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The nLab lists a bunch of examples for internal categories in various categories. If we think of a topos as a "universe" for mathematics the need for internal categories in a topos becomes obvious.

How about if the ambient category is not a topos? There are some examples (given on the linked page), but there are not familiar to me and more importantly I don't see how there are "natural", i.e. how you get interested in them without already knowing internal categories. Hence the question:

Is there an obvious application of internal category theory outside from topoi?

I'd like to see an example, which can be understood by a typical undergraduate who knows a bit of category theory, but no internal category theory, if that is at all possible.

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  • $\begingroup$ The general version of Galois theory found in Borceux and Janelidze's Galois Theories replaces Galois groups with internal presheaves on an internal groupoid. I'm not sure though how to make this understandable to an undergrad without a lot of effort. $\endgroup$ – Vladimir Sotirov Jul 3 '16 at 3:23
  • $\begingroup$ Well, I study the category theory of an alternative set theory whose category of sets and functions do not form a topos; talking about internal category theory helps figure out what you can accomplish mathematically in this theory without resorting to an external universe. And internal Heyting algebras can be used to generate new categories with a topos-like internal logic. $\endgroup$ – Malice Vidrine Jul 3 '16 at 5:03
  • $\begingroup$ @MaliceVidrine Ah, perhaps I should have said "outside from Heyting pretopoi" to be on the "safe side" ;) $\endgroup$ – Stefan Perko Jul 3 '16 at 9:39
  • $\begingroup$ Yeah, sadly I'm a logician to the core, so everything I can think of is in that neighborhood :p $\endgroup$ – Malice Vidrine Jul 3 '16 at 17:09
  • $\begingroup$ @MaliceVidrine Nothing sad about that :) $\endgroup$ – Stefan Perko Jul 3 '16 at 17:14

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