Lawvere's "Some thoughts on the future of category theory."

The notion of a "category of Being" that Lawvere discusses there is the notion that more recently he has been calling a category of cohesion . I'll try to illuminate a bit what's going on .

I'll restrict to the case that the category is a topos and say cohesive topos for short. This is a topos that satisfies a small collection of simple but powerful axioms that are supposed to ensure that its objects may consistently be thought of as geometric spaces built out of points that are equipped with "cohesive" structure (for instance topological structure, or smooth structure, etc.). So the idea is to axiomatize big toposes in which geometry may take place.

Further details and references can be found here:

http://nlab.mathforge.org/nlab/show/cohesive+topos .

Let's walk through the article:

One axiom on a cohesive topos $\mathcal{E}$ is that the global section geometric morphism $\Gamma : \mathcal{E} \to \mathcal{S}$ to the given base topos $\mathcal{S}$ has a further left adjoint $\Pi_0 := \Gamma_! : \mathcal{E} \to \mathcal{S}$ to its inverse image $\Gamma^{\ast}$, which I'll write $\mathrm{Disc} := \Gamma^{\ast}$, for reasons discussed below. This extra left adjoint has the interpretation that it sends any object $X$ to the set $\Pi_0(X)$ "of connected components". What Lawvere calls a connected object in the article (p. 4) is hence one that is sent by $\Pi_0$ to the terminal object.

Another axiom is that $\Pi_0$ preserves finite products. This implies by the above that the collection of connected objects is closed under finite products. This appears on page 6. What he mentions there with reference to Hurewicz is that given a topos with such $\Pi_0$, it becomes canonically enriched over the base topos in a second way, a geometric way.

I believe that this, like various other aspects of cohesive toposes, lives up to its full relevance as we make the evident step to cohesive $\infty$-toposes. More details on this are here

http://nlab.mathforge.org/nlab/show/cohesive+(infinity,1)-topos

(But notice that this, while inspired by Lawvere, is not due to him.)

In this more encompassing context the extra left adjoint $\Pi_0$ becomes $\Pi_\infty$ which I just write $\Pi$: it sends, one can show, any object to its geometric fundamental $\infty$-groupoid, for a notion of geometric paths intrinsic to the $\infty$-topos. The fact that this preserves finite products then says that there is a notion of concordance of principal $\infty$-bundles in the $\infty$-topos.

The next axiom on a cohesive topos says that there is also a further right adjoint $\mathrm{coDisc} := \Gamma^! : \mathcal{S} \to \mathcal{E}$ to the global section functor. This makes in total an adjoint quadruple

$$ (\Pi_0 \dashv \mathrm{Disc} \dashv \Gamma \dashv \mathrm{coDisc}) := (\Gamma_! \dashv \Gamma^* \dashv \Gamma_* \dashv \Gamma^!) : \mathcal{E} \to \mathcal{S} $$

and another axiom requires that both $\mathrm{Disc}$ as well as $\mathrm{coDisc}$ are full and faithful.

This is what Lawvere is talking about from the bottom of p. 12 on. The downward functor that he mentions is $\Gamma : \mathcal{E} \to \mathcal{S}$. This has the interpretation of sending a cohesive space to its underlying set of points, as seen by the base topos $\mathcal{S}$. The left and right adjoint inclusions to this are $\mathrm{Disc}$ and $\mathrm{coDisc}$. These have the interpretation of sending a set of points to the corresponding space equipped with either discrete cohesion or codiscrete (indiscrete) cohesion . For instance in the case that cohesive structure is topological structure, this will be the discrete topology and the indiscrete topology, respectively, on a given set. Being full and faithful, $\mathrm{Disc}$ and $\mathrm{coDisc}$ hence make $\mathcal{S}$ a subcategory of $\mathcal{E}$ in two ways (p. 7), though only the image of $\mathrm{coDisc}$ will also be a subtopos, as he mentions on page 7.

