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Stream: learning: reading & references

Topic: internal categories in a presheaf category

Nathanael Arkor (Aug 06 2025 at 15:30):

I am looking for a standard reference for the fact that there is an isomorphism of 2-categories $\text{Cat}([C, \text{Set}]) \cong [C, \text{Cat}]$ (by "standard", I mean either the earliest reference, or a textbook reference). I looked in Johnstone's books, where this fact is surely mentioned, but I couldn't spot it. Does anyone have any suggestions?

Fernando Yamauti (Aug 06 2025 at 15:36):

I think there's a typo. Do you really want internal cats in a 2-topos?

Nathanael Arkor (Aug 06 2025 at 15:44):

Thanks, I fixed the typo.

Bryce Clarke (Aug 06 2025 at 16:50):

Nathanael Arkor said:

I am looking for a standard reference for the fact that there is an isomorphism of 2-categories $\text{Cat}([C, \text{Set}]) \cong [C, \text{Cat}]$ (by "standard", I mean either the earliest reference, or a textbook reference). I looked in Johnstone's books, where this fact is surely mentioned, but I couldn't spot it. Does anyone have any suggestions?

I think it is in his Topos Theory book, but as an exercise.

Nathanael Arkor (Aug 06 2025 at 16:54):

Ah, that may be why I hadn't spotted it in the book.

Nathanael Arkor (Aug 06 2025 at 16:56):

image.png
This seems the closest, but it's still not quite what I'm looking for.

fosco (Aug 06 2025 at 17:19):

Considering the topic of his thesis, @Enrico Ghiorzi might know if there is a precise reference for this result. I think somewhere, someday, I have seen the statement that for every essentially algebraic theory T, T-models in [C,Set] are functors C -> Mod(T) (is that even true?)

Nathanael Arkor (Aug 06 2025 at 17:33):

It is true, and I know of a reference for that more general result, but I imagine there should be a reference specific to internal categories much earlier.

Bryce Clarke (Aug 07 2025 at 05:39):

Screenshot 2025-08-07 at 7.34.41 am.png
Page 342 in the Elephant has this paragraph (which is still not a reference).

Bryce Clarke (Aug 07 2025 at 05:48):

Nathanael Arkor said:

image.png
This seems the closest, but it's still not quite what I'm looking for.

I think the closest is reading Exercise 5 and Exercise 6 in Topos Theory together, which characterises the image of G as the split opfibrations.

Bryce Clarke (Aug 07 2025 at 05:52):

Screenshot 2025-08-07 at 7.50.33 am.png
Of course! Adrian Miranda's Masters thesis is where to look!
https://doi.org/10.25949/19434626.v1

Nathanael Arkor (Aug 07 2025 at 06:57):

Thanks for investigating! I suppose that, given that Adrian didn't give a reference for the result, he hadn't found a good reference either.