I'm looking for directions to the literature that might contain fairly explicit constructions that might be called (the algebra of functions on) the "derived mapping space" from a simplicial set to a simplicial (affine) scheme. To make the question reasonably self-contained and to give a sense of my background and current understanding, I will begin with some general abstract nonsense, and then point to a construction that I have found in the literature and that does not work to my satisfaction.
A little abstract nonsense
Let $C$ and $D$ be categories. In a bit, I will give them specific values, but for now I will ask only that $D$ be small, and that $C$ have any necessary limits and colimits. A (generalized) $D$-object in $C$ is a presheaf on $D$ valued in $C$, i.e. a functor $X : D^{\mathrm{op}}\to C$. Each $d\in D$ determines (and is determined by) a $D$-object in $\mathrm{SET}$, by the usual Yoneda embedding $d \mapsto \operatorname{hom}_D(-,d)$. It will be convenient for me to denote the presheaf $\operatorname{hom}_D(-,d)$ by $[d]$, and given $k\in D$ and $X : D^{\mathrm{op}}\to C$, I will write $X_k$ for $X(k)$.
For $x\in C$ and $s\in \mathrm{SET}$, there is an object $\operatorname{maps}(s,x) = x^s = \prod_s x \in C$, which is the $s$-fold cartesian product of $x$ with itself. Now, let $X : D^{\mathrm{op}} \to C$ be a $D$-object in $C$, and $S: D^{\mathrm{op}} \to \mathrm{SET}$ a $D$-set. Then there is an object $\operatorname{hom}_D(S,X) \in C$, which is built as a certain limit ranging over the objects $\operatorname{maps}(S_k,X_k)$ for $k\in D$. Even better, the categories of $D$-sets and $D$-objects in $C$ have products — the ("categorical") cartesian product of functors is constructed by taking the product for each — and so we can define an enriched hom by: $$ \underline{\operatorname{hom}}_D(S,X) : D^{\mathrm{op}} \to C, \quad d \mapsto \operatorname{hom}_D(S \times [d],X). $$ Finally, there is one more, much more naive "mapping space" between $D$-objects, which I will denote by $\operatorname{maps}(S,X) : D \times D^{\mathrm{op}} \to C$, sending $(d,k) \mapsto \operatorname{maps}(S_d,X_k)$.
A little concrete nonsense
I will be interested in the situation where $D = \Delta$ is the category of finite nonempty totally-ordered sets (and monotonic maps). It has a skeletalization with objects indexed by the natural numbers, given by $[n] = \lbrace 0 < \dots < n \rbrace$. Note that the $\Delta$-set $[0]$ is terminal, so $\underline{\operatorname{hom}}_D(S,X)_0 = \operatorname{hom}_D(S,X)$. An object $X : D^{\mathrm{op}} \to C$ determines, among other data, two maps $X_1 \rightrightarrows X_0$, corresponding to the two inclusions $[0] \rightrightarrows [1]$. By definition, $\pi_0(X) \in C$ is the coequalizer of the two arrows $X_1 \rightrightarrows X_0$.
Fix a commutative ring $\mathbb K$. I will not be upset if you would like to make further assumptions on $\mathbb K$, e.g. that $\mathbb K \supseteq \mathbb Q$, or that $\mathbb K$ is an algebraically closed field. I believe that I am primarily interested in the following two values for $C$, but I am open to being convinced otherwise:
- $C = \mathrm{CAlg}^{\mathrm{op}}_{\mathbb K}$ is the category of affine schemes over $\mathbb K$.
- $C = \mathrm{Mod}_{\mathbb K}$ is the category of $\mathbb K$-modules.
There is a well-known contravariant forgetful functor $\mathcal O$ from $\mathrm{CAlg}^{\mathrm{op}}_{\mathbb K}$ to $\mathrm{Mod}_{\mathbb K}$.
I will also take inspiration from the case $C = \mathrm{Top}$ of nice enough topological spaces.
There is a further functor $\operatorname{ch}: \Delta\mathrm{Mod}_{\mathbb K} \to \mathrm{DGMod}_{\mathbb K}$ (the category of homologically-graded chain complexes of $\mathbb K$-modules) which sets $\operatorname{ch}(X)_k = X_k$ with differential a certain well-known alternating sum. There is a standard symmetric monoidal structure on $\mathrm{DGMod}_{\mathbb K}$ which sums the homological degrees, and for this structure $\operatorname{ch}$ is not strongly monoidal, but there is a canonical Eilenberg–Zilber map $\operatorname{ch}(X) \otimes \operatorname{ch}(Y) \to \operatorname{ch}(X \otimes Y)$, which sums over all $(k+\ell)$-simplices in a product of a $k$-simplex with an $\ell$-simplex (closely related is the fact that for simplicial sets, the geometric realization of a product is homeomorphic to the product (in the category of compactly-generated spaces) of geometric realizations), making $\operatorname{ch}$ into a "lax symmetric monoidal functor". The Eilenberg–Zilber map is a quasi-isomorphism, and one choice of quasi-inverse is the (non-symmetric) Alexander–Whitney map; if $\mathbb K \supseteq \mathbb Q$, there are other more symmetrical choices. In any case, the Eilenberg–Zilber map means that any simplicial commutative algebra determines canonically a dg commutative algebra. Of course, when $C = \mathrm{CAlg}^{\mathrm{op}}_{\mathbb K}$, a simplicial affine scheme $X : \Delta^{\mathrm{op}} \to \mathrm{CAlg}^{\mathrm{op}}_{\mathbb K}$ determines a cosimplicial commutative algebra $\mathcal{O}(X)$, and so $\operatorname{ch}(\mathcal{O}(X))$ is not quite a dgca (the Alexander–Whitney map makes it into a dga). Anyway, this all won't matter much for me.
