Group actions and descent | Higher Topos Theory notes

Example 2.13. (Group actions)

This example previews the groupoid objects introduced in Subsection 2.2.1. Informally, a groupoid object is a simplicial object encoding objects, invertible morphisms between them, and all the coherences for composition. A group object is the special case in which the object of objects is terminal.

Let $$T$$ be a topos, and let $$G$$ be a group object in $$T$$ , regarded as a groupoid object $G_{\bullet}\colon \simp\catop \to T$ satisfying $G_{0} = *$ . Let $\bB G := \colim_{[n] \in \simp\catop} G_n$ . Then there is an equivalence

T_{/\bB G} \quad \simeq \quad \lim_{[n] \in \simp} T_{/G^n} \quad=:\quad \Mod_G(T).

In other words, a module over $$G$$ is a simplicial object $X_{\bullet}$ with a cartesian transformation $X_{\bullet} \to G_{\bullet}$ , which we can depict (suppressing degeneracies) as a simplicial diagram of face maps:

with each square cartesian. In particular, this implies $X_1 \simeq G \times X_0$ . The two maps $G \times X_0 \simeq X_1 \to X_0$ can then be interpreted as the projection map and an action map. The rest of the simplicial diagram $X_{\bullet}$ records the higher coherences of this $$G$$ -action on $$X_0$$ .