Contents

topos theory

# Contents

## Idea

A geometric morphism $f : E \to F$ between toposes is a functor of the underlying categories that is consistent with the interpretation of $E$ and $F$ as generalized topological spaces.

If $F = Set = Sh(*)$ is the terminal sheaf topos, then $E \to Set$ is essential if $E$ is a locally connected topos. In general, $f$ being essential is a necessary (but not sufficient) condition to ensure that $f$ behaves like a map of topological spaces whose fibers are locally connected: that it is a locally connected geometric morphism.

## Definition

###### Definition

Given a geometric morphism $(f^* \dashv f_*) : E \to F$, it is an essential geometric morphism if the inverse image functor $f^*$ has not only the right adjoint $f_*$, but also a left adjoint $f_!$:

$(f_! \dashv f^* \dashv f_*) \;\;\; : \;\;\; E \stackrel{\overset{f_!}{\longrightarrow}}{\stackrel{\overset{f^*}{\longleftarrow}}{\underset{f_*}{\longrightarrow}}} F \,.$

A point of a topos $x : Set \to E$ which is given by an essential geometric morphism is called an essential point of $E$.

###### Remark

There are various further conditions that can be imposed on a geometric morphism:

• If $f_!$ can be made into an $E$-indexed functor and $f^*$ satisfies some extra conditions, the geometric morphism $f$ is a locally connected geometric morphism (see there for details).

• If $f_!$ preserves finite products then $f$ is called connected surjective.

• If in addition to the above $f$ is a local geometric morphism in that there is a further functor $f^! : F \to E$ which is right adjoint $(f_* \dashv f^!)$ and full and faithful then the geometric morphism $f$ is called cohesive.

###### Remark

Since for a Grothendieck topos $E$ the inverse image $x^*$ of an essential point is of the form $Hom_E(P,-)$ where $P\ncong\emptyset$ is projective and connected, objects satisfying these three conditions are sometimes called essential objects (cf. Johnstone 1977, p.255).

## Properties

### A criterion for Grothendieck toposes

The inverse image $f^\ast$ of an essential geometric morphism preserves small limits since it is a right adjoint. Hence, this provides a minimal requirement to satisfy for a general geometric morphism $f^\ast\dashv f_\ast$ in order to qualify for being essential. In case, the toposes involved are Grothendieck toposes this condition is not only necessary but sufficient.

###### Proposition

Let $f^\ast\dashv f_\ast:\mathcal{E}\to\mathcal{F}$ a geometric morphism between Grothendieck toposes. Then $f$ is essential iff $f^\ast$ preserves small limits iff $f^\ast$ preserves small products.

###### Proof

Grothendieck toposes are locally presentable and $f^\ast$ has rank i.e. it preserves $\alpha$-filtered colimits for some regular cardinal $\alpha$ since it is a left adjoint. But by (Borceux vol. 2 Thm.5.5.7, p.275) a functor between two locally presentable categories has a left adjoint precisely if it has rank and preserves small limits.

Since limits can be constructed from products and equalizers and $f^\ast$ preserves the latter, it preserves small limits precisely when it preserves small products.

### Relation to morphisms of (co)sites

For $C$ and $D$ small categories write $[C,Set]$ and $[D,Set]$ for the corresponding copresheaf toposes. (If we think of the opposite categories $C^{op}$ and $D^{op}$ as sites equipped with the trivial coverage, then these are the corresponding sheaf toposes.)

###### Proposition

This construction extends to a 2-functor

$[-,Set] : Cat_{small}^{co} \to Topos_{ess}$

from the 2-category Cat${}_{small}$ with 2-morphisms reversed) to the sub-2-category of Topos on essential geometric morphisms, where a functor $f : C \to D$ is sent to the essential geometric morphism

$(f_! \dashv f^* \dashv f_*) : [C,Set] \stackrel{\overset{f_! := Lan_f}{\to}}{\stackrel{\overset{f^* := (-) \circ f}{\leftarrow}}{\underset{f_* := Ran_f}{\to}}} [D,Set] \,,$

where $Lan_f$ and $Ran_f$ denote the left and right Kan extension along $f$, respectively.

###### Proposition

This 2-functor is a full and faithful 2-functor when restricted to Cauchy complete categories:

$[-, Set] : Cat^co_{CauchyComp} \hookrightarrow Topos_{ess} \,.$

For all small categories $C,D$ we have an equivalence of categories

$Func(\overline{C},\overline{D})^{op} \simeq Topos_{ess}([C,Set], [D,Set])$

between the opposite category of the functor category between the Cauchy completions of $C$ and $D$ and the the category of essential geometric morphisms between the copresheaf toposes and geometric transformations between them.

In particular, since every poset – when regarded as a category – is Cauchy complete, we have

###### Corollary

The 2-functor

$[-,Set] : Poset \to Topos_{ess}$
###### Remark

Sometimes it is useful to decompose this statement as follows.

There is a functor

$Alex : Poset \to Locale$

which assigns to each poset a locale called its Alexandroff locale. By a theorem discussed there, a morphisms of locales $f : X \to Y$ is in the image of this functor precisely if its inverse image morphism $f^* Op(Y) \to Op(X)$ of frames has a left adjoint in the 2-category Locale.

