View source for Bicategory

''(update), weak <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b1202302.png" />-category''

A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b1202304.png" />-category [[#References|[a7]]], [[#References|[a8]]], [[#References|[a14]]], [[#References|[a19]]], [[#References|[a22]]] is a [[Category|category]] <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b1202305.png" /> consisting of objects <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b1202306.png" /> and arrows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b1202307.png" /> together with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b1202308.png" />-arrows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b1202309.png" />, which can be displayed as

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b120230a.gif" />

Figure: b120230a

and has vertical and horizontal composition operations

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b120230b.gif" />

Figure: b120230b

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b120230c.gif" />

Figure: b120230c

These compositions are required to be associative and unital; moreover, horizontal composition must preserve vertical units and the following interchange law is imposed:

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023010.png" /></td> </tr></table>

<img style="border:1px solid;" src="https://www.encyclopediaofmath.org/legacyimages/common_img/b120230d.gif" />

Figure: b120230d

The basic example of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023011.png" />-category is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023013.png" />, whose objects are (small) categories, whose arrows are the functors (cf. also [[Functor|Functor]]), and whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023014.png" />-arrows are natural transformations. Indeed, the basic  "five rules"  for composition of natural transformations appeared in [[#References|[a11]]], Appendix.

There is a weaker notion of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023015.png" />-category which occurs in practice. A weak <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023017.png" />-category or bicategory [[#References|[a1]]] consists of the data and conditions of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023018.png" />-category except that the associativity and unital equalities for horizontal composition are replaced by the extra data of invertible natural families of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023019.png" />-arrows

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023020.png" /></td> </tr></table>

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023021.png" /></td> </tr></table>

called associativity and unital constraints, such that the associativity pentagon (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023023.png" />-cocycle condition)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023024.png" /></td> </tr></table>

and unit triangle (or normalization condition)

<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023025.png" /></td> </tr></table>

are imposed. In some of the recent literature, bicategories are called <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023026.png" />-categories and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023027.png" />-categories are called strict <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023029.png" />-categories.

A monoidal category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023030.png" /> can be identified with the one-object bicategory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023031.png" /> whose arrows are the objects of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023032.png" />, whose <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023033.png" />-arrows are the arrows of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023034.png" />, whose horizontal composition is the tensor product of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023035.png" />, and whose vertical composition is the composition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023036.png" />.

There is a bicategory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023037.png" /> whose objects are (small) categories and whose arrows are modules [[#References|[a31]]], [[#References|[a34]]] (called pro-functors or distributors in [[#References|[a2]]], and bimodules in [[#References|[a21]]]) between categories.

An arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023038.png" /> in a bicategory is called an equivalence when there is an arrow <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023039.png" /> such that there are invertible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023040.png" />-arrows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023041.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023042.png" />. A weak <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023044.png" />-groupoid is a bicategory in which each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023045.png" />-arrow is invertible and each arrow is an equivalence. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023047.png" />-groupoid is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023048.png" />-category with all arrows and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023049.png" />-arrows invertible. For each space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023050.png" />, there is a homotopy <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023052.png" />-groupoid, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023053.png" /> whose objects are the points of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023054.png" />; it contains the information of the fundamental groupoid <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023055.png" /> and the homotopy groups <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023056.png" /> for each <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023057.png" /> (cf. also [[Homotopy group|Homotopy group]]; [[Crossed module|Crossed module]]; [[Crossed complex|Crossed complex]]; [[Fundamental group|Fundamental group]]). An early application of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023058.png" />-categories to [[Homotopy|homotopy]] theory occurs in [[#References|[a9]]]. In fact, Ch. Ehresmann [[#References|[a7]]] defined double categories and double groupoids, which generalize <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023059.png" />-categories in that they have two types of arrows (see [[#References|[a19]]]), and these also have proved important in homotopy theory [[#References|[a5]]].

While many examples occur naturally as bicategories rather than <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023060.png" />-categories, there is a coherence theorem asserting that every bicategory is equivalent (in the appropriate sense) to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023061.png" />-category [[#References|[a23]]], [[#References|[a12]]].

