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.