Some more edits and additions to internal diagram, mostly expanding the definition.

]]>I’ve moved that section across now.

]]>It is OK with me. Somebody could cite then 1984 paper on coherence by MacLane and Pare where they have a part of the paper dedicated to internalization of concepts in an index category with diagram/category in an indexed category an example. If we denote again by $Cat(F\to C)$ the 2-category of diagrams in a fibered category $F\to C$ and $Cat(C)$ the 2-category of internal category in the base, the natural projection among the two is not necessary a 2-fibration, but it satisfies 2 out of 4 universality conditions from Hermida’s paper on 2-fibrations.

internal presheaves

When we take the default terminology favoring contravariant internal presheaves then they correspond to discrete opfibrations.

]]>Fair point. Perhaps we should move the section on diagrams in an indexed category from internal profunctor to internal diagram, since it would be more at home there.

]]>I don’t think there’s much hope of settling on a single term, given that they’re all used in the literature; we should just make sure that there are plenty of redirects and cross-links.

Of course, “discrete fibration” is more general in that it applies in any 2-category, not just a 2-category of internal categories. And “diagram” is more general in that it applies to diagrams in any indexed category. So each page can treat the natural level of generality for the concept with the appropriate name.

]]>Yes, but what internal profunctor describes as an ’internal presheaf’ is the usual sort of internal diagram. Actually I wrote most of that page and I remember that I didn’t know any unambiguous name for ’diagrams’ in an indexed category, so I just used the heading ’Internal diagrams’ (I think that’s what Johnstone calls them), though that may have been a poor choice.

These things seem to have far too many different names – internal presheaves, diagrams, discrete fibrations… Maybe we should settle on a single term?

]]>Thanks. In internal diagram I treated an internal diagram in a cartesian category. Entry internal profunctor has a section on internal diagrams in an indexed category, what is a bit more general situation and there described in different terms.

]]>Most of this is already at internal profunctor, I think.

]]>New entry internal diagram, generalizing internal functor.

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