Contents

topos theory

# Contents

## Idea

For any geometric theory $\mathbb{T}$ there exists a geometric theory $\mathbb{T}^2$, the theory of $\mathbb{T}$-model homomorphisms, whose models in a Grothendieck topos $\mathcal{E}$ are precisely the homomorphisms between $\mathbb{T}$-models in $\mathcal{E}$.

## Definition

Let $\mathbb{T}$ be a geometric theory over the signature $\Sigma$.

Let $\Sigma^2$ be the signature containing a pair of new sort symbols $X^0$, $X^1$ and a new function symbol $f_X:X^0\to X^1$ for any sort symbol $X$ in $\Sigma$, as well, as pairs $g^0:X_1^0\times\dots\times X_n^0\to Y^0$, $g^1:X_1^1\times\dots\times X_n^1\to Y^1$ of new functions symbols (resp. pairs $R^0\rightarrowtail X_1^0\times\dots\times X_n^0$, $R^1\rightarrowtail X_1^1\times\dots\times X_n^1$ of new relation symbols) for any function symbol $g:X_1\times\dots\times X_n\to Y$ (resp. relation symbol $R\rightarrowtail X_1\times\dots\times X_n$) in $\Sigma$.

The theory of $\mathbb{T}$-model homomorphisms is the theory $\mathbb{T}^2$ over the signature $\Sigma^2$ with the following sequents:

A pair of sequents $\varphi^0\vdash \psi^0$, $\varphi^1\vdash \psi^1$ for any sequent $\varphi\vdash \psi$ in $\mathbb{T}$ where the new sequents result from replacing the function and relation symbols in $\varphi\vdash \psi$ with their 0-indexed, resp. 1-indexed pendants where necessary1. Plus a sequent

$\top\vdash f_Y(g^0(x_1,\dots,x_n))=g^1(f_{X_1}(x_1),\dots,f_{X_n}(x_n))$

resp.

$R^0(x_1,\dots,x_n)\vdash R^1(f_{X_1}(x_1),\dots,f_{X_n}(x_n))$

for any pair $g^0$, $g^1$ of function symbols in $\Sigma^2$ corresponding to $g:X_1\times\dots\times X_n\to Y$ in $\Sigma$ (in case $n=0\,$, the sequent reads $\top\vdash f_Y(g^0)=g^1\,$), resp. any pair $R^0$, $R^1$ of relation symbols in $\Sigma^2$ corresponding to $R\rightarrowtail X_1\times\dots\times X_n$ in $\Sigma$ (in case $n=0\,$, the sequent reads $R^0\vdash R^1\,$).

## Properties

###### Proposition

Let $\mathcal{E}$ be a Grothendieck topos. Then

${\mathbb{T}^2}\text{-}Mod(\mathcal{E})=\mathbb{T}\text{-}Mod(\mathcal{E})^2=\mathbb{T}\text{-}Mod(\mathcal{E}^2)\quad.$

Cf. Johnstone (1977, p. 203; 2002, p.425), Mac Lane-Moerdijk (1994, ex.X.5 p.572).

Note that the second equality determines the class of $\mathbb{T}$-model homomorphisms once the $\mathbb{T}$-models in the arrow categories $\mathcal{E}^2$ are known.

Remark: This e.g. precludes the existence of geometric theories $\mathbb{T}^{op}$ or $\mathbb{T}^\times$ with the property that, given a geometric theory $\mathbb{T}$, the category of models ${\mathbb{T}^{op}}\text{-}Mod(\mathcal{E})$ and ${\mathbb{T}^{\times}}\text{-}Mod(\mathcal{E})$ coincide with the opposite category ${\mathbb{T}}\text{-}Mod(\mathcal{E})^{op}$ or the twisted arrow category ${\mathbb{T}}\text{-}Mod(\mathcal{E})^\times$ for any Grothendieck topos, respectively. In particular, since the (geometric) theory of morphisms $\mathbb{O}^2$ (see the examples below) assigns the arrow category $\mathcal{E}^2$ as category of models to a Grothendieck topos $\mathcal{E}$ no geometric theory can assign all Grothendieck toposes $\mathcal{E}$ their twisted arrow category $\mathcal{E}^\times$ as category of models because the class of models coincides with the class of models for the theory of morphisms. $\qed$

