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topos of algebras over a monad

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Idea

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

Properties

General

Proposition

Let \mathcal{E} be a topos. Then

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

    Moreover,

    • the cofree/forgetful adjunction

      (UF):FUTCoAlg (U \dashv F) : \mathcal{E} \stackrel{\overset{U}{\leftarrow}}{\underset{F}{\to}} T CoAlg

      is a geometric morphism.

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

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

  • Therefore, if a monad T:T : \mathcal{E} \to \mathcal{E} has a right adjoint, then the category of algebras TAlg()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.

Image factorization of toposes

Proposition

The geometric morphisms of the form p=(UF):TCoAlg() 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.

Examples

Observation

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

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

is evidently left exact, hence (f *f *)CoAlg()(f^* f_*) CoAlg(\mathcal{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 /X\mathcal{E}/X being the category of coalgebras of the comonad X×:X \times - \colon \mathcal{E} \to \mathcal{E}).

References

Section V 8. of

Revised on January 5, 2014 09:03:44 by Urs Schreiber (89.204.154.192)