Could not include topos theory - contents
symmetric monoidal (∞,1)-category of spectra
If a monad or comonad $T$ on a topos $\mathcal{E}$ is sufficiently well behaved, then the category of (co)algebras $T Alg(C)$ over the (co)monad is itself an (elementary) topos.
Let $\mathcal{E}$ be a topos. Then
if a comonad $T : \mathcal{E} \to \mathcal{E}$ is left exact, then the category of coalgebras $T CoAlg(\mathcal{E})$ is itself an (elementary) topos.
Moreover,
the cofree/forgetful adjunction
is a geometric morphism.
If $T$ is furthermore accessible and $\mathcal{E}$ is a sheaf topos, then also $T CoAlg(\mathcal{C})$ is a sheaf topos.
Even if $T$ is merely pullback-preserving, the category of coalgebras is a topos.
Therefore, if a monad $T : \mathcal{E} \to \mathcal{E}$ has a right adjoint, then the category of algebras $T Alg(\mathcal{E})$ 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.
The geometric morphisms of the form $p = (U \dashv F) : \mathcal{E} \to T CoAlg(\mathcal{E})$ 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.
For $(f^* \dashv f_*) : \mathcal{E} \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}} \mathcal{F}$ any geometric morphism, the induced comonad
is evidently left exact, hence $(f^* f_*) CoAlg(\mathcal{E})$ is a topos of coalgebras.
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 $\mathcal{E}/X$ being the category of coalgebras of the comonad $X \times - \colon \mathcal{E} \to \mathcal{E}$).
Section V 8. of