topos theory

# Contents

## Idea

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

Moreover,

• $\left(U⊣F\right):ℰ\stackrel{\stackrel{U}{←}}{\underset{F}{\to }}T\mathrm{CoAlg}$(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 $T\mathrm{CoAlg}\left(𝒞\right)$ 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 $T\mathrm{Alg}\left(ℰ\right)$ 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 $p=\left(U⊣F\right):ℰ\to T\mathrm{CoAlg}\left(ℰ\right)$ 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 $\left({f}^{*}⊣{f}_{*}\right):ℰ\stackrel{\stackrel{{f}^{*}}{←}}{\underset{{f}_{*}}{\to }}ℱ$ any geometric morphism, the induced comonad

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

is evidently left exact, hence $\left({f}^{*}{f}_{*}\right)\mathrm{CoAlg}\left(ℰ\right)$ 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 $X×-:ℰ\to ℰ$).

## References

Section V 8. of

Revised on May 9, 2013 18:43:11 by Todd Trimble (67.81.93.26)