nLab
theory of flat functors

Contents

Context

Topos Theory

topos theory

Background

Toposes

Internal Logic

Topos morphisms

Extra stuff, structure, properties

Cohomology and homotopy

In higher category theory

Theorems

Contents

Idea

The theory of J-continuous flat functors on a site (𝒞,J)(\mathcal{C},J) is the geometric theory 𝕋 J 𝒞\mathbb{T}_J^{\mathcal{C}} whose models in a Grothendieck topos \mathcal{E} correpond precisely to J-continuous flat functors 𝒞\mathcal{C}\to\mathcal{E} and, by well known representation theorems, to geometric morphisms Sh(𝒞,J)\mathcal{E}\to Sh(\mathcal{C},J) whence 𝕋 J 𝒞\mathbb{T}_J^{\mathcal{C}} provides a uniform albeit (in general) uneconomical axiomatization of the geometric theory classified by Sh(𝒞,J)Sh(\mathcal{C},J).

Definition

Let (𝒞,J)(\mathcal{C},J) be a small site. The theory of J-continuous flat functors is given by the following geometric theory 𝕋 J 𝒞\mathbb{T}_J^{\mathcal{C}} over the signature Σ\Sigma which has a sort A\lceil A\rceil for every object AA of 𝒞\mathcal{C} and function symbols f:AB\lceil f\rceil :\lceil A\rceil\to\lceil B\rceil for every function f:ABf:A\to B:

  • axioms xi(x)=x\top\vdash_x \lceil i\rceil(x)=x for all identity morphisms ii.

  • axioms xf(x)=h(g(x))\top\vdash_x\lceil f\rceil(x)=\lceil h\rceil(\lceil g\rceil (x)) for all triples f,g,hf,g,h with f=ghf=g\circ h.

  • an axiom A𝒞(x:A)\top\vdash\underset{A\in\mathcal{C}}{\bigvee} (\exists x:\lceil A\rceil) \top.

  • axioms x:A,y:BAfCBg(z:C)(f(z:C)=x:Ag(z:C)=y:B)\top\vdash_{x:A,y:B}\underset{A\overset{f}{\leftarrow} C\overset{g}{\rightarrow B}}{\bigvee}\exists (z:\lceil C\rceil)\big(\lceil f\rceil (z:\lceil C\rceil)=x:\lceil A\rceil\wedge \lceil g\rceil (z:\lceil C\rceil)=y:\lceil B\rceil\big) where the disjunction ranges over all cones on the discrete diagram consisting of AA and BB.

  • axioms f(x:A)=g(x:A) x:A h:CA|fh=gh(z:C)(h(z:C)=x:A)\lceil f\rceil (x:\lceil A\rceil)=\lceil g\rceil (x:\lceil A\rceil)\vdash_{x:\lceil A\rceil}\bigvee_{h:C\to A|f\circ h=g\circ h} \exists(z:\lceil C\rceil)\big(\lceil h\rceil(z:\lceil C\rceil)=x:\lceil A\rceil\big) for any pair f,g:ABf,g:A\to B, the disjunction ranging over all hh equalizing f,gf,g.

  • axioms x:A iI(y i:B i)(f i(y i:B i)=x:A)\top\vdash_{x:\lceil A\rceil}\bigvee_{i\in I}\exists (y_i:\lceil B_i\rceil) \big(\lceil f_i\rceil(y_i:\lceil B_i\rceil)=x:\lceil A\rceil\big) for each J-covering {f i:B iA|iI}\{f_i:B_i\to A|i\in I\}.

Remark

The first two axiom schemata correspond to functoriality, the third to fifth to filteredness (this being a notion equivalent to flatness), and the last to J-continuity (turning J-covers into epimorphic families).

Given a small category 𝒞\mathcal{C}, Set 𝒞 opSh(𝒞,J 0)Set^{\mathcal{C}^{op}}\simeq Sh(\mathcal{C},J_0) where J 0J_0 is the trivial topology. The theory of flat functors on 𝒞\mathcal{C} coincides with 𝕋 J 0 𝒞\mathbb{T}_{J_0}^{\mathcal{C}} since the last axiom schema becomes redundant for the maximal sieves whence the first five axiom schemata axiomatize the notion of a flat functor on 𝒞\mathcal{C} and the resulting theory of flat functors (also called the theory of flat diagrams) is seen to be of presheaf type.

Examples

  • Let (,)(\emptyset,\emptyset) be the empty site. Then the signature Σ\Sigma is empty and only the third condition takes hold, yielding 𝕋 ={}\mathbb{T}_\emptyset^{\emptyset}=\{\top\vdash\bot\} i.e. the empty topos Sh(,)=1Sh(\emptyset,\emptyset)=\mathbf{1} classifies the inconsistent theory.

References

Last revised on July 6, 2020 at 14:08:15. See the history of this page for a list of all contributions to it.