Geodesic
Lax Presheaves and Exponentiability
Theory and applications of categories, Tome 24 (2010), pp. 288-301.

Voir la notice de l'article provenant de la source Theory and Applications of Categories website

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.
Classification : 18A22, 18A25, 18A40, 18B10, 18B25, 18D05, 18F20
Keywords: span, relation, partial map, topos, cartesian closed, exponentiable, presheaf 
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     author = {Susan Niefield},
     title = {Lax {Presheaves} and {Exponentiability}},
     journal = {Theory and applications of categories},
     pages = {288--301},
     publisher = {mathdoc},
     volume = {24},
     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2010_24_a11/}
}
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Susan Niefield. Lax Presheaves and Exponentiability. Theory and applications of categories, Tome 24 (2010), pp. 288-301. http://geodesic.mathdoc.fr/item/TAC_2010_24_a11/