For a small category $B$ and a double category $\mathbb D$, let ${\rm Lax}_N(B,\mathbb D)$ denote the category whose objects are vertical normal lax functors $B\to\mathbb D$ and morphisms are horizontal lax transformations. It is well known that $Lax_N(B, \mathbb Cat) \simeq Cat/B$, where $\mathbb Cat$ is the double category of small categories, functors, and profunctors. We generalized this equivalence to certain double categories, in the case where $B$ is a finite poset. Street showed that $Y\to B$ is exponentiable in $Cat/B$ if and only if the corresponding normal lax functor $B\to \mathbb Cat$ is a pseudo-functor. Using our generalized equivalence, we show that a morphism $Y\to B$ is exponentiable in $ {\mathbb D}_0/B$ if and only if the corresponding normal lax functor $B\to\mathbb D$ is a pseudo-functor plus an additional condition that holds for all $X\to !B$ in $Cat$. Thus, we obtain a single theorem which yields characterizations of certain exponentiable morphisms of small categories, topological spaces, locales, and posets.
Keywords: exponentiable space, function space, lax slice, specialization order
2000 MSC: 18B30, 18A40, 18A25, 54C35, 54F05, 06F30
Theory and Applications of Categories, Vol. 27, 2012, No. 2, pp 10-26.
Published 2012-05-01.
http://www.tac.mta.ca/tac/volumes/27/2/27-02.dvi
http://www.tac.mta.ca/tac/volumes/27/2/27-02.ps
http://www.tac.mta.ca/tac/volumes/27/2/27-02.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/27/2/27-02.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/27/2/27-02.ps