endgroup$ Commented Apr 20, 2021 at 15:54
  • $\begingroup$ A more general theory (more or less) is developed in my paper Enriched indexed categories, with the universal property of presheaf categories being Theorem 10.33. The case of indexed categories is roughly the specialization of this to enrichment over the codomain fibration, but there are size and representability issues in making that completely precise (which are discussed in the paper). $\endgroup$ Commented Apr 20, 2021 at 15:57
  • $\begingroup$ It's possible some of the work on Yoneda structures could be specialized to the case of indexed categories and the result deduced from general facts about Yoneda structures. $\endgroup$ Commented Apr 20, 2021 at 15:59
  • $\begingroup$ Maybe after you work out the details, you can write them up and post them at ncatlab.org/nlab/show/codomain%20fibration. (-: $\endgroup$ Commented Apr 20, 2021 at 16:00

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