Could not include topos theory - contents
symmetric monoidal (∞,1)-category of spectra
Let be a topos. Then
Therefore, if a monad has a right adjoint, then the category of algebras 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.)
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. For (∞,1)-toposes see this MO discussion.
This appears as (MacLaneMoerdijk, VII 4. prop. 4).
is evidently left exact, hence 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 being the category of coalgebras of the comonad ).
Section V 8. of