#
Slicing sites and semireplete factorization systems

##
Thorsten Palm

A factorization system (E, M) on a category A gives rise to the covariant
category-valued pseudofunctor P of A sending each object to its slice category
over M. This article characterizes the P so obtained as follows: their object
images have terminal objects, and they admit bicategorically cartesian
liftings, up to equivalence, of slice-category projections. It is clear that,
and how, (E, M) can be recovered from such a P. The correspondence thus
described is actually the second of three similar ones between certain (E, M)
and certain P that the article presents. In the first one the characterization
of the P has all ultimately bicategorical ingredients replaced with their
categorical analogues. A category A with such a P is precisely what the author
has called a `slicing site'. In the article's terms the associated (E, M) are
again factorization systems - but the concept conveyed extends the standard one
by not obliging isomorphisms to belong to either factor class -, namely those
that are `right semireplete' (isomorphisms do belong to M and `left semistrict'
(morphisms in M are monic relative to E). The third correspondence subsumes the
other two; here the (E, M) are all right-semireplete factorization systems.

Keywords:
factorization system, slice categories, cartesian morphisms, slicing site

2010 MSC:
18D99, 18A32

*Theory and Applications of Categories,*
Vol. 30, 2015,
No. 14, pp 489-526.

Published 2015-04-23.

http://www.tac.mta.ca/tac/volumes/30/14/30-14.pdf

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