Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Exact completions and small sheaves   We prove a general theorem which includes most notions of "exact completion".
The theorem is that "k-ary exact categories" are a reflective sub-2-category of
"k-ary sites", for any regular cardinal k. A k-ary exact category is an exact
category with disjoint and universal k-small coproducts, and a k-ary site is a
site whose covering sieves are generated by k-small families and which
satisfies a weak size condition. For different values of k, this includes the
exact completions of a regular category or a category with (weak) finite
limits; the pretopos completion of a coherent category; and the category of
sheaves on a small site. For a large site with k the size of the universe, it
gives a well-behaved "category of small sheaves". Along the way, we define a
slightly generalized notion of "morphism of sites", and show that k-ary sites
are equivalent to a type of "enhanced allegory".

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The categories of this paper are: nlin
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The categories of this paper are: ao-sci
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The categories of this paper are: hep-ph
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The categories of this paper are: hep-lat
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The categories of this paper are: q-alg
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The categories of this paper are: solv-int
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The categories of this paper are: dg-ga
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The categories of this paper are: atom-ph
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The categories of this paper are: nucl-ex
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The categories of this paper are: q-fin
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The categories of this paper are: adap-org
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The categories of this paper are: cond-mat
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The categories of this paper are: cmp-lg
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The categories of this paper are: stat
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The categories of this paper are: hep-ex
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The categories of this paper are: quant-ph