The Stacks project

Definition 4.29.1. A (strict) $2$-category $\mathcal{C}$ consists of the following data

1. A set of objects $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$.

2. For each pair $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ a category $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y)$. The objects of $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y)$ will be called $1$-morphisms and denoted $F : x \to y$. The morphisms between these $1$-morphisms will be called $2$-morphisms and denoted $t : F' \to F$. The composition of $2$-morphisms in $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y)$ will be called vertical composition and will be denoted $t \circ t'$ for $t : F' \to F$ and $t' : F'' \to F'$.

3. For each triple $x, y, z\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ a functor

   \[ (\circ , \star ) : \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(y, z) \times \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y) \longrightarrow \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, z). \]

   The image of the pair of $1$-morphisms $(F, G)$ on the left hand side will be called the composition of $F$ and $G$, and denoted $F\circ G$. The image of the pair of $2$-morphisms $(t, s)$ will be called the horizontal composition and denoted $t \star s$.

These data are to satisfy the following rules:

1. The set of objects together with the set of $1$-morphisms endowed with composition of $1$-morphisms forms a category.

2. Horizontal composition of $2$-morphisms is associative.

3. The identity $2$-morphism $\text{id}_{\text{id}_ x}$ of the identity $1$-morphism $\text{id}_ x$ is a unit for horizontal composition.

Definition 4.29.2. Let $\mathcal{C}$ be a $2$-category. A sub $2$-category $\mathcal{C}'$ of $\mathcal{C}$, is given by a subset $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C}')$ of $\mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$ and sub categories $\mathop{\mathrm{Mor}}\nolimits _{\mathcal{C}'}(x, y)$ of the categories $\mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(x, y)$ for all $x, y \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C}')$ such that these, together with the operations $\circ $ (composition $1$-morphisms), $\circ $ (vertical composition $2$-morphisms), and $\star $ (horizontal composition) form a $2$-category.

Definition 4.29.4. Two objects $x, y$ of a $2$-category are equivalent if there exist $1$-morphisms $F : x \to y$ and $G : y \to x$ such that $F \circ G$ is $2$-isomorphic to $\text{id}_ y$ and $G \circ F$ is $2$-isomorphic to $\text{id}_ x$.

Definition 4.29.5. Let $\mathcal{A}$ be a category and let $\mathcal{C}$ be a $2$-category.

1. A functor from an ordinary category into a $2$-category will ignore the $2$-morphisms unless mentioned otherwise. In other words, it will be a “usual” functor into the category formed out of 2-category by forgetting all the 2-morphisms.

2. A weak functor, or a pseudo functor $\varphi $ from $\mathcal{A}$ into the 2-category $\mathcal{C}$ is given by the following data

   1. a map $\varphi : \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}) \to \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$,

   2. for every pair $x, y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$, and every morphism $f : x \to y$ a $1$-morphism $\varphi (f) : \varphi (x) \to \varphi (y)$,

   3. for every $x\in \mathop{\mathrm{Ob}}\nolimits (A)$ a $2$-morphism $\alpha _ x : \text{id}_{\varphi (x)} \to \varphi (\text{id}_ x)$, and

   4. for every pair of composable morphisms $f : x \to y$, $g : y \to z$ of $\mathcal{A}$ a $2$-morphism $\alpha _{g, f} : \varphi (g \circ f) \to \varphi (g) \circ \varphi (f)$.

   These data are subject to the following conditions:

   1. the $2$-morphisms $\alpha _ x$ and $\alpha _{g, f}$ are all isomorphisms,

   2. for any morphism $f : x \to y$ in $\mathcal{A}$ we have $\alpha _{\text{id}_ y, f} = \alpha _ y \star \text{id}_{\varphi (f)}$:

      \[ \xymatrix{ \varphi (x) \rrtwocell ^{\varphi (f)}_{\varphi (f)}{\ \ \ \ \text{id}_{\varphi (f)}} & & \varphi (y) \rrtwocell ^{\text{id}_{\varphi (y)}}_{\varphi (\text{id}_ y)}{\alpha _ y} & & \varphi (y) } = \xymatrix{ \varphi (x) \rrtwocell ^{\varphi (f)}_{\varphi (\text{id}_ y) \circ \varphi (f)}{\ \ \ \ \alpha _{\text{id}_ y, f}} & & \varphi (y) } \]

   3. for any morphism $f : x \to y$ in $\mathcal{A}$ we have $\alpha _{f, \text{id}_ x} = \text{id}_{\varphi (f)} \star \alpha _ x$,

   4. for any triple of composable morphisms $f : w \to x$, $g : x \to y$, and $h : y \to z$ of $\mathcal{A}$ we have

      \[ (\text{id}_{\varphi (h)} \star \alpha _{g, f}) \circ \alpha _{h, g \circ f} = (\alpha _{h, g} \star \text{id}_{\varphi (f)}) \circ \alpha _{h \circ g, f} \]

      in other words the following diagram with objects $1$-morphisms and arrows $2$-morphisms commutes

      \[ \xymatrix{ \varphi (h \circ g \circ f) \ar[d]_{\alpha _{h, g \circ f}} \ar[rr]_{\alpha _{h \circ g, f}} & & \varphi (h \circ g) \circ \varphi (f) \ar[d]^{\alpha _{h, g} \star \text{id}_{\varphi (f)}} \\ \varphi (h) \circ \varphi (g \circ f) \ar[rr]^{\text{id}_{\varphi (h)} \star \alpha _{g, f}} & & \varphi (h) \circ \varphi (g) \circ \varphi (f) } \]