(This has, by the way, an important implication that Lawvere does not seem to mention: it implies that we are entitled to the corresponding quasi-topos of separated bipresheaves, induced by the second topology that is induced by the sub-topos. That, one can show, may be identified with the collection of concrete sheaves, hence concrete cohesive spaces (those whose cohesion is indeed supported on their points). In the case of the cohesive topos for differential geometry, the concrete objects in this sense are precisely the diffeological spaces . )

He calls the subtopos given by the image of $\mathrm{coDisc} : \mathcal{S} \to \mathcal{E}$ that of "pure Becoming" further down on p. 7, whereas the subcategory of discrete objects he calls that of "non Becoming". The way I understand this terminology (which may not be quite what he means) is this:

whereas any old $\infty$-topos is a collection of spaces with structure , a cohesive $\infty$-topos comes with the extra adjoint $\Pi$, which I said has the interpretation of sending any space to its path $\infty$-groupoid. Therefore there is an intrinsic notion of geometric paths in any cohesive $\infty$-topos. This allows notably to define parallel transport along paths and higher paths, hence a kind of dynamics . In fact there is differential cohomology in every cohesive $\infty$-topos.

Now, in a discrete object there are no non-trivial paths (formally because $\Pi \; \mathrm{Disc} \simeq \mathrm{Id}$ by the fact that $\mathrm{Disc}$ is full and faithful), so there is "no dynamics" in a discrete object hence "no becoming", if you wish. Conversely in a codiscrete object every sequence of points whatsoever counts as a path, hence the distinction between the space and its "dynamics" disappears and so we have "pure becoming", if you wish.

Onwards. Notice next that every adjoint triple induces an adjoint pair of a comonad and a monad. In the present situation we get

$$ (\mathrm{Disc} \;\Gamma \dashv \mathrm{coDisc}\; \Gamma) : \mathcal{E} \to \mathcal{E} $$

This is what Lawvere calls the skeleton and the coskeleton on p. 7. In the $\infty$-topos context the left adjoint $\mathbf{\flat} := \mathrm{Disc} \; \Gamma$ has the interpretation of sending any object $A$ to the coefficient for cohomology of local systems with coefficients in $A$.

The paragraph wrapping from page 7 to 8 comments on the possibility that the base topos $\mathcal{S}$ is not just that of sets, but something richer. An example of this that I am kind of fond of is that of super cohesion (in the sense of superalgebra and supergeometry): the topos of smooth super-geometry is cohesive over the base topos of bare super-sets.

What follows on page 9 are thoughts of which I am not aware that Lawvere has later formalized them further. But then on the bottom of p. 9 he gets to the axiomatic identification of infinitesimal or formal spaces in the cohesive topos. In his most recent article on this what he says here on p. 9 is formalized as follows: he says an object $X \in \mathcal{E}$ is infinitesimal if the canonical morphism $\Gamma X \to \Pi_0 X$ is an isomorphism. To see what this means, suppose that $\Pi_0 X = *$, hence that $X$ is connected. Then the isomorphism condition means that $X$ has exactly one global point. But $X$ may be bigger: it may be a formal neighbourhood of that point, for instance it may be $\mathrm{Spec} \;k[x]/(x^2)$. A general $X$ for which $\Gamma X \to \Pi_0 X$ is an iso is hence a disjoint union of formal neighbourhoods of points.

Again, the meaning of this becomes more pronounced in the context of cohesive $\infty$-toposes: there objects $X$ for which $\Gamma X \simeq * \simeq \Pi X$ have the interpretation of being formal $\infty$-groupoids , for instance formally exponentiated $L_\infty$-algebras. And so there is $\infty$-Lie theory canonically in every cohesive $\infty$-topos.

I'll stop here. I have more discussion of all this at:

http://nlab.mathforge.org/schreiber/show/differential+cohomology+in+a+cohesive+topos

As a prelude to the answer above: When Lawvere wanted a mathematics of Becoming and not just Being, one thing he wanted is to work in categories that have exponentials.

For example, in the category of manifolds, we can express the motion of a body as a map $T \times B \rightarrow E$, where $T$ is an object of times, $B$ is a body, and $E$ is Euclidean space. Lawvere wants to work in bigger categories where we can also express this as $B \rightarrow E^T$ (the possible paths of an individual part of the body) or $T \rightarrow E^B$ (the collection of instantaneous descriptions of a body's location).

A category which works for these purposes is $\mathbf{Sets^{Z^{\large Op}}}$, where $Z$ is a category of loci that extends the category of manifolds. This leads to topoi.