typical contexts
This page is meant to provide non-technical motivation for the notion of cohesion as in formalized in the notions
A notion of cohesion on a collection $\mathbf{H}$ of spaces is supposed to be a means to specify how points in any space $X \in \mathbf{H}$ “hang together” or “cohere”, analogous to how water molecules in a droplet of water are held together by cohesion (in the sense of chemistry). (The inclined reader may, or may not, recognize this also in WdL, Form und Inhalt, see the discussion there.)
A basic example arises for topological spaces or manifolds: here the “droplet of water” is an open ball of points. Indeed, one of the central examples of cohesive spaces is that of smooth spaces and these are spaces characterized by the fact that they can be probed by smooth open balls (in the sense described at motivation for sheaves, cohomology and higher stacks), such that these smooth open balls are the basic “cohesive droplets” out of which any smooth space is built (this roughly in the sense of basis of a topology, but a bit more generally than that).
That intuition should be evident enough. The question is which formal axioms capture this droplet-intuition accurately and efficiently. The crucial insight of Bill Lawvere (see the references at cohesive topos) was that a rather minimalistic set of axioms already does the job:
there has to be an assignment $\Pi : \mathbf{H} \to Set$ that sends every cohesive space $X$ to its set of cohesively connected components. For instance a single open ball as above, a basic droplet, is sent to the set $\{*\}$ with a single element.
We should be entitled to regard every set $S \in Set$ trivially as a cohesive space in two ways:
either we regard every element in $S$ as an atomic cohesive droplet itself, cohesively disconnected to any other point. Call the resulting cohesive space $Disc(X)$. For smooth cohesion, $Disc(X)$ is simply the discrete manifold being the disjoint union of one point per element in $S$.
or, at the opposite extreme, we regard all the elements in $S$ as being cohesively connected, hence regard all of $S$ itself as one single big cohesive droplet. We write $coDisc(X)$ for $S$ regarded as a cohesive space this way, because in the context of topological cohesion this is the codiscrete space on the given set.
The collection of discrete and codiscrete cohesive spaces should sit nicely inside the collection of all cohesive spaces, essentially in just the obvious way that one intuitively expects. For instance cohesive maps between two discretely cohesive spaces should be simply maps between the underlying sets, and so on.
In particular the notion of “discrete” exhibited by the collection of discrete objects has indeed to be compatible with the notion of “cohesive” as seen by that map $\Pi$ above. This is just the evident consistency condition: for instance if $Disc(S)$ is a set regarded as a discrete cohesive space, then $\Pi(Disc(S))$, which is supposed to be its set of cohesively connected points, should be just $S$ itself, since no point in $Disc(S)$ is supposed to be cohesively connected to any other.
In particular $\Pi(Disc(*))$ should be the point again.
That’s essentially it, already. It sounds very simple (hopefully) and indeed it is very simple. Once one knows the notion of an adjoint functor it is pretty straightforward to formalize the above text in terms of that notion, to arrive at the concept of cohesive topos.
The interesting observation is that simple as this idea is, it has very powerful consequences … once it is formalized not just with adjoint functors but with adjoint (∞,1)-functors. This is conceptually a very simple step. Moreover, in the foundations provided by homotopy type theory this is actually the simpler and more natural step. It leads to cohesive (∞,1)-toposes and cohesive homotopy type theory.
While cohesive spaces subsume several familiar notions of geometry, there are some constraints.
In particular, a cohesive space is always locally contractible or rather locally $\infty$-connected in some sense. The local contractions are those of the “basic cohesive droplets”, in the spirit of the above discussion. (In the (∞,1)-topos theoretic formalization this is reflected in the fact that every cohesive (∞,1)-topos is in particular a locally ∞-connected (∞,1)-topos.)
For instance locally contractible topological ∞-groupoids are cohesive, as are Euclidean-topological ∞-groupoids.
But a geometry modeled on a small full subcategory of Top that contains locally non-contractible topological spaces will in general not be cohesive. In particular for instance general topological stacks do not live in a cohesive $(\infty,1)$-topos. But for instance differentiable stacks do, and generally smooth ∞-groupoids, because every manifold and hence in particular every smooth manifold is locally contractible.