A * generating family,* in a category C is a collection of objects
$\{A_i|i\in I\}$ such that if for any subobject Y >--> X, every
$f: A_i \rightarrow X$ factors through m, then m is an isomorphism -
i.e. the functors $C(A_i, - )$ are collectively conservative.
In this paper, we examine some circumstances under which subobjects of 1
form a generating family. Objects for which subobjects of 1 do form a
generating family are called *partially well-pointed*. For a
Grothendieck topos, it is well known that subobjects of 1 form a
generating family if and only if the topos is localic. For the elementary
case, little more is known. The problem is studied by Borceux,
where it is shown that the result is internally true, an equivalent
condition is found in the boolean case, and certain preservation
properties are shown. We look at two different approaches to the problem,
one based on a generalization of projectivity, and the other based on
looking at the most extreme sorts of counterexamples.

Keywords: Topoi, generating families, cogenerators, semiprojective objects

2000 MSC: 03G30, 18B25

*Theory and Applications of Categories,*
Vol. 16, 2006,
No. 31, pp 896-922.

http://www.tac.mta.ca/tac/volumes/16/31/16-31.dvi

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http://www.tac.mta.ca/tac/volumes/16/31/16-31.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/31/16-31.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/31/16-31.ps