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Exponentiability in Double Categories and the Glueing Construction
Theory and applications of categories, Tome 35 (2020), pp. 1208-1226 Cet article a éte moissonné depuis la source Theory and Applications of Categories website

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We consider pre-exponentiable objects of a pre-cartesian double category D, i.e., objects Y such that the lax functor - x Y: D --> D has a right adjoint in the 2-category LxDbl of double categories and lax functors. When D has 2-glueing, we show that Y is pre-exponentiable in D if and only if Y is exponentiable in D_0 and - x Y is an oplax functor. Thus, such a D is pre-cartesian closed as a double category if and only if D_0 is a cartesian closed category. Applications include the double categories cat, pos, spaces, loc, and topos, whose objects are small categories, posets, topological space, locales, and toposes, respectively.

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Classification : 18N10, 18D15, 18F10, 18F70, 54C35, 16B35
Keywords: double category, exponentiability, cartesian closed, glueing 
@article{TAC_2020_35_a31,
     author = {Susan Niefield},
     title = {Exponentiability in {Double} {Categories}  and the {Glueing} {Construction}},
     journal = {Theory and applications of categories},
     pages = {1208--1226},
     year = {2020},
     volume = {35},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAC_2020_35_a31/}
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Susan Niefield. Exponentiability in Double Categories  and the Glueing Construction. Theory and applications of categories, Tome 35 (2020), pp. 1208-1226. http://geodesic.mathdoc.fr/item/TAC_2020_35_a31/