Lemma 4.41.2 (004B): 2-Yoneda lemma

Lemma 4.41.2 (2-Yoneda lemma). Let $\mathcal{S}\to \mathcal{C}$ be fibred in groupoids. Let $U \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. The functor

\[ \mathop{\mathrm{Mor}}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{C}/U, \mathcal{S}) \longrightarrow \mathcal{S}_ U \]

given by $G \mapsto G(\text{id}_ U)$ is an equivalence.

Proof. Make a choice of pullbacks for $\mathcal{S}$ (see Definition 4.33.6). We define a functor

\[ \mathcal{S}_ U \longrightarrow \mathop{\mathrm{Mor}}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{C}/U, \mathcal{S}) \]

as follows. Given $x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{S}_ U)$ the associated functor is

on objects: $(f : V \to U) \mapsto f^*x$, and
on morphisms: the arrow $(g : V'/U \to V/U)$ maps to the composition

\[ (f \circ g)^*x \xrightarrow {(\alpha _{g, f})_ x} g^*f^*x \rightarrow f^*x \]

where $\alpha _{g, f}$ is as in Lemma 4.35.2.

We omit the verification that this is an inverse to the functor of the lemma. $\square$

Comments (2)

Comment #7202 by Anonymous on April 20, 2022 at 18:47

Perhaps this could be stated for a general fibred category $\mathcal{S} \rightarrow \mathcal{C}$ , rather than just a category fibred in groupoids. I believe the only changes needed would be to replace $\mathop{\mathrm{Mor}}\nolimits _{\textit{Cat}/\mathcal{C}}(\mathcal{C}/U, \mathcal{S})$ with $\mathop{\mathrm{Mor}}\nolimits _{\textit{Fib}/\mathcal{C}}(\mathcal{C}/U, \mathcal{S})$ (two instances), and to replace the reference to Lemma 003V with a reference to Lemma 02XO.

Comment #7323 by Johan on May 04, 2022 at 16:57

OK, I added it as a separate lemma and I put both of them in a new section on the 2 Yoneda lemma. Thanks very much. See this commit.

The Stacks project

Comments (2)

Post a comment