nLab strict 2-category

Idea

The concept of strict 2-categories is the simplest generalization of that of categories to the $n$-categories of higher category theory. It is a one-step categorification of the concept of a category with strict choice of structure morphisms.

More concretely, a strict 2-category is a directed 2-graph equipped with horizontal- and vertical composition of adjacent 1-cells (1-morphisms) and 2-cells (2-morphisms), respectively, which is strictly unital and associative in both directions, and such that both types of composition are compatible (the “interchange law”).

A quick way of making this precise is to say that strict 2-categories are Cat-enriched categories, see below.

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

Strict 2-categories

A strict 2-category, often called simply a 2-category, is a category enriched over Cat, where Cat is regarded as the 1-category of strict categories with functors between them and equipped with the cartesian monoidal-structure given by forming product categories.

Strict 2-groupoids

Similarly, a strict 2-groupoid is a groupoid enriched over the 1-category Grpd. 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 (above), we find that a strict 2-category $K$ is given by

• a class $ob K$ of objects $a,b,c,\ldots$, together with

• a hom-category $K(a,b)$ for each $a,b$, and

• functors

  • (identity morphisms) $1_a : \mathbf{1} \to K(a,a)$

  • (composition) $comp : K(b,c) \times K(a,b) \to K(a,c)$

  for each $a,b,c$, satisfying associativity and unitality axioms (as given at enriched category).

As for ordinary ($Set$-enriched) categories, an object $f \in K(a,b)$ is called a 1-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-morphisms or 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 composition 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 morphisms 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 1-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 the following 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 composition-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

$$K(-, -) \;\colon\; K^{op} \times K \longrightarrow Cat$$

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

• Strict 2-categories are the same as a strict omega-categories which are 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.

In dependent type theory

In dependent type theory, there are multiple notions of a “strict 2-category”, because there are multiple notions of a category:

• A strict 2-category could be a precategory enriched over the cartesian monoidal category Cat of strict categories
• A strict 2-category could be a strict category enriched over the cartesian monoidal category Cat of strict categories
• A strict 2-category could be a univalent category enriched over the cartesian monoidal category Cat of strict categories

The first definition is a naive translation of “strict 2-category” from set theory to dependent type theory, but in the absence of axiom K or a similar axiom, these strict 2-categories behave differently from the strict 2-categories as defined in set theory. The second definition adds a 0-truncation condition to the type of objects to ensure that the strict 2-categories actually behave like the strict 2-categories in set theory. The third definition satisfies the principle of equivalence: equality of objects is the same as isomorphism of strict categories, and ensures that strict 2-categories are h-groupoids.

Related concepts

• strict 2-groupoid

• strict 2-functor

• strict 2-natural transformation

• strict adjoint 2-functor

• strict 2-group

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 later by Bénabou (1965). In any event, the primitive compositional operations in Godement (1958) were what we call vertical composition and whiskering, 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 introducing 2-categories, Bénabou introduced the more general notion of bicategories.

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 (1996) together with the observation that 2-categories are special types of sesquicategories (page 535).

References

The “five rules of functorial calculus” (above) were formulated (as: cinq règles de calcul fonctoriel) in:

• Roger Godement, Appendice (pp. 269) of: Topologie algébrique et theorie des faisceaux, Actualités Sci. Ind. 1252, Hermann, Paris (1958) [webpage, pdf]

The definition of “double categories” (of which, at least in hindsight, strict 2-categories are an immediate special case) is due to:

• Charles Ehresmann, Catégories double et catégories structurées, C.R. Acad. Paris 256 (1963) 1198-1201 [pdf, gallica]

also discussed (according to reviewers who have seen the text) in:

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

However, Ehresmann did not isolate the notion of strict 2-categories as such.

But apparently inspired by Ehresmann (1963) the actual definition of strict 2-categories is due to:

• Jean Bénabou, Example (5) of: Catégories relatives, C. R. Acad. Sci. Paris 260 (1965) 3824-3827 [gallica]

  (conceived as Cat-enriched categories and under the name “2-categories”)

• Jean-Marie Maranda, Def. 1 in: Formal categories, Canadian Journal of Mathematics 17 (1965) 758-801 [doi:10.4153/CJM-1965-076-0, pdf]

  (conceived as Cat-enriched categories and under the name “categories of the second type”)

• Samuel Eilenberg, G. Max Kelly, p. 425 of: Closed Categories, in: S. Eilenberg, D. K. Harrison, S. MacLane, H. Röhrl (eds.): Proceedings of the Conference on Categorical Algebra - La Jolla 1965, Springer (1966) [doi:10.1007/978-3-642-99902-4]

  (expressed entirely in components and under the name hypercategories)

and the notion is invoked for various purposes (such as in speaking of Cat as a 2-category) in:

• Jean Bénabou, Structures algébriques dans les catégories, Cahiers de topologie et géométrie différentielle, 10 1 (1968) 1-126 [doi:CTGDC_1968__10_1_1_0]

Exposition and review:

• Ross Street, p. 535 in: Categorical Structures, in Handbook of Algebra Vol. 1 (ed. M. Hazewinkel), Elsevier Science, Amsterdam (1996) [doi:10.1016/S1570-7954(96)80019-2, pdf, pdf ISBN:978-0-444-82212-3]

• 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]

• 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)

See also

• Wikipedia, Strict 2-category

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)