What I wanted to mention about all this is that if $X : \Delta^{\mathrm{op}} \to \mathrm{CAlg}^{\mathrm{op}}_{\mathbb K}$ is a simplicial affine scheme, then $\operatorname{H}_\bullet(\operatorname{ch}(\mathcal{O}(X)))$ is canonically a graded commutative algebra (supported in nonpositive homological degrees; and there is more algebraic data in the form of Massey products) and $$ \operatorname{H}_0(\operatorname{ch}(\mathcal{O}(X))) = \mathcal{O}(\pi_0(X)). $$
Examples
Let $A$ be a commutative $\mathbb K$-algebra, with corresponding affine scheme $X = \operatorname{spec}(A) \in \mathrm{CAlg}^{\mathrm{op}}_{\mathbb K}$. If you want, you can extend $X$ to a constant functor $X : \Delta^{\mathrm{op}} \to \mathrm{CAlg}^{\mathrm{op}}_{\mathbb K}$. Let $S^1$ denote the simplicial set generated by one nondegenerate $0$-simplex and one nondegenerate $1$-simplex. Then $\operatorname{maps}(S^1, X)$ is a cosimplicial affine scheme (or simplicial cosimplicial, but constant in the simplicial direction), and so $\mathcal{O}(\operatorname{maps}(S^1, X))$ is a simplicial commutative algebra. By definition, $$ \operatorname{HH}_\bullet(A) = \operatorname{H}_\bullet(\operatorname{ch}(\mathcal{O}(\operatorname{maps}(S^1, X)))) $$ is the Hochschild homology of $A$. The complex $\operatorname{ch}(\mathcal{O}(\operatorname{maps}(S^1, X)))$ can be alternately defined by making a certain choice of resolution of $A$ as an $(A\otimes A)$-module, and using this resolution to construct the derived tensor product $A \otimes_{A\otimes A} A$.
Let $G$ be an affine algebraic group over $\mathbb K$ (e.g. a finite group). There is a well-known simplicial affine scheme $X = \mathrm{B}G$ whose space of $k$-simplices is $G^k$, with boundary maps that encode the multiplication. Let $M$ be a simplicial set, and I am primarily interested in the case that $M$ is a simplicial finite set describing the homotopy type of a finite-dimensional compact manifold. The simplicial affine scheme $\underline{\operatorname{hom}}_\Delta(M,\mathrm{B}G)$ is the space if $G$-local systems on $M$. In particular, $\pi_0(\underline{\operatorname{hom}}_\Delta(M,\mathrm{B}G))$ is the character variety of $M$.
My question
I am looking for a general construction, of the flavor above, that incorporates both examples. More specifically, the construction should:
- input a simplicial (finite) set $M$ and a a simplicial affine scheme $X$ over $\mathbb K$
- output a chain complex $V(M,X)$ over $\mathbb K$, supported in both directions, that deserves to be thought of as a "derived space of global functions on the space of maps from $M$ to $X$"
- have good functoriality and monoidality properties in both variables (implying for instance that $V(M,X)$ has a strongly-homotopy commutative dg algebra structure, coming from various diagonal and Eilenberg–Zilber-like maps)
- if $X = \operatorname{spec}(A)$ is a constant simplicial scheme, then $V(M,\operatorname{spec}(A))$ is the generalized Hochschild homology of $A$ determined by $M$
- $\operatorname{H}_0(V(M,X)) = \mathcal{O}(\pi_0(\underline{\operatorname{hom}}_\Delta(M,X)))$
Some near misses
The problem seems to be when $X$ is not "simply connected". In particular, I have not come across a construction that works even when $X = \mathrm{B}G$ for $G$ a finite simple group.
