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.

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.

For more on this see also

The definition of essential geometric morphisms appears before Lemma A.4.1.5 in

Connected surjective and local geometric morphisms are discussed in

Further refinements are in

• Bill Lawvere, Axiomatic cohesion Theory and Applications of Categories, Vol. 19, No. 3, 2007, pp. 41–49. (pdf)

1. See at Aufhebung for further references on essential localizations.