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Lax pullback complements and pullbacks of spans

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Seyed Naser Hosseini, Walter Tholen, and Leila Yeganeh

The formation of the "strict" span category Span(C) of a category C with
pullbacks is a standard organizational tool of category theory.
Unfortunately, limits or colimits in Span(C) are not easily computed in
terms of constructions in C. This paper shows how to form the pullback in
Span(C) for many, but not all, pairs of spans, given the existence of some
specific so-called lax pullback complements in C of the "left legs" of at
least one of the two given spans. For some types of spans we require the
ambient category to be adhesive to be able to form at least a weak
pullback in Span(C). The existence of all lax
pullback complements in C along a given morphism is equivalent
to the exponentiability of that morphism. Since exponentiability is a
rather restrictive property of a morphism, the paper first develops a
comprehensive framework of rules for individual lax pullback complement
diagrams, which resembles the set of pasting and cancellation rules for
pullback diagrams, including their behaviour under pullback. We also
present examples of lax pullback complements along non-exponentiable
morphisms, obtained via lifting along a fibration.

Keywords:
lax pullback complement, exponentiable morphism, partial product, adhesive
category, span category, total spans, cototal spans, partial morphisms,
categories of presheaves, graphs, topological spaces

2010 MSC:
18A99,18B05,18B30,18B35,18D15,18F20

*Theory and Applications of Categories,*
Vol. 33, 2018,
No. 16, pp 445-475.

Published 2018-05-24.

http://www.tac.mta.ca/tac/volumes/33/16/33-16.pdf

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