### big and little toposes

Section 3 summarizes the relation of toposes to geometry as

*a topos is a generalized space*

Strictly speaking, this misses an aspect of topos theory in geometry: not every topos is like a generalized space, some are like *categories of generalized spaces* .

Of course the distinction depends on what one counts as a generalized space. Often it is not made. But that one should make a distinction goes back to remarks by Grothendieck and has been emphasized over many years by Bill Lawvere, first in

Bill Lawvere, *Categories of spaces may not be generalized spaces* .

The two kinds of toposes in geometry are sometimes called *gros and petit toposes* or big and little toposes. A little topos is to be regarded as a generalizeed space itself, a big topos is to be regarded as a category of generalized spaces.

What Lawvere has been proposing ever since the above note is an axiomatics for *big toposes* . The solution is quite beautiful: a Grothendieck topos $\mathcal{E}$ is a *big topos of generalized spaces* – Lawvere introduced for this the nicely descriptive term topos of cohesion – if the terminal geometric morphism $(\Delta \dashv \Gamma) : \mathcal{E} \to Set$ to the base has an extra left adjoint and an extra right adjoint to a total of a quadruple of adjoint functors

$(\Pi_0 \dashv Disc \dashv \Gamma \dashv Codisc) : \mathcal{E} \to Set$

such that $\Pi_0$ preserves finite products. This has the following meaning in terms of the interpretation of *objects* of $\mathcal{E}$ as generalized spaces:

the *underlying set of points* of an object $X \in \mathcal{E}$ is $\Gamma(E)$;

the set of *connected components of points* of $X$ is $\Pi_0(X)$.

This allows us to think of $X \in \mathcal{E}$ as being a set of points that are lumped together by some kind of *cohesion* . For instance by topology, or by smooth structure.

Then

For $S$ a set, $Disc(S)$ is the *discrete cohesion* on this set (for instance the discrete topology on $S$, or the discrete smooth structure);

For $S$ a set, $Codisc(S)$ is the *codiscrete cohesion* on this set (for instance the codiscrete topology on $S$, or the codiscrete smooth structure).

For instance some *smooth toposes* modelling the axioms of *synthetic differential geometry* are cohesive, reflecting the fact that they are categories of generalized smooth spaces (for instance the Cahiers topos).

In his original note Lawvere observes that a topos that satisfies the axioms of a cohesive topos is genuinely big in that it cannot be a localic topos. The meaning of this again depends a bit on which standpoint one takes towards generalized spaces. One can make this a positive statement and observe that as generalized spaces, cohesive toposes are *fat points* , namely the *abstract cohesive lump of points* on wich the given notion of generalized space is modeled.

In the Idea-section of the $n$Lab entry topos we have some general remarks on the usefulness of the two perspectives of big and small toposes in geometry. For instance for every object $X$ in a big/cohesive topos $\mathcal{E}$, the over-topos $\mathcal{E}/X$ is the little topos incarnation of $X$, thus extracting the generalized space $X$ from inside the big topos to a stand-alone little topos.

This and various other aspects of cohesive toposes becomes even clearer and more pronounced as we pass from toposes to $(\infty,1)$-toposes – as we should, lest we are being evil and impose equations where there are only equivalences.

The above definition immediately generalizes to give the notion of cohesive $(\infty,1)$-toposes. It is quite remarkable how much property and structure is implied in and on a cohesive $(\infty,1)$-topos, discussed in detail at the above link.

Notably there is a nice relation to classifying toposes for essentially algebraic theories. Thiss closes a grand circle and shows how this all fits together into one grand general abstract picture of *geometry* as such (all $\infty$-s implicit from now on):

A category $\mathcal{G}$ that both has finite limits and the structure of a site in a compatible way we call a geometry. Examples to keep in mind are

Then the sheaf topos

$\mathbf{H} = Sh(\mathcal{G})$

is the category of generalized spaces modeled on the geometry $\mathcal{G}$. At least in the case of $\mathcal{G} = Manifolds$ this is cohesive.

On the other hand, let $\mathcal{X}$ be a little sheaf topos. Then a finite-limit preserving functor

$\mathcal{O}_{\mathcal{X}} : \mathcal{G} \to \mathcal{X}$

is a $\mathcal{G}$-algebra in $\mathcal{X}$: a $\mathcal{G}$-valued structure sheaf. If this preserves coverings then this makes $(\mathcal{X}, \mathcal{O}_{\mathcal{X}})$ a *locally ringed topos* with respect to the notion of “ring”=algebra encoded by the essentially algebraic theory $\mathcal{G}$.

Every object $X \in \mathbf{H}$ ought to be canonically a locally $\mathcal{G}$-ringed topos, and it is: the little over-topos $\mathbf{H}/X$ sits canonically over $\mathbf{H}$ by an étale geometric morphism

$\mathbf{H}/X \stackrel{\overset{X_!}{\to}}{\stackrel{\overset{X^*}{\leftarrow}}{\underset{X_*}{\to}}}
\mathbf{H}$

and composed with the Yoneda embedding $j$ this gives the canonical structure sheaf (= algebra over the theory $\mathcal{G}$)

$\mathcal{O}_X :
\mathcal{G} \stackrel{j}{\to}
\mathbf{H} \stackrel{X^*}{\to}
\mathbf{H}/X
\,.$

So we see that the two or three roles of toposes as big or little toposes of generalized spaces and as classifying toposes of essentially algebraic theories are really all different aspects of one single general abstract concept: geometry.

Either one probes a generalized space by mapping test spaces in $\mathcal{G}$ *into* it: this gives the notion of a sheaf over $\mathcal{G}$;

or one co-probes a generalized space by mapping *out of it* into test spaces in $\mathcal{G}$: this gives the $\mathcal{G}$-valued structure sheaf on the space and hence an algebra over $\mathcal{G}$.

The unification of geometry and algebra at work here is essentially what is known as Isbell duality. Topos theory, with its duality of big versus little toposes, of cohesive versus locally ringed toposes, classifying toposes versus localic toposes is in a nice realization of this fundamental duality: a space may be detected either by mapping probes into it or by mapping out of it into probe objects. Lawvere called this fundamental duality that of *space and quantity* . Geometry and algebra. Two sides of the same coin.

## Re: An Informal Introduction to Topos Theory

Thanks! Topos theory is not in my personal Christmas tradition, but expository LaTeXing is. As it turns out, the only Christmas present I have given this year is an informal and (I hope) approachably brief introduction to operads, which I finished this morning.

The only Christmas present I have received is now an informal introduction to topos theory - how pleasingly symmetric.

By the way, before anyone tries to pity me about the presents, let me say that I read somewhere that approximately £2 billion was spent on unwanted Christmas presents just in the UK last year. I decided to remove myself from that equation several years ago.

Merry, or suicidal, Christmas, depending on your inclinations!