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Copower objects and their applications to finiteness in topoi

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Toby Kenney

In this paper, we examine a new approach to topos theory - rather than
considering subobjects, look at quotients. This leads to the notion of a
copower object, which is the object of quotients of a given object. We
study some properties of copower objects, many of which are similar to the
properties of power objects. Given enough categorical structure (i.e. in
a pretopos) it is possible to get power objects from copower objects, and
vice versa.

We then examine some new definitions of finiteness arising from the notion
of a copower object. We will see that the most naturally occurring such
notions are equivalent to the standard notions, K-finiteness (at least
for well-pointed objects) and $\tilde{K}$-finiteness, but that this
new way of looking at them gives new information, and in fact gives rise
to another notion of finiteness, which is related to the classical notion
of an amorphous set - i.e. an infinite set that is not the disjoint union
of two infinite sets.

Finally, We look briefly at two similar notions: potency objects and per
objects.

Keywords:
Topoi, finiteness, copower objects

2000 MSC:
03G30, 18B25

*Theory and Applications of Categories,*
Vol. 16, 2006,
No. 32, pp 923-956.

http://www.tac.mta.ca/tac/volumes/16/32/16-32.dvi

http://www.tac.mta.ca/tac/volumes/16/32/16-32.ps

http://www.tac.mta.ca/tac/volumes/16/32/16-32.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/32/16-32.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/32/16-32.ps

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