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Onwards. Notice next that every adjoint triple induces an adjoint pair of monads. In the present situation we get

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

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 a total adjoint quadruple

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

whereas any old $\infty$-topos is a collection of spaces , 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.

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

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

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.

(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 induced by the sub-topos. That, one can show, may be identified with the collection of concrete cohesive spaces. In the case of the cohesive topos for differential geometry, the concrete objects in this sense are precisely the diffeological spaces . )

whereas any old $\infty$-topos is a collection of spaces , 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.

(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 . )

whereas any old $\infty$-topos is a collection of spaces , 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.