The Stacks project

Lemma 4.33.7. Assume $p : \mathcal{S} \to \mathcal{C}$ is a fibred category. Assume given a choice of pullbacks for $p : \mathcal{S} \to \mathcal{C}$.

  1. For any pair of composable morphisms $f : V \to U$, $g : W \to V$ there is a unique isomorphism

    \[ \alpha _{g, f} : (f \circ g)^\ast \longrightarrow g^\ast \circ f^\ast \]

    as functors $\mathcal{S}_ U \to \mathcal{S}_ W$ such that for every $y\in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$ the following diagram commutes

    \[ \xymatrix{ g^\ast f^\ast y \ar[r] & f^\ast y \ar[d] \\ (f \circ g)^\ast y \ar[r] \ar[u]^{(\alpha _{g, f})_ y} & y } \]

  2. If $f = \text{id}_ U$, then there is a canonical isomorphism $\alpha _ U : \text{id} \to (\text{id}_ U)^*$ as functors $\mathcal{S}_ U \to \mathcal{S}_ U$.

  3. The quadruple $(U \mapsto \mathcal{S}_ U, f \mapsto f^*, \alpha _{g, f}, \alpha _ U)$ defines a pseudo functor from $\mathcal{C}^{opp}$ to the $(2, 1)$-category of categories, see Definition 4.29.5.

Proof. In fact, it is clear that the commutative diagram of part (1) uniquely determines the morphism $(\alpha _{g, f})_ y$ in the fibre category $\mathcal{S}_ W$. It is an isomorphism since both the morphism $(f \circ g)^*y \to y$ and the composition $g^*f^*y \to f^*y \to y$ are strongly cartesian morphisms lifting $f \circ g$ (see discussion following Definition 4.33.1 and Lemma 4.33.2). In the same way, since $\text{id}_ x : x \to x$ is clearly strongly cartesian over $\text{id}_ U$ (with $U = p(x)$) we see that there exists an isomorphism $(\alpha _ U)_ x : x \to (\text{id}_ U)^*x$. (Of course we could have assumed beforehand that $f^*x = x$ whenever $f$ is an identity morphism, but it is better for the sake of generality not to assume this.) We omit the verification that $\alpha _{g, f}$ and $\alpha _ U$ so obtained are transformations of functors. We also omit the verification of (3). $\square$

Comments (2)

Comment #6306 by Andrew on

In Hypothesis (1), I'm confused as to how the morphisms and could fail to be composable.

Comment #6418 by on

OK, I think there usually is no problem with having redundant information.

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Your email address will not be published.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 02XO. Beware of the difference between the letter 'O' and the digit '0'.



View Lemma 4.33.7 as pdf