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Stream: learning: questions

Topic: Why are fully faithful functors stable under pushouts?

Nathanael Arkor (Nov 13 2025 at 19:14):

It is known that fully faithful functors are stable under pushout, in the sense that if we have a pushout square of functors:
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if $g$ is fully faithful, then so is $h$ (e.g. see the references on [[embedding of categories]]). I am aware of two flavours of proof for this result: the first makes use of the concrete description of a pushout of categories (which is rather involved), and the second makes use of the presheaf construction to transform the pushout to a pullback, and then analyse that.

I am interested in analogues of this result in which the structures are more complicated than categories (meaning a concrete description of the pushout is not appealing) and for which there is not an analogue of the presheaf construction. It would therefore be useful to have a different abstract proof of the stability of pushouts in Cat. Does anyone know of any other arguments that establish this property?

fosco (Nov 13 2025 at 19:45):

I have no idea if it works, but have you tried with the idea that sometimes, in a category $C$ with a factorization system $(E,M)$, $E$ is stable under pullback? The dual condition is pushout stability of $M$ in the sense you need, when $C=Cat$ and $(E,M)$ is the bo-ff factorization

fosco (Nov 13 2025 at 19:46):

I suspect this path makes things even more difficult, if you don't have access to a way to ensure that the FS you generate, provided it exists, has a pushout-stable right class.

Nathanael Arkor (Nov 14 2025 at 07:17):

Do you have any conditions in mind that guarantee pushout stability?

Nathanael Arkor (Nov 14 2025 at 07:19):

I had thought perhaps there could be a weak factorisation system in which the right class was the fully faithful functors (similarly to those in §4.4 of Bourke–Garner's Algebraic weak factorisation systems I: accessible AWFS), but I'm not familiar with a tool to check this.

James Deikun (Nov 14 2025 at 08:55):

I thought at first there would be a factorization system following the above exactly, but it seems not, because fully faithful morphisms are not always exponentiable in Cat. (This became obvious upon looking at more than just the classifying morphism.)

James Deikun (Nov 14 2025 at 09:16):

It seems, in fact, that the existence of such a factorisation system (at least an algebraic one) for classifiable maps implies that the maps are exponentiable, because $\mathcal P_f$ is in fact identified with the relevant comma category, so you can read pi types off the factorisations. So if there isn't a problem with that argument, such a factorisation system cannot exist.

Nathanael Arkor (Nov 14 2025 at 09:27):

Right, I should have pointed out that the fully faithful functors can't form a factorisation system by the methods of section 4.4 for exactly the reason you point out. However, as far as I know, this argument doesn't eliminate the possibility there may be a factorisation system nonetheless (simply not constructed in this manner).

James Deikun (Nov 14 2025 at 09:33):

It would have to be pretty dissimilar as you couldn't use the pullbacks as the squares of a double category at all.

James Deikun (Nov 14 2025 at 09:38):

Nonetheless, maybe you could lift one somehow from the category of graphs, through the free category monad.