Moreover, for any poset $P$ the sheaf topos over $Alex P$ is naturally equivalent to $[P,Set]$

$[-,Set] \simeq Sh \circ Alex \,.$

With this, the fact that $[-,Set] : Poset \to Topos$ hits precisely the essential geometric morphisms follows with the basic fact about localic reflection, which says that $Sh : Locale \to Topos$ is a full and faithful 2-functor.

### Some morphism calculus

###### Proposition

Let $f : E \to F$ be an essential geometric morphism.

For every $\phi : X \to f^* f_* A$ in $E$ the diagram

$\array{ X &\stackrel{\phi}{\to}& f^* f_* A \\ \downarrow && \downarrow \\ f^* f_! X &\stackrel{}{\to}& A }$

commutes, where the vertical morphisms are unit and counit, respectively, and where the bottom horizontal morphism is the adjunct of $\phi$ under the composite adjunction $(f^* f_! \dashv f^* f_*)$.

###### Proof

The morphism $\phi : X \to f^* f_* A$ is the component of a natural transformation

$\array{ *&&\overset{X}{\to}&& E \\ {}^{\mathllap{A}}\downarrow &\Downarrow^\phi&& {}^{\mathllap{f^*}}\nearrow \\ E &\underset{f_*}{\to}&F } \,.$

The composite $X \stackrel{\phi}{\to} f^* f_* A \to A$ is the component of this composed with the counit $f^* f_* \Rightarrow Id$.

We may insert the 2-identity given by the zig-zag law

$\cdots \;\;\; = \;\;\; \array{ *&&\overset{X}{\to}&& E && = && E \\ {}^{\mathllap{A}}\downarrow &\Downarrow^\phi&& {}^{\mathllap{f^*}}\nearrow &\Downarrow& \searrow^{f_!} &\Downarrow& \nearrow_{\mathrlap{f^*}} \\ E &\underset{f_*}{\to}&F &&=&& F } \,.$

Composing this with the counit $f^* f_* \Rightarrow Id$ produces the transformation whose component is manifestly the morphism $X \to f^* f_! X \to A$.

## Examples

### Logical functors and etale geometric morphisms

A logical functor $E\to F$ with a left adjoint has automatically also a right adjoint whence is the inverse image part of an essential geometric morphism $F\to E$.

A particularly important instance of this situation is the following:

For any morphism $f\colon A\to B$ in a topos $E$, the induced geometric morphism $f\colon E/A \to E/B$ of overcategory toposes is essential. Here, the logical functor is given by the pullback functor $f^*:E/B\to E/A$ of course.

In the special case $B = *$ the terminal object, the essential geometric morphism

$\pi : E/A \to E$

is also called an etale geometric morphism.

Conversely, (almost) any essential geometric morphism $\pi$ whose inverse image $\pi^*$ is a logical functor has the form of an etale geometric morphism:

###### Proposition

If the left adjoint $\pi_!$ of a logical functor $\pi^*:E\to F$ furthermore preserves equalizers, then the corresponding essential geometric morphism is up to equivalence an etale geometric morphism $\pi : F\simeq E/A \to E$ for an object $A$ in $E$ that is determined up to isomorphism.

For a proof see e.g. Johnstone (1977, p.37).

### Locally connected toposes

A locally connected topos $E$ is one where the global section geometric morphism $\Gamma : E \to Set$ is essential.

$(f_! \dashv f^* \dashv f_*) \;\;\; : \;\;\; E \stackrel{\overset{\Pi_0}{\longrightarrow}}{\stackrel{\overset{LConst}{\longleftarrow}}{\underset{\Gamma}{\longrightarrow}}} Set \,.$

In this case, the functor $\Gamma_! = \Pi_0 : E \to Set$ sends each object to its set of connected components. More on this situation is at homotopy groups in an (∞,1)-topos.

Note, though that if $p\colon E\to S$ is an arbitrary geometric morphism through which we regard $E$ as an $S$-topos, i.e. a topos “in the world of $S$,” the condition for $E$ to be locally connected as an $S$-topos is not just that $p$ is essential, but that the left adjoint $p_!$ can be made into an $S$-indexed functor (which is automatically true for $p^*$ and $p_*$). This is automatically the case for $Set$-toposes (at least, when our foundation is material set theory—and if our foundation is structural set theory, then our large categories and functors all need to be assumed to be $Set$-indexed anyway). For more see locally connected geometric morphism.

### Tiny objects

The tiny objects of a presheaf topos $[C,Set]$ are precisely the essential points $Set \to [C,Set]$. See tiny object for details.

## References

Like many other things, it all started as an exercise in

Speaking of exercises, consider the results of Roos on essential points reported in exercise 7.3 of

• Peter Johnstone, Topos Theory , Academic Press New York 1977. (Dover reprint 2014, pp.254f)

The case of sheaves valued in FinSet is considered in

• J. Haigh, Essential geometric morphisms between toposes of finite sets , Math. Proc. Phil. Soc. 87 (1980) pp.21-24.

The standard reference for essential localizations 1, aka levels, is

• G. M. Kelly, F. W. Lawvere, On the Complete Lattice of Essential Localizations , Bull. Soc. Math. de Belgique XLI (1989) pp.289-319.