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Stream: learning: questions

Topic: Properties of the pullback of categories

Ralph Sarkis (Jan 17 2023 at 13:21):

Given a cospan in a category of categories $\mathbf{C} \to \mathbf{E} \leftarrow \mathbf{D}$, I am looking for statements of the form: if $\mathbf{C}$ and $\mathbf{D}$ have property $P$ (and optionally, the legs of the cospan preserve $P$) then the pullback $\mathbf{C} \times_{\mathbf{E}} \mathbf{D}$ also has property $P$ (and optionally, the legs of the span preserve $P$).

I guess $\mathbf{C} \times \mathbf{D}$ is so nice that there will be many such results when $\mathbf{E} = \mathbf{1}$ (e.g. (co)limits are taken coordinate-wise), but I am not sure what happens when you look at the induced functor $\mathbf{C} \times_{\mathbf{E}} \mathbf{D} \rightarrow \mathbf{C}\times \mathbf{D}$. Maybe it reflects or creates (co)limits.

Mike Shulman (Jan 17 2023 at 15:49):

Are you interested in strict pullbacks or pseudo ones?

Mike Shulman (Jan 17 2023 at 16:09):

If you're interested in "categorical" properties, then pseudo pullbacks will work much better. Or strict pullbacks that happen to be equivalent to pseudo ones, like if one of the legs of the cospan is an isofibration.

Mike Shulman (Jan 17 2023 at 16:13):

In that case, you can say that if $C,D,E$ are algebras for some 2-monad $T$, and the legs of the cospan are pseudo $T$-morphisms, then the pullback is also a $T$-algebra and its projections are pseudo $T$-morphisms, and jointly reflect $T$-morphism structure — hence your induced functor $C\times_E D \to C\times D$ also reflects it. This includes limits and colimits, since they can be expressed as algebra structures for a 2-monad, where the pseudo algebra morphisms are those that "preserve" limits/colimits in the usual up-to-isomorphism sense.

Ralph Sarkis (Jan 17 2023 at 17:37):

I think I vaguely get the idea. Trying to get closer to my application, I would need a 2-monad whose algebras are categories with coequalizers. Are pseudomorphisms of these algebras functors that preserve coequalizers? Or that reflect coequalizers?

I am taking the strict pullback so I would need to check if it is equivalent to the pseudo pullback.

Mike Shulman (Jan 17 2023 at 17:47):

Pseudomorphisms will be functors that preserve coequalizers.

fosco (Jan 18 2023 at 07:40):

For strict pullbacks you can find something about creation of limits in CWM, but you need to be a bit finicky about the requirements on the legs of the cospan. image.png

Mike Shulman (Jan 18 2023 at 15:29):

Regarding that, it should be noted that there are multiple meanings of [[creation of limits]]. That fact is only true for the strict notion, which amounts to a sort of isofibrational condition.