Since $\mathcal{E}^2=\mathcal{E}\times Set^2$ in general and the Sierpinski topos $Set^2$ is exponentiable, one gets

${\mathbb{T}^2}\text{-}Mod(\mathcal{E})=\mathbb{T}\text{-}Mod(\mathcal{E}^2)=Hom(\mathcal{E}^2,Set[\mathbb{T}])=Hom(\mathcal{E}\times Set^2, Set[\mathbb{T}])=Hom(\mathcal{E},Set[\mathbb{T}]^{(Set^2)})$

or in other words

###### Corollary

The classifying topos for the theory $\mathbb{T}^2$ of $\mathbb{T}$-model homorphisms is $Set[\mathbb{T}^2]= Set[\mathbb{T}]^{(Set^2)}$.

Remark: It is worthwhile to muse a bit how “squaring” fares as a theory operator with respect to the various subdoctrines of geometric logic: since it does not introduce logical operators like infinitary disjunctions or quantifiers unless the operators were already present in $\mathbb{T}$, one sees that the formats of the more expressive subdoctrines like Horn, regular, and coherent logic are respected by the passage to $\mathbb{T}^2$. Also, the square of a propositional theory i.e. one over a signature lacking sort symbols, is again propositional - this implies e.g. via the above corollary that the exponentiation of a localic topos by $Set^2$ is again a localic topos. The order sensitive subdoctrines like cartesian or disjunctive logic are also respected. One case where squaring does not preserve the subdoctrine is 1-sorted algebraic logic since then $\mathbb{T}^2$ is necessarily 2-sorted. $\qed$

By an observation of Hébert (2010, p.1), the finitely presentable objects in arrow categories $\mathcal{K}^2$, are precisely the morphisms between finitely presentable objects in $\mathcal{K}$ whence, denoting the subcategory of finitely presentable models of a theory $\mathbb{T}$ in a topos $\mathcal{E}$ by $\mathbb{T}\text{-}Mod_{fp}(\mathcal{E})$ one has

$\mathbb{T}^2\text{-}Mod_{fp}(\mathcal{E})=\mathbb{T}\text{-}Mod_{fp}(\mathcal{E})^2\;.$

But since classifying toposes of cartesian theories are given by the presheaf toposes on the opposite of the category of their finitely presentable models in $Set$ (cf. Johnstone 2002b, p. 891) one has

###### Proposition

Let $\mathbb{T}$ be a cartesian theory. Then the theory $\mathbb{T}^2$ of $\mathbb{T}$-model homorphisms is cartesian and its classifying topos is $Set[\mathbb{T}^2]=Set^{\mathbb{T}^2\text{-}Mod_{fp}(Set)}=Set^{(\mathbb{T}\text{-}Mod_{fp}(Set)^2)}=\big (Set^{\mathbb{T}\text{-}Mod_{fp}(Set)}\big)^{(Set^2)}=Set[\mathbb{T}]^{(Set^2)}$.

For an easy application of this result see at the theory $\mathbb{O}$ of objects in the examples below!

## Examples

• Let $\mathbb{T}^\emptyset_\emptyset$ and $\mathbb{T}^\emptyset_1$ be the empty theory, resp., the inconsistent theory, over the empty signature. Then $\mathbb{T}^{\emptyset 2}_\emptyset=\mathbb{T}^\emptyset_\emptyset$ and $\mathbb{T}^{\emptyset 2}_1=\mathbb{T}^\emptyset_1$ in accordance with the fact that these theories have only empty models whence all model homomorphisms are identity morphisms of empty models. In other words, squaring theories obeys the laws of arithmetic at 0 and 1.