Greg Ginot and collaborators (see e.g. Higher order Hochschild cohomology, Derived Higher Hochschild Homology, Topological Chiral Homology and Factorization algebras, and A Chen model for mapping spaces and the surface product) have extended work by Pirashvili defining the generalized Hochschild homology. Let $A$ be a cdga over $\mathbb K \supseteq \mathbb Q$ and let $M$ be a simplicial set. Then there is a simplicial cdga $\int_M A = \mathcal{O}(\operatorname{maps}(M,\operatorname{spec}(A)))$ with good functoriality and monoidality properties, which agrees up to quasi-isomorphism with Lurie's "topological chiral homology."
By definition, a quasi-isomorphism of cdgas is a morphism that induces isomorphisms on homology. One of the things that Ginot et al prove is that a quasi-isomorphism $A \to B$ induces a quasi-isomorphism $\int_M A \to \int_M B$. Thus in particular when $A = \mathcal{O}(\mathrm{B}G) = \operatorname{Ext}_G(\mathbb K,\mathbb K)$, for any meaning of this, and $G$ is a finite simple group, then the canonical map $\mathbb K \to A$ is a quasi-isomorphism, and so the chain complex $\int_M A$ will never contain data. So this construction fails my last condition, e.g.: $\pi_0(\underline{\operatorname{hom}}_\Delta(S^1,\mathrm{B}G)) = G/G^{\mathrm{conj}}$ and $\mathcal{O}(\pi_0(\underline{\operatorname{hom}}_\Delta(S^1,\mathrm{B}G))) = \mathcal{O}(G)^G$ is the algebra of class functions on $G$, whereas $\operatorname{H}_0(\int_{S^1}\mathcal{O}(\mathrm{B}G)) = \mathbb K$.
Ben-Zvi and Nadler have discussed loop spaces and connections their relationships to Hochschild homology and representations. They run into what I believe are related issues, but work primarily with not the space of loops in a derived scheme, but rather the infinitesimal neighborhood of the constant loops within that space. I should also mention that for my particular application, I really am looking for an explicit one-categorical construction (akin to the Pirashvili-style work), rather than quickly moving to model or $\infty$ categories.
Finally, perhaps the result I should have started with is one I learned from a review by Loday (original references are included there). Suppose that $M$ is a simplicial approximation of an $n$-dimensional manifold, and that $X$ is a simplicial set which is $n$-connected, in $\pi_{\leq n}(X)$ is trivial. (So $1$-connected means connected simply-connected.) I can build a cosimplicial simplicial set $\operatorname{maps}(M,X)$, and a simplicial set $\underline{\operatorname{hom}}_\Delta(M,X)$, as discussed above. There is a _free $\mathbb K$-module_ functor $\mathbb K : \mathrm{SET} \to \mathrm{Mod}_{\mathbb K}$, and with is I get a cosimplicial simplicial $\mathbb K$-module $\mathbb K\operatorname{maps}(M,X)$ and a simplicial $\mathbb K$-module $\mathbb K\underline{\operatorname{hom}}_\Delta(M,X)$. Of course, given a cosimplicial simplicial $\mathbb K$-module, I can apply the "alternating sum of boundaries" functor $\operatorname{ch}$ to get a bicomplex, which I can then totalize. Unless I have made a mistake, I believe the statement is that under the conditions on $M$ and $X$, the canonical map of chain complexes between $\operatorname{ch}(\mathbb K\operatorname{maps}(M,X))$ and $\operatorname{ch}(\mathbb K\underline{\operatorname{hom}}_\Delta(M,X))$ is a quasi-isomorphism. This example specifically does not include classifying spaces of finite groups.
Final comments and examples
Truth be told, I am most interested in the case $X = \mathrm B G$ for $G = \mathrm{SL}(2)$ and $M$ a simplicial approximation of a three-manifold. Note that the topological space $\mathrm{B}(\mathrm{SL}(2,\mathbb C))$ is $3$-connected (it is homotopy equivalent to $\mathrm{B}(\mathrm{Spin}(3,\mathbb R))$), but I don't have a good sense about notions like "3-connected" for algebraic stacks. And, besides, I would like a robust construction.
Let me end with an example that does work. A much easier category than $\mathrm{CAlg}^{\mathrm{op}}_{\mathbb K}$ is the category $\mathrm{CCog}_{\mathbb K}$ of cocommutative coalgebras (or "cogebres" in French, hence the name). A group object in $\mathrm{CCog}_{\mathbb K}$ is a cocommutative Hopf algebra, and a good example is the universal enveloping algebra $U\mathfrak g$ of a Lie algebra $\mathfrak g$. Then $\mathrm B U\mathfrak g = \mathrm B \mathfrak g$ is a simplicial cocommutative coalgebra, which is quasi-isomorphic to the dg cocomutative coalgebra $\mathrm{CE}(\mathfrak g)$ of Chevalley–Eilenberg cochains with trivial coefficients. I believe that it _is_ true that Hochschild homology of $\mathrm{CE}(\mathfrak g)$ is the Chevalley–Eilenberg cochain complex with coefficients in $U\mathfrak g$, as it should be if you think about loop spaces.
