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bicategory

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Bicategories

Idea

A bicategory is a particular algebraic notion of weak 2-category (in fact, the earliest to be formulated, and still the one in most common use). The idea is that a bicategory is a category weakly enriched over Cat: the hom-objects of a bicategory are hom-categories, but the associativity and unity laws of enriched categories hold only up to coherent isomorphism.

Definition

A bicategory BB consists of

  • A collection of objects x,y,z,x,y,z,\dots, also called 00-cells;
  • For each pair of 00-cells x,yx,y, a category B(x,y)B(x,y), whose objects are called morphisms or 11-cells and whose morphisms are called 2-morphisms or 22-cells;
  • For each 00-cell xx, a distinguished 11-cell 1 xB(x,x)1_x\in B(x,x) called the identity morphism or identity 11-cell at xx;
  • For each triple of 00-cells x,y,zx,y,z, a functor :B(y,z)×B(x,y)B(x,z){\circ}\colon B(y,z)\times B(x,y) \to B(x,z) called horizontal composition;
  • For each pair of 00-cells x,yx,y, natural isomorphisms called unitors: id B(x,y)const 1 xid B(x,y)const 1 yid B(x,y):B(x,y)B(x,y)id_{B(x,y)} \circ const_{1_x} \cong id_{B(x,y)} \cong const_{1_y} \circ id_{B(x,y)}\colon B(x,y) \to B(x,y); and
  • For each quadruple of 00-cells w,x,y,zw,x,y,z, a natural isomorphism called the associator between the two functors from B y,z×B x,y×B w,xB_{y,z} \times B_{x,y} \times B_{w,x} to B w,zB_{w,z} built out of {\circ}; such that
  • Such that the pentagon identity is satisfied by the associators.

If there is exactly one 00-cell, say **, then the definition is exactly the same as a monoidal structure on the category B(*,*)B(*,*). This is one of the motivating examples behind the delooping hypothesis and the general notion of k-tuply monoidal n-category.

Details

Here we spell out the above definition in full detail. Compare to the detailed definition of strict 22-category, which is written in the same style but is simpler.

A bicategory BB consists of

  • a collection ObBOb B or Ob BOb_B of objects or 00-cells,
  • for each object aa and object bb, a collection B(a,b)B(a,b) or Hom B(a,b)Hom_B(a,b) of morphisms or 11-cells aba \to b, and
  • for each object aa, object bb, morphism f:abf\colon a \to b, and morphism g:abg\colon a \to b, a collection B(f,g)B(f,g) or 2Hom B(f,g)2Hom_B(f,g) of 22-morphisms or 22-cells fgf \Rightarrow g or fg:abf \Rightarrow g\colon a \to b,

equipped with

  • for each object aa, an identity 1 a:aa1_a\colon a \to a or id a:aa\id_a\colon a \to a,
  • for each a,b,ca,b,c, f:abf\colon a \to b, and g:bcg\colon b \to c, a composite f;g:acf ; g\colon a \to c or gf:acg \circ f\colon a \to c,
  • for each f:abf\colon a \to b, an identity or 22-identity 1 f:ff1_f\colon f \Rightarrow f or Id f:ff\Id_f\colon f \to f,
  • for each f,g,h:abf,g,h\colon a \to b, η:fg\eta\colon f \Rightarrow g, and θ:gh\theta\colon g \Rightarrow h, a vertical composite θη:fh\theta \bullet \eta\colon f \Rightarrow h,
  • for each a,b,ca,b,c, f:abf\colon a \to b, g,h:bcg,h\colon b \to c, and η:gh\eta\colon g \Rightarrow h, a left whiskering ηf:gfhf\eta \triangleleft f\colon g \circ f \Rightarrow h \circ f,
  • for each a,b,ca,b,c, f,g:abf,g\colon a \to b, h:bch\colon b \to c, and η:fg\eta\colon f \Rightarrow g, a right whiskering hη:hfhgh \triangleright \eta \colon h \circ f \Rightarrow h \circ g,
  • for each f:abf\colon a \to b, a left unitor λ f:fid af\lambda_f\colon f \circ \id_a \Rightarrow f and an inverse left unitor λ¯ f:ffid a\bar{\lambda}_f\colon f \Rightarrow f \circ \id_a,
  • for each f:abf\colon a \to b, a right unitor ρ f:id bff\rho_f\colon \id_b \circ f \Rightarrow f, and an inverse right unitor ρ¯ f:fid bf\bar{\rho}_f\colon f \Rightarrow \id_b \circ f, and
  • for each afbgchda \overset{f}\to b \overset{g}\to c \overset{h}\to d, an associator α f,g,h:h(gf)(hg)f\alpha_{f,g,h}\colon h \circ (g \circ f) \Rightarrow (h \circ g) \circ f and an inverse associator α¯ f,g,h:(hg)fh(gf)\bar{\alpha}_{f,g,h}\colon (h \circ g) \circ f \Rightarrow h \circ (g \circ f),