• Let $\mathbb{O}$ be the theory of objects i.e. the theory with no sequents over the signature with one sort symbol $O$. Then $\mathbb{O}^2$ is the theory with no sequents over the signature with two sort symbols $O^0$, $O^1$ and a function symbol $f_O:O^0\to O^1$. Clearly, its models in a Grothendieck topos $\mathcal{E}$ are the morphisms of $\mathcal{E}$ and, accordingly, $\mathbb{O}^2$ is called the theory of morphisms.

It is the dual theory of the theory classified by the Sierpinski topos $Set^2$, or in other words, its classifying topos called the morphism classifier (cf. Johnstone 1977, p.184; 2002, p.426) is

$Set[\mathbb{O}^2]=Set[\mathbb{O}]^{Set^2}=(Set^{FinSet})^{Set^2}=Set^{(FinSet^2)}\quad .$

The generic morphism is given by the natural transformation $\eta\,:\,\mathbf{O}\circ dom\to\mathbf{O}\circ cod$ with $\mathbf{O}$ the generic object i.e. the inclusion functor $FinSet\hookrightarrow Set$, and $dom\,:\,FinSet^2\to FinSet\, ,\, (X\to Y)\mapsto X$ the domain projection functor, and $cod\,:\,FinSet^2\to FinSet\,,\, (X\to Y)\mapsto Y$ the codomain projection functor, with components $\eta_{(X\to Y)}=\mathbf{O}(X\to Y)$.

• Let $\mathbb{K}$ be the theory of categories (e.g. Johnstone 1977, p.202) whose models in a Grothendieck topos $\mathcal{E}$ are the internal categories $\mathbf{C}\in cat(\mathcal{E})$. Then $\mathbb{K}^2$ is the theory of functors. $\mathbb{K}$ likely being the most famous cartesian theory, one has $Set[\mathbb{K}^2]=Set^{(\mathbb{K}\text{-}Mod_{fp}(Set)^2)}$.

## The connection to natural homotopy

The Sierpinski topos $Set^2$ is a connected, locally connected and local topos. As a result of Artin gluing along $Set\overset{id}{\to}Set$ it has two topos points (corresponding to the two points of the underlying Sierpinski space). Whence one can take $Set^2$ as an abstract interval object for a homotopy theory of Grothendieck toposes correlated with their nature as “generalized spaces” and view, accordingly, the exponential $\mathcal{E}^{Set^2}$ as a path space for $\mathcal{E}$.

(Cf. Beke 2000, p.11f)

• Tibor Beke, Homotopoi , ms. University of Massachusetts Lowell (2000). (dvi)

• Michel Hébert?, Finitely presentable morphisms in exact sequences , TAC 24 no.9 (2010) pp.209-220. (abstract)

• Peter Johnstone, Topos Theory , Academic Press New York (1977). (Also available as Dover Reprint, Mineola 2014)

• Peter Johnstone, Sketches of an Elephant vol.1 , Oxford UP 2002.

• Peter Johnstone, Sketches of an Elephant vol.2 , Oxford UP 2002.

• André Joyal, Gavin Wraith, Eilenberg-Mac Lane Toposes and Cohomology , pp.117-131 in Cont. Math. 92 AMS 1984.

• Saunders Mac Lane, Ieke Moerdijk, Sheaves in Geometry and Logic , Springer Heidelberg 1994.

1. It is understood that variables and contexts are updated with the new sorts e.g. the sequent $\top\vdash_{x:S} x=x$ yields two sequents $\top\vdash_{x:S^0} x=x$ and $\top\vdash_{x:S^1} x=x$ and $\top\vdash \big (\forall x:S\big ) x=x$ yields $\top\vdash \big (\forall x:S^0\big ) x=x$ and $\top\vdash \big (\forall x:S^1\big ) x=x$ etc. In case, the sequent $\varphi\vdash\psi$ contains nothing to update (e.g. the sequent $\top\vdash\bot$) one just takes the old sequent as the “new pair”.