#
Exact completions and small sheaves

##
Michael Shulman

We prove a general theorem which includes most notions of "exact
completion" as special cases. The theorem is that
"κ-ary exact
categories" are a reflective sub-2-category of
"κ-ary sites",
for any regular cardinal κ. A κ-ary exact category is an
exact
category with disjoint and universal κ-small coproducts, and a
κ-ary site is a site whose covering sieves are generated by
κ-small families and which satisfies a solution-set condition for
finite limits relative to κ.

In the unary case, this includes the exact completions of a regular
category, of a category with (weak) finite limits, and of a category
with a factorization system. When κ=ω it includes the
pretopos completion of a coherent category. And when κ=∞ is
the size of the universe, it includes the category of sheaves on a
small site, and the category of small presheaves on a locally small
and finitely complete category. The ∞-ary exact completion of a
large nontrivial site gives a well-behaved "category of small
sheaves".

Along the way, we define a slightly generalized notion of "morphism
of sites" and show that κ-ary sites are equivalent to a type
of
"enhanced allegory". This enables us to construct the exact
completion in two ways, which can be regarded as decategorifications
of "representable profunctors" (i.e. entire functional
relations)
and "anafunctors", respectively.

Keywords:
exact completion, site, sheaf, exact category, pretopos, topos

2010 MSC:
18B25

*Theory and Applications of Categories,*
Vol. 27, 2012,
No. 7, pp 97-173.

Published 2012-09-04.

http://www.tac.mta.ca/tac/volumes/27/7/27-07.dvi

http://www.tac.mta.ca/tac/volumes/27/7/27-07.ps

http://www.tac.mta.ca/tac/volumes/27/7/27-07.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/27/7/27-07.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/27/7/27-07.ps

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