* Cocompletions of Categories
@ 1997-11-26 17:42 categories
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From: categories @ 1997-11-26 17:42 UTC (permalink / raw)
To: categories
Date: Wed, 26 Nov 97 10:19:37 +0100
From: Jiri Velebil <velebil@math.feld.cvut.cz>
Dear Colleagues,
I wonder whether the following description of a
free F-conservative completion of any category under
C-colimits (where F and C are classes of small categories
such that F is a subclass of C and "F-conservative"
means "preserving existing F-colimits").
The description of the cocompletion is as follows:
Suppose C is a class of small categories and F is
a subclass of C.
Let X be any category.
Denote by [X^op,Set] the quasicategory of all
functors and all natural transformations between them.
Denote by F^op-[X^op,Set] the quasicategory of all
functors which preserve F^op-limits, i.e. limits
of functors d : D -> X^op with D^op in F.
Claim 1.
F^op-[X^op,Set] is reflective in [X^op,Set]
(The proof uses the fact that the above Claim
holds for the case when X is small -
Korollar 8.14 in Gabriel, Ulmer: Lokal
pr"asentierbare Kategorien.)
By Claim 1., F^op-[X^op,Set] has all small colimits.
Denote by D(X) the closure of X (embedded by Yoneda)
in F^op-[X^op,Set] under C-colimits. Then one can
prove that D(X) is a legitimate category.
The codomain-restriction I: X -> D(X) of the
Yoneda embedding fulfills the following:
1. D(X) has C-colimits.
2. I preserves F-colimits.
3. D(X) has the following universal property:
for any functor H : X -> Y which preserves
F-colimits and the category Y has C-colimits
there is a unique (up to an isomorphism)
functor H* : D(X) -> Y such that H* preserves
C-colimits and H*.I = H.
In fact, this gives a 2-adjunction between
C-CAT_C : the 2-quasicategory of all categories having C-colimits,
all functors preserving C-colimits and all natural
transformations
and
CAT_F : the 2-quasicategory of all categories, all functors
preserving F-colimits and all natural transformations.
The result also holds for V-categories, instead of a class C
of small categories one has to work with a class of small indexing
types.
Thank you,
Jiri Velebil
velebil@math.feld.cvut.cz
Department of Mathematics
FEL CVUT
Technicka 2
Praha 6
Czech Republic
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