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2.1.1 $2$-Categories

Let $\operatorname{\mathcal{C}}$ be a strict $2$-category (Definition 2.1.0.1). Then the composition of $1$-morphisms in $\operatorname{\mathcal{C}}$ is strictly associative: that is, given a triple of composable $1$-morphisms

\[ f: W \rightarrow X \quad \quad g: X \rightarrow Y \quad \quad h: Y \rightarrow Z \]

of $\operatorname{\mathcal{C}}$, we have an equality $h \circ (g \circ f) = (h \circ g) \circ f$. Our goal in this section is to introduce the more general notion of (non-strict) $2$-category, where we weaken the associativity requirement: rather than demand that the $1$-morphisms $h \circ (g \circ f)$ and $(h \circ g) \circ f$ are identical, we instead ask for a specified isomorphism $\alpha _{h,g,f}: h \circ (g \circ f) \xRightarrow {\sim } (h \circ g) \circ f$ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(W, Z)$. In order to obtain a sensible theory, we must require that these isomorphisms satisfy a certain “higher-order” associative law.

Definition 2.1.1.1 (Bénabou). A $2$-category $\operatorname{\mathcal{C}}$ consists of the following data:

  • A collection of objects $\{ X, Y, Z, \cdots \} $; we will write $X \in \operatorname{\mathcal{C}}$ to indicate that $X$ is an object of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $X, Y \in \operatorname{\mathcal{C}}$, a category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y)$. We refer to objects $f$ of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ as $1$-morphisms from $X$ to $Y$ and write $f: X \rightarrow Y$ to indicate that $f$ is a $1$-morphism from $X$ to $Y$. Given a pair of $1$-morphisms $f,g \in \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$, we refer to morphisms from $f$ to $g$ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ as $2$-morphisms from $f$ to $g$. We will sometimes write $\gamma : f \Rightarrow g$ or $f \xRightarrow {\gamma } g$ to indicate that $\gamma $ is a $2$-morphism from $f$ to $g$.

  • For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, a composition functor

    \[ \circ : \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( Y, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z). \]
  • For every object $X \in \operatorname{\mathcal{C}}$, an identity $1$-morphism $\operatorname{id}_{X} \in \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, a natural isomorphism $\lambda $ from the functor

    \[ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \quad \quad f \mapsto \operatorname{id}_{Y} \circ f \]

    to the identity functor, which we refer to as the left unit constraint. That is, $\lambda $ associates to each $1$-morphism $f: X \rightarrow Y$ an invertible $2$-morphism $\lambda _{f}: \operatorname{id}_{Y} \circ f \xRightarrow {\sim } f$.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, a natural isomorphism $\rho $ from the functor

    \[ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \quad \quad f \mapsto f \circ \operatorname{id}_ X \]

    to the identity functor, which we refer to as the right unit constraint. That is, $\rho $ associates to each $1$-morphism $f: X \rightarrow Y$ an invertible $2$-morphism $\rho _{f}: f \circ \operatorname{id}_ X \xRightarrow {\sim } f$.

  • For every quadruple of objects $W, X, Y, Z \in \operatorname{\mathcal{C}}$, a natural isomorphism $\alpha $ from the functor

    \[ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(W,X) \rightarrow \underline{\operatorname{Hom}}(W, Z) \quad \quad (h,g,f) \mapsto h \circ (g \circ f) \]

    to the functor

    \[ \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Y) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(W,X) \rightarrow \underline{\operatorname{Hom}}(W, Z) \quad \quad (h,g,f) \mapsto (h \circ g) \circ f, \]

    which we refer to as the associativity constraint. We denote the value of $\alpha $ on a triple $(h,g,f)$ by $\alpha _{h,g,f}: h \circ (g \circ f) \xRightarrow {\sim } (h \circ g) \circ f$.

These data are required to satisfy the following pair of conditions:

$(T)$

For every pair of composable $1$-morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$ in $\operatorname{\mathcal{C}}$, the diagram of isomorphisms

\[ \xymatrix@C =50pt@R=50pt{ g \circ (\operatorname{id}_ Y \circ f) \ar@ {=>}[rr]^{ \alpha _{g, \operatorname{id}_ Y, f} }_{\sim } \ar@ {=>}[dr]_{ \operatorname{id}_ g \circ \lambda _ f}^{\sim } & & (g \circ \operatorname{id}_ Y) \circ f \ar@ {=>}[dl]^{ \rho _{g} \circ \operatorname{id}_{f} }_{\sim } \\ & g \circ f & \\ } \]

commutes in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( X, Z)$.

