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).

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.