There are several purely categorical motivations for the development of bicategory theory. The first is to study bicategories following the theory of categories but taking account of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023062.png" />-dimensionality; this is the spirit of [[#References|[a10]]], [[#References|[a14]]], [[#References|[a18]]], [[#References|[a29]]]. A given concept in category theory typically has several generalizations, stemming from the fact that equalities between arrows can be replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023063.png" />-arrow constraints (lax generalization), by invertible <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023064.png" />-arrow constraints (pseudo-generalization), or by keeping the equalities; further equalities are required on the constraints. A bicategory can thus be regarded as a pseudo-category, an equivalence as a pseudo-isomorphism, and a stack (a  "champ"  in French) as a pseudo-sheaf. In lax cases there are also choices of direction for the equality-breaking constraints. All this applies to functors: there are lax functors (also called morphisms) and pseudo-functors (also called homomorphisms) between bicategories; there are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023065.png" />-functors between <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023066.png" />-categories having equality constraints. This also applies to limits, adjunctions, Kan extensions, etc. [[#References|[a10]]], [[#References|[a14]]]. One can use the fact that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023067.png" />-categories are categories with homs enriched in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023068.png" />, that is, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023070.png" />-categories where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023071.png" /> [[#References|[a8]]]. Some laxness is even accounted for in this way: lax limits are enriched limits for a suitable weight (or index) [[#References|[a28]]].

A second motivation comes from the fact that bicategories are  "monoidal categories with several objects" . Included in this is the study of categories enriched in a bicategory which leads to a unification of category theory, [[Sheaf|sheaf]] theory, Boolean-valued logic, and the theory of metric spaces (cf. [[Metric space|Metric space]]) [[#References|[a38]]], [[#References|[a31]]], [[#References|[a34]]], [[#References|[a3]]], [[#References|[a24]]]. The generalization of Cauchy completion (cf. also [[Cauchy sequence|Cauchy sequence]]) from the case of metric spaces is fundamental [[#References|[a21]]].

A third impetus is the formalization of properties of the bicategory <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023072.png" /> (as in the part of category theory which abstracts properties of the category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023073.png" /> of sets) allowing the use of bicategories as organizational tools for studying categories with extra structure (in the way that categories themselves organize sets with structure). This leads to the study of arrow categories [[#References|[a13]]], adjunctions [[#References|[a17]]], monads (also called triples; cf. [[Triple|Triple]]) [[#References|[a26]]], Kan extensions [[#References|[a37]]], [[#References|[a27]]] factorization systems (cf. also [[E-M-factorization-system-in-a-category|<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023074.png" />-factorization system in a category]]) [[#References|[a31]]], [[#References|[a32]]], [[#References|[a33]]], [[#References|[a6]]], etc., as concepts belonging within a fixed bicategory. Familiar constructions (such as comma categories and Eilenberg–Moore categories for monads) made with these concepts turn out to be limits of the kind arising in other motivations. In this spirit, one can mimic the construction of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023075.png" /> from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023076.png" /> starting with a bicategory (satisfying certain exactness conditions) much as one constructs a category of relations in a regular category or [[Topos|topos]] [[#References|[a29]]], [[#References|[a6]]], [[#References|[a25]]]. The size needs of category theory add extra challenges to the subject [[#References|[a27]]], [[#References|[a37]]].

