Global actions on topoi

Context: defn - Symtr(X, C), category of symmetries of an object

The $\infty$-category $P(C)$ of presheaves on some small category $C$ is equipped with a natural right action by the $E_1$-monoidal $\infty$-category $End(C):=Fun(C,C)$. More explicitly, the action looks like pre-composition

$$Fun(C^{op},S)\times End(C)\to Fun(C^{op},S),(F,\varphi )\mapsto F\circ \varphi$$

That is, we have $P(C)\in RMo{d}_{End(C)}(Cat)$ Example

Let $O$ be an $\infty$-operad, $C:=Al{g}_O(S{)}^{op}:=Af{f}_O$ , we have $End(C\simeq Fun(Al{g}_O(S),Al{g}_O(S))\simeq Al{g}_O(P(C))$ We get

$$P(Af{f}_O)\in RMo{d}_{Al{g}_O(P(Af{f}_O))}(Cat)$$

In words, $O$-algebras in stacks acts on $O$-stacks. Internalizing, this says it means something for an $O$-algebra in stacks to act on any $O$-stack.

In our case of interest (prismatic cohomology), this discussion accounts for the role of ring stacks.

Example In the above example, a “$O$-group-stack” $X\in Al{g}_O(P(Af{f}_O))$ is equivalent to data of an endomorphism ${\varphi }_X:Af{f}_O\to Af{f}_O$. The action above, restricted to $X$

$$P(C)\stackrel{(-,X)}{\to }P(C)\times End(C)\to P(C)$$

is an instance of “transmutation” as in Bhatt’s notes on F-gauges. For example, taking group stacks associated to crystals produces the corresponding relative prismatization functors.

Geometric Interpretation

$P(C)$ is the category of objects formally built out of $C$-objects by small colimits.

Geometrically, we can think of this as a category of moduli problems indexed by $C$. The category $C$ specifies what kind of “parameter spaces” families of objects are indexed by, and furthermore it specify how these parameter spaces can be related. A prestack must specify spaces of families for each parameter space, and also specify what it means to restrict along morphisms of parameter spaces. This in turn tells what it means to deform a family along a map of parameter spaces.

The above action says that any such moduli space $M\in P(Af{f}_O)$ admits a symmetry: that it’s equivariant along changes of coordinates in the parameter spaces - this is specified by a $\varphi \in End(Af{f}_O)\simeq Al{g}_O(P(Af{f}_O))$.

A takeaway from this picture deserves to be stated again: a group-stack is a “globally-compatible” change of coordinates for parameter spaces.