nLab
strict 2-category

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Strict 22-categories

Idea

  • A strict 2-category is a directed 2-graph equipped with a composition operation on adjacent 1-cells and 2-cells which is strictly unital and associative.

  • The concept of a strict 2-category is the simplest generalization of a category to a higher category. It is the one-step categorification of the concept of a category.

The term 2-category implicitly refers to a globular structure. By contrast, double categories are based on cubes instead. The two notions are closely related, however: every strict 2-category gives rise to several strict double categories, and every double category has several underlying 2-categories.

Notice that double category is another term for 2-fold category. Strict 2-categories may be identified with those strict 2-fold/double categories whose category of vertical morphisms is discrete, or those whose category of horizontal morphisms is discrete.

(And similarly, strict globular n-categories may be identified with those n-fold categories for which all cube faces “in one direction” are discrete. A similar statement for weak nn-categories is to be expected, but little seems to be known about this.)

Definition

A strict 2-category, often called simply a 2-category, is a category enriched over Cat, where CatCat is treated as the 1-category of strict categories.

Similarly, a strict 2-groupoid is a groupoid enriched over groupoids. This is also called a globular strict 2-groupoid, to emphasise the underlying geometry. The category of strict 2-groupoids is eqivalent to the category of crossed modules over groupoids. It is also equivalent to the category of (strict) double groupoids with connections.

They are also special cases of strict globular omega-groupoids, and the category of these is equivalent to the category of crossed complexes.

Details

Working out the meaning of ’CatCat-enriched category’, we find that a strict 2-category KK is given by

  • a collection obKob K of objects a,b,c,a,b,c,\ldots, together with
  • a hom-category K(a,b)K(a,b) for each a,ba,b, and
  • a functor 1 a:1K(a,a)1_a : \mathbf{1} \to K(a,a) and a functor comp:K(b,c)×K(a,b)K(a,c)comp : K(b,c) \times K(a,b) \to K(a,c) for each a,b,ca,b,c

satisfying associativity and identity axioms (given here).

As for ordinary (SetSet-enriched) categories, an object fK(a,b)f \in K(a,b) is called a morphism or 1-cell from aa to bb and written f:abf:a\to b as usual. But given f,g:abf,g:a\to b, it is now possible to have non-trivial arrows α:fgK(a,b)\alpha:f\to g \in K(a,b), called 2-cells from ff to gg and written as α:fg\alpha : f \Rightarrow g. Because the hom-objects K(a,b)K(a,b) are by definition categories, 2-cells carry an associative and unital operation called vertical composition. The identities for this operation, of course, are the identity 2-cells 1 f1_f given by the category structure on K(a,b)K(a,b).

The functor compcomp gives us an operation of horizontal composition on 2-cells. Functoriality of compcomp then says that given α:fg:ab\alpha : f \Rightarrow g : a\to b and β:fg:bc\beta : f' \Rightarrow g' : b\to c, the composite comp(β,α)\comp(\beta,\alpha) is a 2-cell βα:ffgg:ac\beta \alpha : f'f \Rightarrow g'g : a \to c. Note that the boundaries of the composite 2-cell are the composites of the boundaries of the components.

We also have the interchange law: because compcomp is a functor it commutes with composition in the hom-categories, so we have (writing vertical composition with \circ and horizontal as juxtaposition):

(ββ)(αα)=(βα)(βα) (\beta' \circ \beta)(\alpha' \circ \alpha) = (\beta' \alpha') \circ (\beta \alpha)

The axioms for associativity and unitality of compcomp ensure that horizontal composition behaves just like composition of 1-cells in a 1-category. In particular, the action of compcomp on objects f,gf,g of hom-categories (i.e. 1-cells of KK) is the usual composite of morphisms.

More details

In even more detail, a strict 22-category KK consists of

  • a collection ObKOb K or Ob KOb_K of objects or 00-cells,
  • for each object aa and object bb, a collection K(a,b)K(a,b) or Hom K(a,b)Hom_K(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 K(f,g)K(f,g) or 2Hom K(f,g)2 Hom_K(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 1 f:ff1_f\colon f \Rightarrow f or Id f:ff\Id_f\colon f \Rightarrow 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, and
  • 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,

such that

  • for each f:abf\colon a \to b, the composites fid af \circ \id_a and id bf\id_b \circ f each equal ff,
  • for each afbgchda \overset{f}\to b \overset{g}\to c \overset{h}\to d, the composites h(gf)h \circ (g \circ f) and (hg)f(h \circ g) \circ f are equal,
  • 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 η:fg:ab\eta\colon f \Rightarrow g\colon a \to b, the whiskerings ηid a\eta \triangleleft \id_a and id bη\id_b \triangleright \eta equal η\eta,
  • 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 afbgca \overset{f}\to b \overset{g}\to c and η:hi:cd\eta\colon h \Rightarrow i\colon c \to d, the left whiskerings η(gf)\eta \triangleleft (g \circ f) and (ηg)f(\eta \triangleleft g) \triangleleft f 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 whiskerings i(ηf)i \triangleright (\eta \triangleleft f) and (iη)f(i \triangleright \eta) \triangleleft f 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 right whiskerings i(hη)i \triangleright (h \triangleright \eta) and (ih)η(i \circ h) \triangleright \eta are equal, and
  • 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) \circ (\theta \triangleleft f) and (θg)(hη)(\theta \triangleleft g) \circ (h \triangleright \eta) are equal.

The construction in the last axiom is the horizontal composite θη:hfig\theta \circ \eta\colon h \circ f \to i \circ g. It is possible (and probably more common) to take the horizontal composite as basic and the whiskerings as derived operations. This results in fewer, but more complicated, axioms.

Remarks

  • A strict 2-category is the same as a strict omega-category which is trivial in degree n3n \geq 3.

  • This is to be contrasted with a weak 2-category called a bicategory. In a strict 2-category composition of 1-morphisms is strictly associative and comoposition with identity morphisms strictly satisfies the required identity law. In a weak 2-category these laws may hold only up to coherent 2-morphisms.

Revised on August 31, 2013 20:16:13 by Anonymous Coward (74.102.92.190)