nLab strict 2-category

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 $n$-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 $Cat$ is treated as the 1-category of strict categories with functors between them.

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 ‘$Cat$-enriched category’, we find that a strict 2-category $K$ is given by

• a collection $ob K$ of objects $a,b,c,\ldots$, together with
• a hom-category $K(a,b)$ for each $a,b$, and
• a functor $1_a : \mathbf{1} \to K(a,a)$ and a functor $comp : K(b,c) \times K(a,b) \to K(a,c)$ for each $a,b,c$ satisfying associativity and identity axioms (given here).

As for ordinary ($Set$-enriched) categories, an object $f \in K(a,b)$ is called a morphism or 1-cell from $a$ to $b$ and written $f:a\to b$ as usual. But given $f,g:a\to b$, it is now possible to have non-trivial arrows $\alpha:f\to g \in K(a,b)$, called 2-cells from $f$ to $g$ and written as $\alpha : f \Rightarrow g$. Because the hom-objects $K(a,b)$ 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 $1_f$ given by the category structure on $K(a,b)$.

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

We also have the interchange law (also called Godement law or middle 4 interchange law): because $comp$ is a functor it commutes with composition in the hom-categories, so we have (writing vertical composition with $\circ$ and horizontal as juxtaposition):

$$(\beta' \circ \beta)(\alpha' \circ \alpha) = (\beta' \alpha') \circ (\beta \alpha)$$

The axioms for associativity and unitality of $comp$ ensure that horizontal composition behaves just like composition of 1-cells in a 1-category. In particular, the action of $comp$ on objects $f,g$ of hom-categories (i.e. 1-cells of $K$) 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 $2$-category $K$ consists of stuff:

• a collection $Ob K$ or $Ob_K$ of objects or $0$-cells,
• for each object $a$ and object $b$, a collection $K(a,b)$ or $Hom_K(a,b)$ of morphisms or $1$-cells $a \to b$, and
• for each object $a$, object $b$, morphism $f\colon a \to b$, and morphism $g\colon a \to b$, a collection $K(f,g)$ or $2 Hom_K(f,g)$ of $2$-morphisms or $2$-cells $f \Rightarrow g$ or $f \Rightarrow g\colon a \to b$,

that is equipped with structure:

• for each object $a$, an identity $1_a\colon a \to a$ or $\id_a\colon a \to a$,
• for each $a,b,c$, $f\colon a \to b$, and $g\colon b \to c$, a composite $f ; g\colon a \to c$ or $g \circ f\colon a \to c$,
• for each $f\colon a \to b$, an identity $1_f\colon f \Rightarrow f$ or $\Id_f\colon f \Rightarrow f$,
• for each $f,g,h\colon a \to b$, $\eta\colon f \Rightarrow g$, and $\theta\colon g \Rightarrow h$, a vertical composite $\theta \bullet \eta\colon f \Rightarrow h$,
• for each $a,b,c$, $f\colon a \to b$, $g,h\colon b \to c$, and $\eta\colon g \Rightarrow h$, a left whiskering $\eta \triangleleft f\colon g \circ f \Rightarrow h \circ f$, and
• for each $a,b,c$, $f,g\colon a \to b$, $h\colon b \to c$, and $\eta\colon f \Rightarrow g$, a right whiskering $h \triangleright \eta \colon h \circ f \Rightarrow h \circ g$,

satisfying the following properties:

1. for each $f\colon a \to b$, the composites $f \circ \id_a$ and $\id_b \circ f$ each equal $f$,
2. for each $a \overset{f}\to b \overset{g}\to c \overset{h}\to d$, the composites $h \circ (g \circ f)$ and $(h \circ g) \circ f$ are equal,
3. for each $\eta\colon f \Rightarrow g\colon a \to b$, the vertical composites $\eta \bullet \Id_f$ and $\Id_g \bullet \eta$ both equal $\eta$,
4. for each $f \overset{\eta}\Rightarrow g \overset{\theta}\Rightarrow h \overset{\iota}\Rightarrow i\colon a \to b$, the vertical composites $\iota \bullet (\theta \bullet \eta)$ and $(\iota \bullet \theta) \bullet \eta$ are equal,
5. for each $a \overset{f}\to b \overset{g}\to c$, the whiskerings $\Id_g \triangleleft f$ and $g \triangleright \Id_f$ both equal $\Id_{g \circ f }$,
6. for each $\eta\colon f \Rightarrow g\colon a \to b$, the whiskerings $\eta \triangleleft \id_a$ and $\id_b \triangleright \eta$ equal $\eta$,
7. for each $f\colon a \to b$ and $g \overset{\eta}\Rightarrow h \overset{\theta}\Rightarrow i\colon b \to c$, the vertical composite $(\theta \triangleleft f) \bullet (\eta \triangleleft f)$ equals the whiskering $(\theta \bullet \eta) \triangleleft f$,
8. for each $f \overset{\eta}\Rightarrow g \overset{\theta}\Rightarrow h\colon a \to b$ and $i\colon b \to c$, the vertical composite $(i \triangleright \theta) \bullet (i \triangleright \eta)$ equals the whiskering $i \triangleright (\theta \bullet \eta)$,
9. for each $a \overset{f}\to b \overset{g}\to c$ and $\eta\colon h \Rightarrow i\colon c \to d$, the left whiskerings $\eta \triangleleft (g \circ f)$ and $(\eta \triangleleft g) \triangleleft f$ are equal,
10. for each $\eta\colon f \Rightarrow g\colon a \to b$ and $b \overset{h}\to c \overset{i}\to d$, the right whiskerings $i \triangleright (h \triangleright \eta)$ and $(i \circ h) \triangleright \eta$ are equal,
11. for each $f\colon a \to b$, $\eta\colon g \Rightarrow h\colon b \to c$, and $i\colon c \to d$, the whiskerings $i \triangleright (\eta \triangleleft f)$ and $(i \triangleright \eta) \triangleleft f$ are equal, and
12. for each $\eta\colon f \Rightarrow g\colon a \to b$ and $\theta\colon h \Rightarrow i\colon b \to c$, the vertical composites $(i \triangleright \eta) \bullet (\theta \triangleleft f)$ and $(\theta \triangleleft g) \bullet (h \triangleright \eta)$ are equal.

The construction in the last axiom is the horizontal composite $\theta \circ \eta\colon h \circ f \to i \circ g$. 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 (1958) for the special case Cat.

So to say it again, but a little differently: a sesquicategory consists of a category $K$ (giving the $0$-cells and $1$-cells) together with a functor

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

such that composing $K(-, -)$ with the functor $ob: Cat \to Set$ (the one sending a category to its set of objects) gives $\hom_K: K^{op} \times K \to Set$, the hom-functor for the category $K$. So: for $0$-cells $a, b$, the objects of the category $K(a, b)$ are $1$-cells $f \in \hom_K(a, b)$. The morphisms of $K(a, b)$ are $2$-cells (with $0$-source $a$ and $0$-target $b$). Composition within the category $K(a, b)$ corresponds to vertical composition.

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

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 $Cat$).

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 $n \geq 3$.

• 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 (1958), 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 (1965). 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

Original articles:

• Roger Godement, Topologie algébrique et theorie des faisceaux, Hermann, Paris (1958) [webpage]

• Charles Ehresmann, Catégories et structures, Dunod, Paris (1965)

parts of which is available in:

• Charles Ehresmann, Catégories et structures : extraits, Séminaire Ehresmann. Topologie et géométrie différentielle 6 (1964) 1-31 [eudml:112200]

Exposition and review:

• Saunders MacLane, §XII.3 of: Categories for the Working Mathematician, Graduate Texts in Mathematics 5 Springer (second ed. 1997) [doi:10.1007/978-1-4757-4721-8]

• Ross Street, Categorical Structures, in Handbook of Algebra Vol. 1 (ed. M. Hazewinkel), Elsevier Science, Amsterdam 1996. (pdf)

• Steve Lack, A 2-categories companion, In: Baez J., May J. (eds.) Towards Higher Categories. The IMA Volumes in Mathematics and its Applications, vol 152. Springer 2010 (arXiv:math.CT/0702535, doi:10.1007/978-1-4419-1524-5_4)

• Birgit Richter, Section 9.5 of: From categories to homotopy theory, Cambridge Studies in Advanced Mathematics 188, Cambridge University Press 2020 (doi:10.1017/9781108855891, book webpage, pdf)

• Niles Johnson, Donald Yau, Section 2.3 of: 2-Dimensional Categories, Oxford University Press 2021 (arXiv:2002.06055, doi:10.1093/oso/9780198871378.001.0001)

The special case of strict (2,1)-categories:

• Peter H. H. Fantham, Eric J. Moore, Groupoid Enriched Categories and Homotopy Theory, Canadian Journal of Mathematics 35 3 (1983) 385-416 (doi:10.4153/CJM-1983-022-8)