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$G$ -sets are strictly monadic over sets

Statement

For a fixed group $G$ , the underlying-set functor from the category of left $G$ -sets and equivariant maps to $S e t$ is strictly monadic.

Facts & Assumptions

Given: A fixed group $G$ with identity $e$ .

[L1]

A left $G$ -action satisfies $e \cdot x = x$ and $(g h) \cdot x = g \cdot (h \cdot x)$ (Left group actions, transitive actions, and faithful actions).

[L2]

A map of $G$ -sets is equivariant when $f (g \cdot x) = g \cdot f (x)$ for every $g, x$ (Equivariant maps and isomorphisms of group actions).

[L3]

A $T$ -algebra structure $a : T X \to X$ satisfies $a η_{X} = 1_{X}$ and $a T (a) = a μ_{X}$ , and an algebra homomorphism satisfies the corresponding square (Algebra and algebra homomorphism for a monad).

[L4]

A functor is strictly monadic when its comparison with the Eilenberg–Moore category is an isomorphism (Monadic and strictly monadic functors).

Proof

technique · direct

1.1

Define $F (X) = G \times X$ with action $h \cdot (g, x) = (h g, x)$ . A function $u : X \to U Y$ into a $G$ -set extends uniquely to the equivariant map $\overset{ˉ}{u} (g, x) = g \cdot u (x)$ , whose inverse correspondence evaluates at $(e, x)$ . This natural bijection gives the free-action adjunction $F ⊣ U$ .

L1construct

2.1

Its induced monad is $T (X) = G \times X$ , with $T (f) = 1_{G} \times f$ , $η_{X} (x) = (e, x)$ , and $μ_{X} (g, (h, x)) = (g h, x)$ . The group identity and associativity laws verify the monad equations.

step 1.1L1algebra

3.1

A map $a : G \times X \to X$ satisfies the two algebra laws in [L3] exactly when $a (e, x) = x$ and $a (g, a (h, x)) = a (g h, x)$ , which are the action laws in [L1]. This includes the empty set, the trivial group, and trivial actions.

step 2.1L1L3

4.1

The algebra-homomorphism equation is $f (a (g, x)) = b (g, f (x))$ , exactly the equivariance condition in [L2].

step 3.1L2L3

5.1

By steps 3.1 and 4.1, the comparison for the adjunction constructed in step 1.1 is bijective on objects and morphisms, with inverse given by the same action structure and underlying functions. It is therefore an isomorphism over $S e t$ , so the underlying-set functor is strictly monadic by [L4].

step 1.1step 3.1step 4.1L4∎

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Sources

E. Riehl, Category Theory in Context, 2nd ed., Corollary 5.5.3(ii) and Exercise 5.5.iv (standard reference, not scraped)

GG-sets are strictly monadic over sets