nLab
topos of algebras over a monad

Idea

If a monad or comonad $T$ on a topos $ℰ$ is sufficiently well behaved, then the category of (co)algebras $T\mathrm{Alg}(C)$ over the (co)monad is itself an (elementary) topos.

Properties

General

Proposition

Let $ℰ$ be a topos. Then

• if a comonad $T:ℰ\to ℰ$ is left exact, then the category of coalgebras $T\mathrm{CoAlg}(ℰ)$ is itself an (elementary) topos.

  Moreover,

  • the cofree/forgetful adjunction

    $$(U⊣F):ℰ\stackrel{\stackrel{U}{\leftarrow }}{\underset{F}{\to }}T\mathrm{CoAlg}$$(U \dashv F) : \mathcal{E} \stackrel{\overset{U}{\leftarrow}}{\underset{F}{\to}} T CoAlg

    is a geometric morphism.

  • If $T$ is furthermore accessible and $ℰ$ is a sheaf topos, then also $T\mathrm{CoAlg}(𝒞)$ is a sheaf topos.

  • Even if $T$ is merely pullback-preserving, the category of coalgebras is a topos.

• Therefore, if a monad $T:ℰ\to ℰ$ has a right adjoint, then the category of algebras $T\mathrm{Alg}(ℰ)$ is itself an (elementary) topos. (Because the right adjoint of a monad carries a comonad structure, evidently a left exact comonad, and there is a canonical equivalence between the category of algebras over the monad and the category of coalgebras over the comonad.)

• If a monad on a topos is pullback-preserving and idempotent, then the category of algebras is a subtopos (hence the category of sheaves for some Lawvere-Tierney topology).

The result for left exact comonads appears for instance as (MacLaneMoerdijk, V 8. theorem 4); the result for monads possessing a right adjoint appears in op. cit. as corollary 7. The statement on pullback-preserving comonads is given in The Elephant, A.4.2.3.

Image factorization of toposes

Proposition

The geometric morphisms of the form $p=(U⊣F):ℰ\to T\mathrm{CoAlg}(ℰ)$ from prop. 1 are precisely, up to equivalence, the geometric surjections.

This appears as (MacLaneMoerdijk, VII 4. prop. 4).

This way the geometric surjection/embedding factorization in Topos is constructed. See there for more.

Examples

Observation

For $(f^{*}⊣f_{*}):ℰ\stackrel{\stackrel{f^{*}}{\leftarrow }}{\underset{f_{*}}{\to }}ℱ$ any geometric morphism, the induced comonad

$$f^{*}{f}_{*}:ℰ\to ℰ$$f^* f_* : \mathcal{E} \to \mathcal{E}

is evidently left exact, hence $(f^{*}{f}_{*})\mathrm{CoAlg}(ℰ)$ is a topos of coalgebras.

Observation

The so-called “fundamental theorem of topos theory”, that an overcategory of a topos is a topos, is a corollary of the result that the category of coalgebras of a pullback-preserving comonad on a topos is a topos (the slice $ℰ/X$ being the category of coalgebras of the comonad $X\times -:ℰ\to ℰ$).

Related concepts

• Also a category of algebras over a commutative finitary algebraic theory in Set has properties very close to the properties of a Grothendieck topos, in fact only one axiom has to be modified. This is one of the themes of the theory of vectoids of Nikolai Durov.

References

Section V 8. of

• Saunders MacLane, Ieke Moerdijk, Sheaves in Geometry and Logic