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On Generic Separable Objects

##
Robbie Gates

The notion of {\em separable} (alternatively {\em unramified},
or {\em decidable}) objects and their place in a categorical
theory of space have been described by Lawvere (see \cite{lawvere:como}),
drawing on notions of separable from algebra and unramified from geometry.
In \cite{schanuel:halifax}, Schanuel constructed the
generic separable object in an extensive category with products
as an object of the free category with finite sums on the dual of the
category of finite sets and injections.

We present here a generalization of the work of \cite{schanuel:halifax},
replacing the category of finite sets and injections by a
category $\cat A$ with a suitable factorization system. We describe
the analogous construction, and identify and prove a universal property
of the constructed category for both extensive categories and
extensive categories with products (in the case $\cat A$ admits sums).

In constructing the machinery for proving the required universal property,
we recall briefly the boolean algebra structure of the summands of an object
in an extensive category. We further present a notion
of direct image for certain maps in an extensive category, to allow
construction of left adjoints to the inverse image maps obtained
from pullbacks.

Please note the electronically available References at
http://www.tac.mta.ca/tac/volumes/1998/n10/reference.html

Keywords: extensive category, separable object, generic solution, direct image.

1991 MSC: 18D99 (18B15, 18C10).

*Theory and Applications of Categories*, Vol. 4, 1998, No. 10, pp 208-248.

http://www.tac.mta.ca/tac/volumes/1998/n10/n10.dvi

http://www.tac.mta.ca/tac/volumes/1998/n10/n10.ps

http://www.tac.mta.ca/tac/volumes/1998/n10/n10.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1998/n10/n10.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/1998/n10/n10.ps

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