The Stacks project

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

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

Comments (2)

Comment #2042 by Matthew Emerton on

In the displayed diagram for condition (b), the domain of is labelled as , rather than as

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 003N. Beware of the difference between the letter 'O' and the digit '0'.