theory of presheaf type



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Theories of presheaf type though being geometric theories of a particular “simple” and tractable type are yet ubiquitous in the sense that every geometric theory is a quotient theory? of some theory of presheaf type.



A geometric theory 𝕋\mathbb{T} is of presheaf type if its classifying topos Set[𝕋]Set[\mathbb{T}] is equivalent to a presheaf topos.


  • Any cartesian theory 𝕋\mathbb{T} being (modulo neglectable size issues1) classified by Set 𝕋Mod fp(Set)Set^{\mathbb{T}-Mod_{fp}(Set)} is of presheaf type with 𝕋-Mod fp(Set)\mathbb{T}\text{-}Mod_{fp}(Set) the category of finitely presentable 𝕋\mathbb{T}-models in SetSet.

  • More concretely, e.g. the theory of objects or the theory of intervals are of presheaf type being classified by the object classifier respectively by the topos of simplicial sets.

  • The inconsistent theory with axiom \top\vdash\bot is of presheaf type since it is classified by the initial Grothendieck topos 1Pr()\mathbf{1}\simeq Pr(\emptyset), the presheaf topos on the empty category.



For a category \mathcal{M} the following are equivalent:

  • \mathcal{M} is finitely accessible.

  • Pts(Set 𝒞)\mathcal{M}\simeq Pts(Set^{\mathcal{C}}) the category of points of some presheaf topos.

  • 𝕋-Mod(Set)\mathcal{M}\simeq \mathbb{T}\text{-}Mod(Set) for some theory 𝕋\mathbb{T} of presheaf type.

  • Ind-𝒞\mathcal{M}\simeq Ind \text{-}\mathcal{C} for some small category 𝒞\mathcal{C}.

Cf. Beke (2004, p.923) and the references given there. In fact, these equivalences are mostly (direct consequences of) classical results in the theory of accessible categories or Grothendieck toposes.

Note that despite the above equivalences the finite accessibility of 𝕋-Mod(Set)\mathbb{T}\text{-}Mod(Set) does not imply that 𝕋\mathbb{T} itself is of presheaf type! One sees this already in case 𝕋-Mod(Set)=\mathbb{T}\text{-}Mod(Set)=\emptyset since there famously are non trivial (Boolean sheaf) toposes lacking points (“non empty generalized spaces without points”) yet up to Morita equivalence the only theory of presheaf type corresponding to the finite accessibility of the empty category is the inconsistent theory.

The following proposition shows in which sense theories of presheaf type are still determined by their (finitely presentable) models in SetSet:


A geometric theory 𝕋\mathbb{T} is of presheaf type iff (modulo neglectable size issues)

Set[𝕋][𝕋-Mod fp(Set),Set].Set[\mathbb{T}]\simeq [\mathbb{T}\text{-}Mod_{fp}(Set),Set]\, .

Proof. “\Rightarrow”:

(Cf. Caramello 2018, pp.198f)

By assumption Set[𝕋][𝒞,Set]Set[\mathbb{T}]\simeq [\mathcal{C}, Set]. Since [𝒞,Set][𝒞^,Set][\mathcal{C},Set]\simeq [\hat{\mathcal{C}}, Set] (by Johnstone 2002, p.10) we can assume that 𝒞\mathcal{C} is Cauchy complete.

We have:

  • Ind-𝒞Flat(𝒞 op,Set)Ind\text{-}\mathcal{C}\simeq Flat(\mathcal{C}^{op}, Set) (by Johnstone 2002, p.723, or Caramello 2018, p.198)

  • (Ind-𝒞) fp𝒞(Ind\text{-}\mathcal{C})_{fp}\simeq \mathcal{C} (by Johnstone 2002 4.2.2.(iii), p.724)

  • 𝕋-Mod(Set)Flat(𝒞 op,Set)\mathbb{T}\text{-}Mod(Set)\simeq Flat(\mathcal{C}^{op}, Set) (from Diaconescu’s theorem).

Whence (Ind-𝒞) fp(𝕋-Mod(Set)) fp=𝕋-Mod fp(Set)𝒞(Ind\text{-}\mathcal{C})_{fp}\simeq (\mathbb{T}\text{-}Mod (Set))_{fp} =\mathbb{T}\text{-}Mod_{fp}(Set)\simeq \mathcal{C} and, accordingly, [𝒞,Set]][𝕋-Mod fp(Set),Set][\mathcal{C},Set]]\simeq [\mathbb{T}\text{-}Mod_{fp}(Set),Set]. \qed

In other words, theories of presheaf type are precisely those geometric theories 𝕋\mathbb{T} such that their classifying toposes Set[𝕋]Set[\mathbb{T}] can be represented as presheaf toposes Set 𝕋-Mod fp(Set)Set^{\mathbb{T}\text{-}Mod_{fp}(Set)}.

This implies e.g. that any consistent theory of presheaf type has models in SetSet.


  1. This means here (and in the following) that the essentially small category 𝕋-Mod fp(Set)\mathbb{T}\text{-}Mod_{fp}(Set) has to be replaced by a skeleton.

Last revised on June 2, 2021 at 06:50:19. See the history of this page for a list of all contributions to it.