[ALGTOP-L] Internal Category References

Anders Kock kock at imf.au.dk
Wed Mar 17 06:08:44 EDT 2010

Philip Hackney wrote:
> Is there a good general reference for internal category theory? 
> Also, is there a way to get from enriched to internal categories, or
> vice versa? If you can't do this in general, are there reasonable
> examples where you can?
> --Philip Hackney
The notion of internal category was studied already in the 1950s in 
France (Grothendieck, Ehresmann), mainly in terms of simplicial objects 
with certain properties (the _nerve_ of the internal category).

A concise account is in Johnstone's "Topos Theory", Chapter 2 (1977)

As for the second question, these issues are best considered through the 
comprehensive notion of indexed, or fibered, category, see again "Topos 
Theory" (Appendix on Locally Internal Categories).

A more elaborate text is Chapter B.2 in Johnstone's "Sketches of an 
Elephant" (2002).

Anders Kock

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