nLab theory of flat functors



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topos theory



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In higher category theory




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).


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\}.


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.


  • 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.


Last revised on July 6, 2020 at 18:08:15. See the history of this page for a list of all contributions to it.