$(P)$

For every quadruple of composable $1$-morphisms

\[ V \xrightarrow {e} W \xrightarrow {f} X \xrightarrow {g} Y \xrightarrow {h} Z \]

in $\operatorname{\mathcal{C}}$, the diagram of isomorphisms

\[ \xymatrix@C =-10pt@R=30pt{ & h \circ ((g \circ f) \circ e) \ar@ {=>}[rr]^{ \alpha _{h, g \circ f, e} }_{\sim } & & (h \circ (g \circ f)) \circ e \ar@ {=>}[dr]^{ \alpha _{h, g, f} \circ \operatorname{id}_ e}_{\sim } & \\ h \circ (g \circ (f \circ e) ) \ar@ {=>}[ur]^{ \operatorname{id}_ h \circ \alpha _{g,f,e} }_{\sim } \ar@ {=>}[drr]_{ \alpha _{h,g,f\circ e}}^{\sim } & & & & ((h \circ g) \circ f) \circ e \\ & & (h \circ g) \circ (f \circ e) \ar@ {=>}[urr]_{\alpha _{ h \circ g, f, e} }^{\sim } & & } \]

commutes in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( V, Z)$.

Remark 2.1.1.2. Definition 2.1.1.1 was introduced by Bénabou in [MR0220789]. Beware that Bénabou uses the term bicategory for what we call a $2$-category.

Remark 2.1.1.3. In the situation of Definition 2.1.1.1, we will refer to axiom $(T)$ as the triangle identity and axiom $(P)$ as the pentagon identity.

Example 2.1.1.4 (Strict $2$-Categories). Let $\operatorname{\mathcal{C}}$ be any strict $2$-category (in the sense of Definition 2.1.0.1). Then $\operatorname{\mathcal{C}}$ can be viewed as a $2$-category (in the sense of Definition 2.1.1.1) by taking each left unit constraints $\lambda _{f}$, right unit constraints $\rho _{f}$, and associativity constraints $\alpha _{h,g,f}$ to be the identity (in this case, the triangle and pentagon identities are automatically satisfied). Conversely, if $\operatorname{\mathcal{C}}$ is a $2$-category in which each of the isomorphisms $\lambda _{f}$, $\rho _{f}$, and $\alpha _{h,g,f}$ is an identity $2$-morphism of $\operatorname{\mathcal{C}}$, then $\operatorname{\mathcal{C}}$ can be viewed as a strict $2$-category.

Variant 2.1.1.5 (Strictly Unitary $2$-Categories). We say that a $2$-category $\operatorname{\mathcal{C}}$ is strictly unitary if, for every $1$-morphism $f: X \rightarrow Y$ in $\operatorname{\mathcal{C}}$, we have equalities

\[ \operatorname{id}_{Y} \circ f = f = f \circ \operatorname{id}_{X}, \]

and the left and right unit constraints $\lambda _{f}$, $\rho _{f}$ are the identity $2$-morphisms from $f$ to itself. Every strict $2$-category is strictly unitary, but the converse is false: we will see later that every $2$-category is isomorphic (in an appropriate sense) to a strictly unitary $2$-category (see Example 2.1.6.14).

Remark 2.1.1.6. Let $\operatorname{\mathcal{C}}$ be a strictly unitary $2$-category. Then the triangle identity $(T)$ of Definition 2.1.1.1 can be stated more simply as the assertion that for every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$, the associativity constraint $\alpha _{g, \operatorname{id}_ Y, f}$ coincides with the identity map from $g \circ (\operatorname{id}_{Y} \circ f) = g \circ f = (g \circ \operatorname{id}_{Y} ) \circ f$ to itself.

Example 2.1.1.7 (Ordinary Categories). Every ordinary category can be regarded as a $2$-category. More precisely, to each category $\operatorname{\mathcal{C}}$ we can associate a (strict) $2$-category $\operatorname{\mathcal{C}}'$ as follows:

  • The objects of $\operatorname{\mathcal{C}}'$ are the objects of $\operatorname{\mathcal{C}}$.

  • For every pair of objects $X,Y \in \operatorname{\mathcal{C}}$, objects of the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X,Y)$ are elements of the set $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y)$, and every morphism in $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X,Y)$ is an identity morphism.

  • For every triple of objects $X,Y,Z \in \operatorname{\mathcal{C}}$, the composition functor

    \[ \circ : \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}( Y, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X,Z) \]

    is given on objects by the composition map $\operatorname{Hom}_{\operatorname{\mathcal{C}}}(Y,Z) \times \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,Z)$.

  • For every object $X \in \operatorname{\mathcal{C}}$, the identity object $\operatorname{id}_{X} \in \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}'}(X,X)$ coincides with the identity morphism $\operatorname{id}_ X \in \operatorname{Hom}_{\operatorname{\mathcal{C}}}(X,X)$.

In this situation, we will generally abuse terminology by identifying the $2$-category $\operatorname{\mathcal{C}}'$ with the ordinary category $\operatorname{\mathcal{C}}$ (see Example 2.1.5.8).

