nLab
bicategory of maps

Definition

If $K$ is a bicategory, then a morphism $f:a\to b$ is called a map if it has a right adjoint $f^{*}:b\to a$. (This is in slight contrast to the common usage of “map” to denote simply a morphism in any category.)

The bicategory $\mathrm{Map}K$ is the locally full sub-2-category of $K$ determined by the maps.

Examples

• In the bicategory Rel of sets and relations, a relation is a map if and only if it is the graph of a function. Consequently, $\mathrm{Map}\mathrm{Rel}$ is equivalent to Set.

• Similarly, if $C$ is a category with finite limits, then there is a bicategory $\mathrm{Span}C$ of spans in $C$. The bicategory $\mathrm{Map}\mathrm{Span}C$ is equivalent to $C$.

• In the bicategory Prof of categories and profunctors (perhaps enriched), if $B$ is a Cauchy complete category, then a profunctor $A\to B$ is a map if and only if it is represented by a functor $A\to B$. If $B$ is not Cauchy complete, then maps $A\to B$ correspond to functors from $A$ to the Cauchy completion of $B$.

Properties

If every map in $K$ is comonadic? and $\mathrm{Map}K$ has a terminal object, then $\mathrm{Map}K$ is equivalent to a $1$-category. If in addition $K$ is a cartesian bicategory and every comonad in $K$ has an Eilenberg--Moore object, then $K$ is biequivalent to $\mathrm{Span}\mathrm{Map}K$, $\mathrm{Map}K$ having finite limits. The converse is true if pullback squares in $\mathrm{Map}K$ satisfy the Beck–Chevalley condition in $K$, i.e. if their mates are invertible (see [LWW10]).

$\mathrm{Map}K$ is a regular category if and only if $K$ is a unitary tabular allegory, equivalently a bicategory of relations in which every coreflexive morphism? splits. In that case $\mathrm{Rel}\mathrm{Map}K\simeq K$.

Similarly, $\mathrm{Map}K$ is a topos if and only if $K$ is a unitary tabular power allegory.

Maps and equipments

A 2-category equipped with proarrows is, by definition, a bijective-on-objects pseudofunctor $K\to M$ such that the image of every arrow in $K$ is a map in $M$. Equivalently, therefore, it is a bijective-on-objects pseudofunctor $K\to \mathrm{Map}M$.

Hence the inclusion $\mathrm{Map}M\to M$ is the “universal” proarrow equipment that can be constructed with a given bicategory $M$ as its bicategory of proarrows. More precisely, there is a forgetful functor from $\mathrm{Equip}$ to $\mathrm{Bicat}$ which remembers only the bicategory $M$ of proarrows, and the assignment of $M$ to $\mathrm{Map}M\to M$ is its right adjoint.

Mike Shulman: This is obviously morally true, but I can’t be bothered right now to check which 1-, 2-, or 3-categories of equipments and bicategories one has to use to make it precisely correct.

A lot of work in bicategories that makes use of maps could easily be reformulated in a proarrow equipment, and conversely. Thus, it is to some extent a question of aesthetics which is preferred. One advantage of proarrow equipments is they can distinguish between a category and its Cauchy completion (as objects of Prof), while maps in bicategories are perhaps simpler in some ways.

References

• Carboni, Walters, Cartesian bicategories I, JPAA 49, 1987.

• Lack, Walters, Wood, Bicategories of spans as cartesian bicategories, TAC 24(1), 2010.