nLab
bicategory of maps

Bicategory of maps

Definition

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

The bicategory MapKMap K is the locally full sub-2-category of KK 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, MapRelMap Rel is equivalent to Set.

  • Similarly, if CC is a category with finite limits, then there is a bicategory SpanCSpan C of spans in CC. The bicategory MapSpanCMap Span C is equivalent to CC.

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

Properties

If every map in KK is comonadic? and MapKMap K has a terminal object, then MapKMap K is equivalent to a 11-category. If in addition KK is a cartesian bicategory and every comonad in KK has an Eilenberg--Moore object, then KK is biequivalent to SpanMapKSpan Map K, MapKMap K having finite limits. The converse is true if pullback squares in MapKMap K satisfy the Beck–Chevalley condition in KK, i.e. if their mates are invertible (see [LWW10]).

MapKMap K is a regular category if and only if KK is a unitary tabular allegory, equivalently a bicategory of relations in which every coreflexive morphism? splits. In that case RelMapKKRel Map K \simeq K.

Similarly, MapKMap K is a topos if and only if KK is a unitary tabular power allegory.

Maps and equipments

A 2-category equipped with proarrows is, by definition, a bijective-on-objects pseudofunctor KMK\to M such that the image of every arrow in KK is a map in MM. Equivalently, therefore, it is a bijective-on-objects pseudofunctor KMapMK\to Map M.

Hence the inclusion MapMMMap M \to M is the “universal” proarrow equipment that can be constructed with a given bicategory MM as its bicategory of proarrows. More precisely, there is a forgetful functor from EquipEquip to BicatBicat which remembers only the bicategory MM of proarrows, and the assignment of MM to MapMMMap 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.

Revised on November 30, 2010 22:48:58 by Finn Lawler (86.41.17.117)