Hi David, The characterization Steve describes is nice and applies more generally to enriched categories. You can eliminate the dependence on Prof by first constructing Prof from Cat: profunctors can be identified with two-sided discrete fibrations, or two-sided discrete cofibrations, in the bicategory Cat. Mark Weber's paper "Yoneda structures in 2-toposes" studies duality involutions characterized using discrete fibrations. However, this property in Prof only characterizes the opposite up to Morita equivalence (i.e. up to Cauchy completion, aka idempotent-splitting), since Morita-equivalent categories are equivalent objects of the bicategory Prof. There is, in fact, also a way to describe opposites purely in terms of Cat. First we note that Cat is an "exact 2-category" -- this means that among other things, every "2-congruence" has a quotient, where a 2-congruence is an internal category for which (source,target) is a two-sided discrete fibration. Now for any category A, let core(A) be its core, i.e. its maximal subgroupoid. The core can be characterized in 2-categorical terms as a right adjoint to the inclusion of Gpd into Cat_g, where Gpd is the 2-category of groupoids (= the category of groupoidal objects in Cat) and Cat_g is the 2-category of categories, functors, and natural isomorphisms. However, of more importance for this discussion is that core(A) is groupoidal and the map p:core(A) --> A is "strong epic" in the sense that it is left orthogonal (in the 2-categorical sense) to all (representably) fully-faithful morphisms. (In Cat, this condition is equivalent to p being essentially surjective on objects.) Since Cat is exact, this implies that A is equivalent to the quotient of the "kernel" of p, which is the 2-congruence given by the comma object (p/p) with its two projections to core(A). However, since core(A) is groupoidal, we can switch these two projections and still have a 2-congruence. The quotient of this "opposite" 2-congruence, which exists since Cat is exact, is (up to equivalence) the opposite category of A. This construction generalizes to any exact 2-category which "has cores," in the sense that every object admits a strong epic from a groupoidal one. The notion of exact 2-category is due to Ross Street in "A characterization of bicategories of stacks." I have not seen the above construction of opposites written down anywhere, so I wrote it down myself here: http://ncatlab.org/michaelshulman/show/duality+involution#GpdFixUniq but if anyone has seen it before, I would love to hear some references. Best, Mike David Leduc wrote:

Dear all,

The same way monads on categories can be generalized to monads on objects of a bicategory, is there a way to generalize opposites of categories to opposites of objects in a bicategory?

David

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