Remark 2.1.1.8. Let $\operatorname{\mathcal{C}}$ be a $2$-category. Then $\operatorname{\mathcal{C}}$ can be obtained from an ordinary category (via the construction of Example 2.1.1.7) if and only if every $2$-morphism in $\operatorname{\mathcal{C}}$ is an identity $2$-morphism.

Notation 2.1.1.9. Let $\operatorname{\mathcal{C}}$ be a $2$-category. We will generally follow the convention of denoting objects of $\operatorname{\mathcal{C}}$ by capital Roman letters, $1$-morphisms of $\operatorname{\mathcal{C}}$ by lowercase Roman letters, and $2$-morphisms of $\operatorname{\mathcal{C}}$ by lowercase Greek letters. However, we will often violate this convention when discussing specific examples. For instance, when studying the (strict) $2$-category $\mathbf{ Cat }$ of small categories (Example 2.1.0.2), we denote objects using calligraphic letters (such as $\operatorname{\mathcal{C}}$ and $\operatorname{\mathcal{D}}$) and $1$-morphisms using uppercase Roman letters (such as $F$ and $G$).

Warning 2.1.1.10. Let $\operatorname{\mathcal{C}}$ be a $2$-category. Then there are two different notions of composition for the $2$-morphisms of $\operatorname{\mathcal{C}}$:

$(V)$

Let $X$ and $Y$ be objects of $\operatorname{\mathcal{C}}$. Suppose we are given $1$-morphisms $f,g,h: X \rightarrow Y$ and a pair of $2$-morphisms

\[ \gamma : f \Rightarrow g \quad \quad \delta : g \Rightarrow h. \]

We can then apply the composition law in the ordinary category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ to obtain a $2$-morphism $f \Rightarrow h$, which we refer to as the vertical composition of $\gamma $ and $\delta $.

$(H)$

Let $X$, $Y$, and $Z$ be objects of $\operatorname{\mathcal{C}}$. Suppose we are given $2$-morphisms $\gamma : f \Rightarrow g$ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y)$ and $\gamma ': f' \Rightarrow g'$ in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(Y,Z)$. Then the image of $(\gamma ', \gamma )$ under the composition law

\[ \circ : \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}( Y, Z) \times \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Y) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X, Z), \]

is a $2$-morphism from $f' \circ f$ to $g' \circ g$, which will refer to as the horizontal composition of $\gamma $ and $\gamma '$.

The terminology is motivated by the following graphical representations of the data described in $(V)$ and $(H)$:

\[ \xymatrix@C =50pt{ X \ar@ /^30pt/[r]^{f} \ar [r]^{g} \ar@ /_30pt/[r]_{h} \ar@ {=>}[]+<32pt,20pt>;+<32pt,10pt>^{\gamma } \ar@ {=>}[]+<32pt,-10pt>;+<32pt,-20pt>^{\delta } & Y & X \ar@ /^15pt/[r]^{f} \ar@ /_15pt/[r]_{g} \ar@ {=>}[]+<32pt,10pt>;+<32pt,-10pt>^{\gamma } & Y \ar@ /^15pt/[r]^{f'} \ar@ /_15pt/[r]_{g'} \ar@ {=>}[]+<32pt,10pt>;+<32pt,-10pt>^{\gamma '}& Z. } \]

To avoid confusion, we will generally denote the vertical composition of $2$-morphisms $\gamma $ and $\delta $ by $\delta \gamma $ and the horizontal composition of $2$-morphisms $\gamma $ and $\gamma '$ by $\gamma ' \circ \gamma $.

We conclude this section by recording a few consequences of the axiomatics of Definition 2.1.1.1.

Proposition 2.1.1.11. Let $\operatorname{\mathcal{C}}$ be a $2$-category containing a pair of composable $1$-morphisms $f: X \rightarrow Y$ and $g: Y \rightarrow Z$. Then:

$(1)$

The associativity constraint $\alpha _{\operatorname{id}_ Z,g,f}: \operatorname{id}_{Z} \circ (g \circ f) \Rightarrow (\operatorname{id}_ Z \circ g) \circ f$ is given by the (vertical) composition

\[ \operatorname{id}_{Z} \circ (g \circ f) \xRightarrow { \lambda _{g \circ f} } g \circ f \xRightarrow {\lambda ^{-1}_{g} \circ \operatorname{id}_{f} } (\operatorname{id}_ Z \circ g) \circ f. \]
$(2)$

The associativity constraint $\alpha _{g,f,\operatorname{id}_ X}: g \circ (f \circ \operatorname{id}_ X) \Rightarrow (g \circ f) \circ \operatorname{id}_ X$ is given by the (vertical) composition

\[ g \circ (f \circ \operatorname{id}_ X) \xRightarrow { \operatorname{id}_{g} \circ \rho _{f}} g \circ f \xRightarrow { \rho ^{-1}_{g \circ f} } (g \circ f) \circ \operatorname{id}_ X \]

Example 2.1.1.12. Let $\operatorname{\mathcal{C}}$ be a strictly unitary $2$-category (Variant 2.1.1.5). Then Proposition 2.1.1.11 can be formulated more simply as follows: for every pair of composable $1$-morphisms $X \xrightarrow {f} Y \xrightarrow {g} Z$, the associativity constraints $\alpha _{ \operatorname{id}_ Z, g, f}$ and $\alpha _{g,f,\operatorname{id}_ X}$ are equal to the identity (as $2$-morphisms from $g \circ f$ to itself).

