The category of **Set**-valued presheaves on a small category *B* is
a topos. Replacing **Set** by a bicategory **S** whose objects are sets
and morphisms are spans, relations, or partial maps, we consider a category
Lax(*B*, **S**) of **S**-valued lax functors on *B*. When
**S** = **Span**, the resulting category is equivalent to
**Cat**/*B*, and hence, is rarely even cartesian closed. Restricting
this equivalence gives rise to exponentiability characterizations for
Lax(*B*, **Rel**) by Niefield and for Lax(*B*, **Par**) in
this paper. Along the way, we obtain a characterization of those *B*
for which the category **UFL**/*B* is a coreflective subcategory of
**Cat**/*B*, and hence, a topos.

Keywords: span, relation, partial map, topos, cartesian closed, exponentiable, presheaf

2000 MSC: 18A22, 18A25, 18A40, 18B10, 18B25, 18D05, 18F20

*Theory and Applications of Categories,*
Vol. 24, 2010,
No. 12, pp 288-301.

http://www.tac.mta.ca/tac/volumes/24/12/24-12.dvi

http://www.tac.mta.ca/tac/volumes/24/12/24-12.ps

http://www.tac.mta.ca/tac/volumes/24/12/24-12.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/24/12/24-12.dvi

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