The n-Category Café

Where is the rational?

There is of course the choice of ground field $k$ in $\mathbf{L} := (Alg_k^\Delta)^{op}$.

The relation to genuine rational homotopy as in Toën’s theorem 2.5.1 is obtained with $k = \mathbb{Q}$.

But other choices of $k$ give analogous interesting structure. For instance for $k$ of finite characteristic and algebraically closed, one still gets that the unit $X \to \Gamma Spec \mathcal{O} LConst X$ of the total adjunction

$$\mathbf{L} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} \mathbf{H} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd \simeq Top_{cg}$$

is an equivalence on $k$-cohomology groups, hence is “rationalization” with respect to $k$.

Alternatively, why isn’t this just Homotopy theory in an (oo,1)-topos?

Right, for the above reason.

The full homotopy theory of $\mathbf{H}$ is instead encoded differently, as described at homotopy groups in an $(\infty,1)$-topos.

One way to say it is to assume that the terminal geometric morphism $\mathbf{H} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}} \infty Grpd$ is essential meaning that there is a further left adjoint $\Pi$ to $LConst$

$$\mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd \,.$$

Then for any $X \in \mathbf{H}$ we have its geometric homotopy groups encoded in the fundamental $\infty$-groupoid $\Pi(X)$ and $|\Pi(X)| \in Top_{cg}$ is the geometric realization of the object in $\mathbf{H}$.

So then the rational homotopy of $X \in \mathbf{H}$ is computed by chasing $X$ along the full S-shaped path of the total diagram

$$\mathbf{L} \stackrel{\overset{\mathcal{O}}{\leftarrow}}{\underset{Spec}{\to}} \mathbf{H} \stackrel{\overset{\Pi}{\to}}{\stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}} \infty Grpd \simeq Top_{cg} \,.$$

How does a reflexive embedding give us a characteristic 0 field?

Well, so far the reflective embedding is built after a $k$ has been chosen by hand.

But I think that one should be able to go the other way round, as you seem to indicate, and determine the $k$ from an abstract property of $\mathbf{H}$. It should be determined by the line object that is encoded by $\Pi$. I can say more about this a little later.