subgroupoids of V-categories

[Emily Riehl <eriehl@math.harvard.edu>]

Hi all,

For general V (closed symmetric monoidal, bicomplete), is there a general
way to construct the maximal subgroupoid of a V-category C?

I think I know how to *detect* the maximal subgroupoid. A map in C is an
isomorphism iff it is representably so: Writing 1 for the monoidal unit,
we say f : 1 -> C(x,y) is an iso iff the induced map f_* : C(z,x) ->
C(z,y) is an iso in V for all z. So we might say that a V-category G with
the same objects and C and an identity-on-objects local monomorphism G ->
C is the maximal subgroupoid provided that a morphism f factors through
G(x,y) -> C(x,y) just when f is an isomorphism.

In examples, this is probably good enough, but I still would feel better
if I had a general construction of the maximal subgroupoid. I feel like
this should be some sort of weighted limit, perhaps with some additional
structure on V?

Thanks,
Emily


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