such that

  • for each η:fg:ab\eta\colon f \Rightarrow g\colon a \to b, the vertical composites ηId f\eta \bullet \Id_f and Id gη\Id_g \bullet \eta both equal η\eta,
  • for each fηgθhιi:abf \overset{\eta}\Rightarrow g \overset{\theta}\Rightarrow h \overset{\iota}\Rightarrow i\colon a \to b, the vertical composites ι(θη)\iota \bullet (\theta \bullet \eta) and (ιθ)η(\iota \bullet \theta) \bullet \eta are equal,
  • for each afbgca \overset{f}\to b \overset{g}\to c, the whiskerings Id gf\Id_g \triangleleft f and gId fg \triangleright \Id_f both equal Id gf\Id_{g \circ f },
  • for each f:abf\colon a \to b and gηhθi:bcg \overset{\eta}\Rightarrow h \overset{\theta}\Rightarrow i\colon b \to c, the vertical composite (θf)(ηf)(\theta \triangleleft f) \bullet (\eta \triangleleft f) equals the whiskering (θη)f(\theta \bullet \eta) \triangleleft f,
  • for each fηgθh:abf \overset{\eta}\Rightarrow g \overset{\theta}\Rightarrow h\colon a \to b and i:bci\colon b \to c, the vertical composite (iθ)(iη)(i \triangleright \theta) \bullet (i \triangleright \eta) equals the whiskering i(θη)i \triangleright (\theta \bullet \eta),
  • for each η:fg:ab\eta\colon f \Rightarrow g\colon a \to b, the vertical composites λ g(ηid a)\lambda_g \bullet (\eta \triangleleft \id_a) and ηλ f\eta \bullet \lambda_f are equal,
  • for each η:fg:ab\eta\colon f \Rightarrow g\colon a \to b, the vertical composites ρ g(id bη)\rho_g \bullet (\id_b \triangleright \eta) and ηρ f\eta \bullet \rho_f are equal,
  • for each afbgca \overset{f}\to b \overset{g}\to c and η:hi:cd\eta\colon h \Rightarrow i\colon c \to d, the vertical composites α f,g,i(η(gf))\alpha_{f,g,i} \bullet (\eta \triangleleft (g \circ f)) and ((ηg)f)α f,g,h((\eta \triangleleft g) \triangleleft f) \bullet \alpha_{f,g,h} are equal,
  • for each f:abf\colon a \to b, η:gh:bc\eta\colon g \Rightarrow h\colon b \to c, and i:cdi\colon c \to d, the vertical composites α f,h,i(i(ηf))\alpha_{f,h,i} \bullet (i \triangleright (\eta \triangleleft f)) and ((iη)f)α f,g,i((i \triangleright \eta) \triangleleft f) \bullet \alpha_{f,g,i} are equal,
  • for each η:fg:ab\eta\colon f \Rightarrow g\colon a \to b and bhcidb \overset{h}\to c \overset{i}\to d, the vertical composites α g,h,i(i(hη))\alpha_{g,h,i} \bullet (i \triangleright (h \triangleright \eta)) and ((ih)η)α f,h,i((i \circ h) \triangleright \eta) \bullet \alpha_{f,h,i} are equal,
  • for each η:fg:ab\eta\colon f \Rightarrow g\colon a \to b and θ:hi:bc\theta\colon h \Rightarrow i\colon b \to c, the vertical composites (iη)(θf)(i \triangleright \eta) \bullet (\theta \triangleleft f) and (θg)(hη)(\theta \triangleleft g) \bullet (h \triangleright \eta) are equal,
  • for each f:abf\colon a \to b, the vertical composites λ fλ¯ f:ff\lambda_f \bullet \bar{\lambda}_f\colon f \Rightarrow f and λ¯ fλ f:fid afid a\bar{\lambda}_f \bullet \lambda_f\colon f \circ \id_a \Rightarrow f \circ \id_a equal the appropriate identity 22-morphisms,
  • for each f:abf\colon a \to b, the vertical composites ρ fρ¯ f:ff\rho_f \bullet \bar{\rho}_f\colon f \Rightarrow f and ρ¯ fρ f:id bfid bf\bar{\rho}_f \bullet \rho_f\colon \id_b \circ f \Rightarrow \id_b \circ f equal the appropriate identity 22-morphisms,
  • for each afbgchda \overset{f}\to b \overset{g}\to c \overset{h}\to d, the vertical composites α f,g,hα¯ f,g,h:(hg)f(hg)f\alpha_{f,g,h} \bullet \bar{\alpha}_{f,g,h}\colon (h \circ g) \circ f \Rightarrow (h \circ g) \circ f and α¯ f,g,hα f,g,h:h(gf)h(gf)\bar{\alpha}_{f,g,h} \bullet \alpha_{f,g,h}\colon h \circ (g \circ f) \Rightarrow h \circ (g \circ f) equal the appropriate identity 22-morphisms,
  • for each afbgca \overset{f}\to b \overset{g}\to c, the vertical composite (λ gf)α f,id b,g(\lambda_g \triangleleft f) \bullet \alpha_{f,\id_b,g} equals the whiskering gρ fg \triangleright \rho_f, and
  • for each afbgchdiea \overset{f}\to b \overset{g}\to c \overset{h}\to d \overset{i}\to e, the vertical composites ((α g,h,if)α f,hg,i)(iα f,g,h)((\alpha_{g,h,i} \triangleleft f) \bullet \alpha_{f,h \circ g,i}) \bullet (i \triangleright \alpha_{f,g,h}) and α f,g,ihα gf,h,i\alpha_{f,g,i \circ h}\bullet \alpha_{g \circ f,h,i} are equal.

