Simplicial spaces internally to simplicial sets

Question

I am a master’s student with interest in topos theory and its applications (motivated by Ingo Blechschmidt’s thesis, as seems to be usual).

After finding out about some of the uses of simplicial spaces (as nice “resolutions” to ordinary spaces), particularly for understanding mixed Hodge theory, I was wondering if there is a way to understand (or replace) the theory of simplicial spaces with a theory of internal locales to $\mathbf{sSet}$ or something similar.

Question. Can you create a constructive version of classical Hodge theory, such that internalized to simplicial sets we retrieve the theory of mixed Hodge theory? Is there some place where the internal language of simplicial sets has been developed?

Perhaps the answer to the question comes from infinity topos theory or similar. I don’t know much about this subject but would still appreciate an answer following this route.

Answer 1

I don't think there is a correspondence between simplicial spaces and certain internal locales in $\mathbf{sSet}$.

Constructing an internal locale is the same thing as defining an internal frame, only the notion of morphism is different. An internal frame in $\mathbf{sSet}$ is a functor $\Delta^\mathrm{op} \to \mathbf{Frm}$ satisfying certain conditions (see Sketches of an Elephant, C1.6, Lemma 1.6.9). Here $\mathbf{Frm}$ is the category of frames. On the other hand, a simplicial space is a functor $\Delta^\mathrm{op} \to \mathbf{Top}$, with $\mathbf{Top}$ the category of topological spaces. You can associate with a topological space its frame of open sets, but then you get a functor $\Delta^\mathrm{op} \to \mathbf{Frm}^\mathrm{op}$ instead of a functor $\Delta^\mathrm{op} \to \mathbf{Frm}$.

On the other hand, what you can do is look at the topos $[\Delta, \mathbf{Set}]$ rather than $\mathbf{sSet} = [\Delta^\mathrm{op},\mathbf{Set}]$. Then each internal locale in $[\Delta, \mathbf{Set}]$ is given by a functor $\Delta \to \mathbf{Frm}$. If you are lucky, then all frames in the image of this functor are spatial, and then you can associate to this internal locale a functor $\Delta^\mathrm{op} \to \mathbf{Top}$, or in other words a simplicial space. It would be interesting to see which simplicial spaces come from internal locales in $[\Delta,\mathbf{Set}]$, but as far as I know this is very difficult.

Answer 2

This is a small clarification of Jens Hemelaer's answer, but it is essentially the same point: Internal locales in a presheaf category are actually very different from "presheaf of locales".

I'm denoting by $\mathcal{Loc}$ the category of locales, with morphisms in the geometric direction (i.e. if $f:X \to Y$ is a morphism in $\mathcal{Loc}$, then we have a morphism of frame $f^*: \mathcal{O}(Y) \to \mathcal{O}(X)$.

In An extension of the Galois theory of Grothendieck (Memoirs of the American Mathematical Society vol 51 (1984) — Which I unfortunately cannot find an online version) Joyal and Tierney prove the following (it also appears as pointed out by Jens as Lemma C.1.6.9 in Sketches of an Elephant):

Theorem Let $A$ be a category with finite limits. Then a locale in the presheaf topos $Psh(A)$ is given by a functor $X: A \to Loc$ such that

(1) For each arrow $f:A \to B$, the induced map of locales $X(A) \to X(B)$ is an open map. That is:

(1a) the map $f^*:\mathcal{O}(X(B)) \to \mathcal{O}(X(A))$ has a further left adjoint $f_!$ and

(1b) it satisfies the Frobenius identity $f_!(f^*(x) \wedge y ) = x \wedge f_!(y)$.

(2) For each pullback square you have a Beck–Chevalley condition: $g^* f_! = k_! h^*$

However, that doesn't apply to $\Delta$ immediately as it does not have finite limits.

What I wanted to say is that, using the same methods their results can be improved as follows:

Proposition: If one remove the assumption that $A$ has finite limits in the theorem, then the result still holds if we replace condition (2) by:

(2') for every copan $X \overset{f} \to B \overset{g}{\leftarrow} Y$ we have $$ g^* f_! = \sup k_! h^*$$

where the supremum is over all pairs of maps $(k,h)$ in $A$ that complete the cospan $(f,g)$ into a commutative square.

Note that it is easy to recover condition (2) from condition (2') when the pullback exists, as the $k_! f^*$ corresponding to the pullback is easily seen to be maximal amongst all these corresponding to arbitrary commutative square.

I worked that out myself a long time ago and never published it, but it might very well be somewhere else in the literature, I'd be happy to add a reference if someone knows one! (to some extent, it is relatively easy to deduce it from Corollary C;1.6.10 of the Elephant, though)

I suspect that condition (2) and (2') have some sort of geometric interpretation, but I haven't worked it out explicitly.

For these interested one can even generalize to the situation where $A$ has a Grothendieck topology, in this case internal locales in the sheaf topos are these that further satisfy the condition:

(3) For each covering family $(B_i \to B)$ in $A$, the maps $X(B_i) \to X(B)$ are jointly a covering.

Here I mean that the open map $\left( \coprod X(B_i) \right) \to X(B)$ is actually an open surjection.

So in any case, internal locales in simplicial sets are fairly different from simplicial spaces. They are cosimplicial spaces whose structural map are open and that further satisfies this condition (2').