Low-dimensional topology enters bicategory theory from two dual directions. The commutative diagrams familiar in a category (cf. also [[Diagram|Diagram]]) laxify in a bicategory to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023077.png" />-dimensional diagrams with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023078.png" />-arrows in the regions; and these diagrams, if well formed, can be evaluated, using the compositions, to yield a unique <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023079.png" />-arrow called the pasted composite of the diagram [[#References|[a1]]], [[#References|[a14]]], [[#References|[a19]]]. Two-dimensional graph-like structures called computads were designed to formalize pasting [[#References|[a28]]]. The planar Poincaré-dual view replaces pasting diagrams with string diagrams; the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023080.png" />-arrows label nodes, the arrows label strings (intervals embedded in the Euclidean plane), and the objects label regions [[#References|[a16]]], [[#References|[a36]]]. The planar geometry of string diagrams under deformation is faithful to the algebra of bicategories. See also [[Higher-dimensional category|Higher-dimensional category]]; [[Bicategory(2)|Bicategory]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  J. Bénabou,   "Introduction to bicategories" , ''Lecture Notes Math.'' , '''47''' , Springer  (1967)  pp. 1–77</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  J. Bénabou,   "Les distributeurs"  ''Sem. Math. Pure Univ. Catholique de Louvain'' , '''33'''  (1973)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  R. Betti,   A. Carboni,   R. Walters,   "Variation through enrichment"  ''J. Pure Appl. Algebra'' , '''29'''  (1983)  pp. 109–127</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  G.J. Bird,   G.M. Kelly,   A.J. Power,   R. Street,   "Flexible limits for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023081.png" />-categories"  ''J. Pure Appl. Algebra'' , '''61'''  (1989)  pp. 1–27</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  R. Brown,   "Higher dimensional group theory"  R. Brown (ed.)  T.L. 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Algebra'' , '''37'''  (1985)  pp. 59–80</TD></TR><TR><TD valign="top">[a24]</TD> <TD valign="top">  A. Pitts,   "Applications of sup-lattice enriched category theory to sheaf theory"  ''Proc. London Math. Soc.'' , '''(3)57'''  (1988)  pp. 433–480</TD></TR><TR><TD valign="top">[a25]</TD> <TD valign="top">  R.D. Rosebrugh,   R.J. Wood,   "Proarrows and cofibrations"  ''J. Pure Appl. Algebra'' , '''53'''  (1988)  pp. 271–296</TD></TR><TR><TD valign="top">[a26]</TD> <TD valign="top">  R. Street,   "The formal theory of monads"  ''J. Pure Appl. Algebra'' , '''2'''  (1972)  pp. 149–168</TD></TR><TR><TD valign="top">[a27]</TD> <TD valign="top">  R. Street,   "Elementary cosmoi 1" , ''Lecture Notes Math.'' , '''420''' , Springer  (1974)  pp. 134–180</TD></TR><TR><TD valign="top">[a28]</TD> <TD valign="top">  R. Street,   "Limits indexed by category-valued <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023084.png" />-functors"  ''J. Pure Appl. Algebra'' , '''8'''  (1976)  pp. 149–181</TD></TR><TR><TD valign="top">[a29]</TD> <TD valign="top">  R. Street,   "Fibrations in bicategories"  ''Cah. Topol. Géom. Diff.'' , '''21; 28'''  (1980; 1987)  pp. 111–160; 53–56</TD></TR><TR><TD valign="top">[a30]</TD> <TD valign="top">  R. Street,   "Conspectus of variable categories"  ''J. Pure Appl. Algebra'' , '''21'''  (1981)  pp. 307–338</TD></TR><TR><TD valign="top">[a31]</TD> <TD valign="top">  R. Street,   "Cauchy characterization of enriched categories"  ''Rend. Sem. Mat. Fis. Milano'' , '''51'''  (1981)  pp. 217–233</TD></TR><TR><TD valign="top">[a32]</TD> <TD valign="top">  R. Street,   "Two dimensional sheaf theory"  ''J. Pure Appl. Algebra'' , '''23'''  (1982)  pp. 251–270</TD></TR><TR><TD valign="top">[a33]</TD> <TD valign="top">  R. Street,   "Characterization of bicategories of stacks" , ''Lecture Notes Math.'' , '''962''' , Springer  (1982)  pp. 282–291</TD></TR><TR><TD valign="top">[a34]</TD> <TD valign="top">  R. Street,   "Enriched categories and cohomology"  ''Quaest. Math.'' , '''6'''  (1983)  pp. 265–283</TD></TR><TR><TD valign="top">[a35]</TD> <TD valign="top">  R. Street,   "Categorical structures"  M. Hazewinkel (ed.) , ''Handbook of Algebra'' , '''I''' , Elsevier  (1996)  pp. 529–577</TD></TR><TR><TD valign="top">[a36]</TD> <TD valign="top">  R. Street,   "Higher categories, strings, cubes and simplex equations"  ''Appl. Categorical Struct.'' , '''3'''  (1995)  pp. 29–77 and 303</TD></TR><TR><TD valign="top">[a37]</TD> <TD valign="top">  R. Street,   R.F.C. Walters,   "Yoneda structures on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b120/b120230/b12023085.png" />-categories"  ''J. Algebra'' , '''50'''  (1978)  pp. 350–379</TD></TR><TR><TD valign="top">[a38]</TD> <TD valign="top">  R.F.C. Walters,   "Sheaves on sites as Cauchy-complete categories"  ''J. Pure Appl. Algebra'' , '''24'''  (1982)  pp. 95–102</TD></TR></table>