### Cohesive ∞-Toposes

#### Posted by Urs Schreiber

The concept of a

axiomatizes properties of a general abstract context of *geometric spaces*. This has been proposed by Bill Lawvere.

Below is

a brief introduction to the definition of

*cohesive topos*an appreciation of Bill Lawvere’s work in this direction.

More is behind the above link.

## The notion of Cohesive Topos

The definition of *cohesive topos* or *category of cohesion* aims to axiomatize properties of a topos that make it a *gros topos* of spaces inside of which geometry may take place.

The idea behind the term is that a *geometric space* is roughly something consisting of points or pieces that are *held together* by some cohesion - for instance by topology, by smooth structure, etc.

The canonical global section geometric morphism $\Gamma : E \to Set$ of a cohesive topos over Set may be thought of as sending a space $X$ to its underlying set of points $\Gamma(X)$. Here $\Gamma(X)$ is $X$ with all *cohesion forgotten* (for instance with the topology or the smooth structure forgotten)

Conversely, the left adjoint and right adjoint of $\Gamma$

$E \stackrel{\overset{Disc}{\leftarrow}}{\stackrel{\overset{\Gamma}{\to}}{\underset{Codisc}{\leftarrow}}} Set$

send a set $S$ either to the *discrete space* $Disc(S)$ with *discrete* cohesive structure (for instance with discrete topology) or to the codiscrete space $Codisc(S)$ with the *codiscrete* cohesive structure (for instance with codiscrete=indiscrete topology) .

Moreover, the idea is that cohesion makes points lump together to *connected pieces* . This is modeled by one more functor $\Pi : E \to S$ left adjoint to $Disc$. In the context of 1-topos theory this sends a cohesive space to its connected components $(\Pi = \pi_0)$. More generally in an $(n,1)$-topos-theory context, $\Pi$ sends a cohesive space $X$ to the $(n-1)$-truncation of its geometric fundamental $\infty$-groupoid $\Pi(X)$.

In total this gives a quadruple of adjoint functors

$(\Pi \dashv Disc \dashv \Gamma \dashv Codisc) : E \stackrel{\stackrel{\overset{\Pi}{\to}}{\overset{Disc}{\leftarrow}}}{\stackrel{\underset{\Gamma}{\to}}{\underset{Codisc}{\leftarrow}}} Set \;$

A cohesive topos is a topos whose terminal geometric morphism admits an extenson to such a quadruple of adjoints, satisfying some further properties.

## A remark on Lawvere’s work

I am very much impressed by what Lawvere is doing here. His work on cohesion is a direct continuation of his thoughts that led to synthetic differential geometry and those expressed in his “Categorical dynamics”-lecture:

He is really at heart a physicist, in the following sense: he is deeply interested in the mathematical model building of reality He is searching for those structures in abstract category theory that do reflect the world. He is asking: What is a space in which physics can take place? Concretely: What is the abstract context in which one can talk about continuum dynamics? I gather even though he is an extraordinary mind, he did not push beyond continuum mechanics, otherwise he would also be asking: What is the abstract context in which quantum field theory takes place?

Because he is working so close to the fundamental root of everything, there is quite a distance from his “very fundamental physics” to the “fundamental physics” that most people will recognize as such. Which conversely leads to the interesting effect that one can see him try to find a dictionary between category theory and ontological concepts in philosophy. What is the precise abstract definition of space, quantity, quality etc? He gives definitions for all this. And useful ones. There is a connection between *ontology* and *fundamental physic* constructed out of hard math.

Then he continues further towards the more tangible world, defining extensive and intensive quantities, as if writing a book on thermodynamics. And indeed, in his article on cohesion he uses all these analogies with heating and cooling! One probably has to be careful with how to say these things in the (ignorant) public, but I think it is absolutely admirable how here a pure category theorist is actively working on unravelling the very foundations of reality. I was hoping for many years that more category theorists would see the immense applicability of the theory to theory-modelling in physics. As Jacob Lurie says rightly: Higher category theory is not theory for its own sake, but for the sake of other theory. And fundamental theoretical physics is all about scanning the space of theories for those that fit reality (as opposed to the physics that most theoretical physicists do, which is scanning the phenomena of one *fixed* theory.)

We can see that Lawvere is pushing in this direction, I think. Which is why I wanted to emphasize what his axioms for a cohesive topos are like if we generalize them to *cohesive $\infty$-toposes* . Because then out of the very same set of axioms springs a structure that all by itself gets even closer to being a model for physical reality: this is the point I kept emphasizing here and there: that *just* the assumption that we have an $\infty$-connected $\infty$-topos, and hence also *just* the assumption that we have a cohesive $\infty$-topos (which implies the former) gives rise to a refinement of the intrinsic cohomology of the $\infty$-topos (which is always there) to intrinsic differential cohomology. In a better world I would walk over to Lawvere and try to tell him that *that’s* what models real-world physics fundamentally: a bare topos or $\infty$-topos with its notion of cohomology is a context for *kinematics* (just the configurations, no forces, no dynamics) while *differential* cohomology encodes the forces and the dynamics. (In case this statement is raising eyebrows with anyone: let me know and we can discuss this in detail, look at examples, etc. This is an important story).

For instance the differential equations in a synthetic smooth topos that Lawvere considered around “Categorical dynamics” can be understood without intervention “by hand” as coming from differential cocycles in the corresponding $\infty$-topos. This is really what the theory of (derived) D-modules etc. is about. But even though it all flows by itself out of a general abstract source of concepts, unwinding it is a long story. I wish there were more people like Lawvere around, with his perspective on the general abstract basis of everything and at the same time with the overview over modern derived $\infty$-topos theory and the understanding that the richer structure people are seeing in these is Lawvere’s observation that reality springs out of topos theory taken to full blossoming: reality springs out of $\infty$-topos-theory.

(Reproduced from this posting. )

## Re: Cohesive ∞-Toposes

So what are some other cohesive toposes? Is Johnstone’s topological topos cohesive? The main thing I’m not sure about there is the existence and behavior of $f_!$.