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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 equivalent 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

(See also the section below on sesquicategories, which provide a conceptual package for the stuff and structure described below.)

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

  • 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,

that is equipped with structure:

  • 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,

satisfying the following properties:

  1. 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,
  2. 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,
  3. 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,
  4. 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,
  5. 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 },
  6. 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,
  7. 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,
  8. 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),
  9. 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,
  10. 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,
  11. 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
  12. 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.

2-categories as sesquicategories

The fine-grained description in the previous subsection can be concisely repackaged by saying that a 2-category is a sesquicategory that satisfies the interchange axiom, i.e., the last axiom (12) which gives the horizontal composite construction. This description is essentially patterned after the “five rules of functorial calculus” introduced by Godement for the special case Cat.

So to say it again, but a little differently: a sesquicategory consists of a category KK (giving the 00-cells and 11-cells) together with a functor

K(,):K op×KCatK(-, -): K^{op} \times K \to Cat

such that composing K(,)K(-, -) with the functor ob:CatSetob: Cat \to Set (the one sending a category to its set of objects) gives hom K:K op×KSet\hom_K: K^{op} \times K \to Set, the hom-functor for the category KK. So: for 00-cells a,ba, b, the objects of the category K(a,b)K(a, b) are 11-cells fhom K(a,b)f \in \hom_K(a, b). The morphisms of K(a,b)K(a, b) are 22-cells (with 00-source aa and 00-target bb). Composition within the category K(a,b)K(a, b) corresponds to vertical composition.

For each object aa of KK and each morphism h:bch: b \to c of KK, there is a functor K(a,h):K(a,b)K(a,c)K(a, h): K(a, b) \to K(a, c). This is right whiskering; it sends a 2-cell η\eta (a morphism of K(a,b)K(a, b)) to a morphism hηh \triangleright \eta of K(b,c)K(b, c). Similarly, for each object cc and morphism f:abf: a \to b, there is a functor K(f,c):K(b,c)K(a,c)K(f, c): K(b, c) \to K(a, c). This is left whiskering; it sends a 2-cell η\eta (a morphism of K(b,c)K(b, c)) to a morphism ηf\eta \triangleleft f of K(a,c)K(a, c).

The long list of compatibility properties enumerated in the previous subsection, all except the last, are concisely summarized in the definition of sesquicategory as recalled above. For example, property (8) just says that left whiskering preserves vertical composition, as it must since it is a functor (a morphism in CatCat).

In summary, a sesquicategory consists of “stuff” and structure as described in the previous subsection, satisfying properties 1-11. A 2-category is then a sesquicategory that further satisfies the interchange axiom (12). Some further illumination of this point of view can be obtained by contemplating string diagrams for 2-categories, where the interchange axiom corresponds to isotopies of (planar, progressive) string diagrams during which the relative heights of nodes labeled by 2-cells are interchanged.

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 composition with identity morphisms strictly satisfies the required identity law. In a weak 2-category these laws may hold only up to coherent 2-morphisms.

History

As intimated above, the essential rules which abstractly govern the behavior of functors and natural transformations and their various compositions were made explicit by Godement, in his “five rules of functorial calculus”. He did not however go as far as use these rules to define the abstract notion of 2-category; this step was taken a few years later by Ehresmann, who in fact defined double categories, and 2-categories as a special case. In any event, the primitive compositional operations in Godement’s account were what we call vertical compositions and whiskerings, with horizontal composition of natural transformations being a derived operation (made unambiguous in the presence of the interchange axiom). Indeed, horizontal composition is often called the Godement product.

A few years after that, Bénabou introduced the notion of bicategory.

Literature references for the abstract notion of sesquicategory, a structure in which vertical compositions and whiskerings are primitive, do not seem to be abundant, but they are mentioned for example in Street together with the observation that 2-categories are special types of sesquicategories (page 535).

References

  • Roger Godement, Topologie algébrique et theorie des faisceaux, Hermann, Paris, 1958.
  • Ross Street, Categorical Structures, in Handbook of Algebra Vol. 1 (ed. M. Hazewinkel), Elsevier Science, Amsterdam 1996. (pdf)

Revised on July 15, 2016 14:01:36 by Todd Trimble (67.81.95.215)