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Ruby Khondaker (she/her) (Apr 28 2026 at 11:04):I’ve come across the category of elements of construction which works for functors valued in Set. Also the actual “geothendieck construction” works for pseudofunctors valued in Cat. And straightening/unstraightening makes this work for infinity categories, from what little I know of it.
However as far as I’m aware, if I just have a functor C -> D there’s not a general way to somehow transform this into a functor E -> C as a kind of projection from a total space. Hence my Q - what kinds of categories (and functors?) support a notion of grothendieck construction? I’d also be interested to hear enriched or internal generalisations if there are some.
fosco (Apr 28 2026 at 11:14):For enriched categories, there is a paper trying to answer that. "The enriched grothendieck construction". https://arxiv.org/abs/1804.03829
Max New (Apr 28 2026 at 13:02):The most general one possible for plain categories is that a category over (i.e. a functor ) is equivalent to a lax functor into the bicategory Span (sets, spans, span transformations), or equivalently a normal lax functor into the bicategory of Profunctors (categories, profunctors, natural transformation). The cases you describe can be seen as instances of this construction if you compose with functors from Cat to Prof/Span and Set to Prof/Span. Cat and Cat^op both map to Prof so this is how you get the Grothendieck construction for both fibrations and opfibrations.
A lax functor into Span recently goes by the name "displayed category": https://ncatlab.org/nlab/show/displayed+category though the observation can be traced back to some old lectures by Benabou. Displayed categories are used heavily in mechanizations of category theory because they make fibered notions nicer to work with in dependent type theory
Ruby Khondaker (she/her) (Apr 28 2026 at 13:06):Ah yes I've come across displayed categories before! It was nice seeing the most general possible way to convert a "fibred" category, so a functor E -> C, into an "indexed" one. I've been looking at grothendieck fibrations recently too and if I understand correctly, these are roughly "bundles with connection"?
Kevin Carlson (Apr 28 2026 at 17:47):Very roughly, yes, but there are at least four different reasonable ways of fleshing that phrase out in CT, corresponding to (discrete) (op)fibrations. There are also (discrete) Conduché fibrations which are more general still while nevertheless closer to the connection concept than general functors.
Joe Moeller (May 05 2026 at 21:17):The Grothendieck construction is also a lax colimit in the 2-category Cat. For reference, see section 10.2 of 2-Dimensional Categories. So I guess you could also take C to be an arbitrary bicategory and ask which lax colimits exist there, and how similar these feel to the classical Grothendieck construction.