It is quite possible that there are errors or omissions in this list, although they should be easy to correct. The point is not that one would want to write out the definition in such elementary terms (although apparently I just did anyway) but rather that one can.

Examples

  • Any strict 2-category is a bicategory in which the unitors and associator are identities. This includes Cat, MonCat?, the algebras for any strict 2-monad, and so on, at least as classically conceived.

  • Categories, anafunctors, and natural transformations, which is a more appropriate definition of Cat in the absence of the axiom of choice, form a bicategory that is not a strict 2-category. Indeed, without the axiom of choice, the proper notion of bicategory is anabicategory.

  • Rings, bimodules, and bimodule homomorphisms are the prototype for many similar examples. Notably, we can generalize from rings to enriched categories.

  • Objects, spans, and morphisms of spans in any category with pullbacks also form a bicategory.

  • The fundamental 2-groupoid? of a space is a bicategory which is not necessarily strict (although it can be made strict fairly easily when the space is Hausdorff by quotienting by thin homotopy, see path groupoid and fundamental infinity-groupoid). When the space is a CW-complex, there are easier and more computationally amenable equivalent strict 2-categories, such as that arising from the fundamental crossed complex.

Coherence theorems

One way to state the coherence theorem for bicategories is that every bicategory is equivalent to a strict 22-category. This “strictification” is not obtained naively by forcing composition to be associative, but (at least in one construction) by freely adding new composites which are strictly associative. Another way to state the coherence theorem is that every formal diagram of the constraints (associators and unitors) commutes.

Note that n=2n=2 is the greatest value of nn for which every weak nn-category is equivalent to a fully strict one; see semi-strict infinity-category and Gray-category.

The proof of the coherence theorem is basically the same as the proof of the coherence theorem for monoidal categories. An abstract approach can be found in Power’s paper “A general coherence result.”

Terminology

Classically, “2-category” meant strict 2-category, with “bicategory” used for the weak notion. This led to the more general use of the prefix “2-” for strict (that is, strictly Cat-enriched) notions and “bi-” for weak ones. For example, classically a “2-adjunction” means a Cat-enriched adjunction, consisting of two strict 2-functors F,GF,G and a strictly Cat-natural isomorphism of categories D(FX,Y)C(X,GY)D(F X, Y)\cong C(X, G Y), while a “biadjunction” means the weak version, consisting of two weak 2-functors and a pseudo natural equivalence D(FX,Y)C(X,GY)D(F X, Y)\simeq C(X, G Y). Similarly for “2-equivalence” and “biequivalence,” and “2-limit” and “bilimit.”

We often use “2-category” to mean a strict or weak 2-category without prejudice, although we do still use “bicategory” to refer to the particular classical algebraic notion of weak 2-category. We try to avoid the more general use of “bi-” meaning “weak,” however. For one thing, is it confusing; a “biproduct” could mean a weak 2-limit, but it could also mean an object which is both a product and a coproduct (which happens quite frequently in additive categories).

Moreover, in most cases the prefix is unnecessary, since once we know we are working in a bicategory, there is usually no point in considering strict notions at all. Fully weak limits are really the only sensible ones to ask for in a bicategory, and likewise for fully weak adjunctions and equivalences. Even in a strict 2-category, while we might need to say “strict” sometimes to be clear, we don't need to say “22-”, since we know that we are not working in a mere category. (Max Kelly pushed this point.)

When we do have a strict 2-category, however, other strict notions can be quite technically useful, even if our ultimate interest is in the weak ones. This is somewhat analogous to the use of strict structures to model weak ones in homotopy theory; see here and here for good introductions to this sort of thing.

Discussion about the use of the term “weak enrichment” above is at weak enrichment.

References

See also the references at 2-category.

Power, A. J. A general coherence result. J. Pure Appl. Algebra 57 (1989), no. 2, 165–173. doi:10.1016/0022-4049(89)90113-8 MR0985657

Revised on October 30, 2013 14:54:54 by David Roberts (129.127.252.5)