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topos of types

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topos theory

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Definition

Let CC be a coherent category. Write τC\tau C for the category whose objects are pairs (X,F)(X,F) where XX is an object of CC and FF is a prime filter on Sub C(X)Sub_C(X). This is the full subcategory of the category of filters? of CC spanned by the prime filters. Jet J pJ_p be the Grothendieck topology on τC\tau C induced from the coherent topology on the category of filters.

As defined by Makkai, the topos of types T(C)\mathbf{T}(C) of CC is the sheaf topos

T(C):=Sh J p(τC). \mathbf{T}(C) := Sh_{J_p}(\tau C) \,.

Note: The word “types” here has nothing to do with type theory. Instead, refers to the notion of type in model theory.

Properties

In terms of the coherent hyperdoctrine

Let CC be a coherent category. The canonical extension 𝒮 C δ\mathcal{S}_C^\delta of its coherent hyperdoctrine 𝒮 C\mathcal{S}_C is naturally an internal locale in the presheaf topos [C op,Set][C^{op}, Set] (as well as in the sheaf topos Sh coh(C)Sh_{coh}(C) for the coherent topology on CC).

The topos of types is equivalent to the topos of internal sheaves over this internal locale.

T(C)Sh(𝒮 C δ). \mathbf{T}(C) \simeq Sh(\mathcal{S}_C^\delta) \,.

This is due to (Coumans, theorem 25).

In particular this implies that there is a localic geometric morphism

T(C)Sh coh(C). \mathbf{T}(C) \to Sh_{coh}(C) \,.

Let Mod(C)Func coh(C,Set)Mod(C) \simeq Func_{coh}(C,Set) be the category of models of the coherent theory given by CC in Set.

There is a full subcategory 𝒦Mod(C)\mathcal{K} \hookrightarrow Mod(C) such that the Yoneda extension of the evaluation map ev:CFunc(𝒦,Set)ev : C \to Func(\mathcal{K}, Set) is the inverse image of a geometric morphism

ϕ ev:Set 𝒦Sh coh(C). \phi_{ev} : Set^{\mathcal{K}} \to Sh_{coh}(C) \,.

The hyperconnected / localic factorization of this morphism is through the topos of types of CC:

ϕ ev:Set 𝒦hyperconn.T(C)localicSh coh(C). \phi_{ev} : Set^{\mathcal{K}} \stackrel{hyperconn.}{\to} \mathbf{T}(C) \stackrel{localic}{\to} Sh_{coh}(C) \,.

This is (Coumans, theorem 39).

References

The notion was introduced in

  • Michael Makkai, The topos of types, Logic Year 1979-80 (Proc. Seminars and Conf. Math. Logic, Univ. Connecticut, Storrs, Conn., 1979/80), Lecture Notes in Math. 859, Springer, Berlin (1981) pp. 157-201, doi.

The relation to canonical extension is made explicit in

  • Dion Coumans, Generalizing canonical extensions to the categorical setting, pdf, to appear in Annals of Pure and Applied Logic

Revised on July 24, 2012 04:01:55 by Zoran Škoda (190.26.78.124)