Proof of Proposition 2.1.1.11. We will prove $(2)$; the proof of $(1)$ is similar. Set $e = \operatorname{id}_ X$, and consider the diagram of isomorphisms

\[ \xymatrix@C =0pt@R=40pt{ & g \circ ((f \circ e) \circ e) \ar@ {=>}[rr]^{ \alpha _{g, f \circ e, e} } \ar@ {=>}[d]^-{\rho _ f} & & (g \circ (f \circ e)) \circ e \ar@ {=>}[dr]^{ \alpha _{g, f, e} } \ar@ {=>}[dl]_-{\rho _ f} & \\ g \circ (f \circ (e \circ e) ) \ar@ {=>}[ur]^{ \alpha _{f,e,e} } \ar@ {=>}[drr]_{ \alpha _{g,f,e \circ e}} \ar@ {=>}[r]^-{\lambda _ e} & g \circ (f \circ e) \ar@ {=>}[r]^-{\alpha _{g,f,e}} & (g \circ f) \circ e & & ((g \circ f) \circ e) \circ e \ar@ {=>}[ll]_-{\rho _{g \circ f}} \\ & & (g \circ f) \circ (e \circ e). \ar@ {=>}[urr]_{\alpha _{ g \circ f, e, e} } \ar@ {=>}[u]_-{\lambda _ e } & & } \]

Here the outer cycle of the diagram commutes by the pentagon identity for $\operatorname{\mathcal{C}}$, the triangles on the upper left and lower right commute by the triangle identity for $\operatorname{\mathcal{C}}$, and the upper and lower square diagrams commute by the functoriality of the associativity constraints. It follows that the triangle on the upper left commutes: that is, the identity $\alpha _{g,f,\operatorname{id}_ X} = \rho ^{-1}_{g \circ f} (\operatorname{id}_ g \circ \rho _{f} )$ holds after applying the functor $(\bullet \circ \operatorname{id}_ X): \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z)$. Since this functor is an equivalence of categories (it is isomorphic to the identity functor by means of the right unit constraint $\rho $), we conclude that the identity $\alpha _{g,f,\operatorname{id}_ X} = \rho _{g \circ f}^{-1} (\operatorname{id}_ g \circ \rho _{f} )$ holds in $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,Z)$ itself. $\square$

Corollary 2.1.1.13. Let $\operatorname{\mathcal{C}}$ be a $2$-category and let $X$ be an object of $\operatorname{\mathcal{C}}$. Then the left and right unit constraints

\[ \lambda _{\operatorname{id}_ X}, \rho _{\operatorname{id}_ X}: \operatorname{id}_{X} \circ \operatorname{id}_ X \Rightarrow \operatorname{id}_ X \]

are the same.

Proof. Set $e = \operatorname{id}_ X$. From the functoriality of the right unit constraint, we deduce that the diagram

\[ \xymatrix { (e \circ e) \circ e \ar@ {=>}[r]^{\rho _{e \circ e}} \ar@ {=>}[d]^{\rho _ e \circ \operatorname{id}_ e} & e \circ e \ar@ {=>}[d]^{ \rho _{e} } \\ e \circ e \ar@ {=>}[r]^-{ \rho _ e} & e } \]

commutes (in the category $\underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$). Since $\rho _{e}$ is an isomorphism, it follows that we have an equality. $\rho _{e \circ e} = \rho _{e} \circ \operatorname{id}_ e$. We now compute

\[ \operatorname{id}_{e} \circ \lambda _ e = (\rho _ e \circ \operatorname{id}_ e) \alpha _{e,e,e} = \rho _{e \circ e} \alpha _{e,e,e} = \operatorname{id}_{e} \circ \rho _ e, \]

where the first equality follows from the triangle identity in $\operatorname{\mathcal{C}}$ and the third equality from Proposition 2.1.1.11. The functor $(e \circ \bullet ): \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X) \rightarrow \underline{\operatorname{Hom}}_{\operatorname{\mathcal{C}}}(X,X)$ is an equivalence of categories (since it is isomorphic to the identity functor via the left unit constraint $\lambda $), so we must have $\lambda _{e} = \rho _{e}$. $\square$