model category

for ∞-groupoids

Contents

Idea

A model category structure on cosimplicial objects in unital, commutative algebras over some field $k$.

The model structure

Let $k$ be a field of characteristic zero.

Definition

Write $cAlg_k^\Delta$ for the category of cosimplicial objects in the category of unital, commutative algebras over $k$.

Remark

Sending $k$-algebras to their underlying $k$-modules yields a forgetful functor

$U \colon cAlg_k^\Delta \longrightarrow k Mod^\Delta$

from cosimplicial $k$-allgebras (def. 1) to cosimplicial objects in $k$-vector spaces.

Moreover, the Dold-Kan correspondence provides the normalized cochain complex functor

$N \colon k Mod_k^\Delta \to Ch^{\geq 0}(k)$

from cosimplicial $k$-vector spaces to cochain complexes (i.e. with differential of degree +1) in non-negative degrees.

Proposition

Say that morphism $f \colon A \to B$ in $cAlg_k^{\Delta}$ (def. 1) is

1. a weak equivalence if its image $N(U(f)) \colon N(U(A)) \to N(U(B))$ under the comparison functors from remark 1 is a quasi-isomorphism in $Ch^{\geq 0}(k)$;

1.a fibration if $f$ is an epimorphism (i.e. degreewise a surjection).

Then

1. this defines a model category structure, to be called the projective model structure on comsimplicial commutative $k$-algebras. $(cAlg_k^\Delta)_{poj}$.

2. this is a cofibrantly generated model category

Proof

The first two statements follow by observing that $(cAlg_k^{\Delta})_{proj} is$the transferred model structure along the forgetful functor $U \circ N$ from remark 1 of the projective model structure on chain complexes, by this prop..

The third statement is the content of prop. 2 below.

Properties

Simplicial model category structure

There is also the structure of an sSet-enriched category on $cAlg_k^\Delta$ (def. 2)

Definition

For $X$ a simplicial set and $A \in Alg_k$ let $A^X \in Alg_k^\Delta$ be the corresponding $A$-valued cochains on simplicial sets

$A^X \;\colon\; [n] \mapsto (A_n)^{X_n} = \underset{X_n}{\prod} A_n \,.$
Remark

If we write $C(X) \coloneqq Hom_{Set}(X_\bullet,k)$ for the cosimplicial algebra of cochains on simplicial sets then for $X$ degreewise finite this may be written as

$A^X = A \otimes C(X)$

where the tensor product is the degreewise tensor product of $k$-algebras.

Definition

For $A,B \in Alg_k^\Delta$ define the sSet-hom-object $Alg_k^\Delta(A,B)$ by

$Alg_k^\Delta(A,B) \coloneqq Hom_{sSet}(A, B^{\Delta[\bullet]}) = Hom_{sSet}(A, B \otimes C(\Delta[\bullet])) \in sSet \,.$
Remark

For $B \in Alg_k$ regarded as a constant cosimplicial object under the canonical embedding $Alg_k \hookrightarrow Alg_k^\Delta$ we have

$Alg_k^\Delta(A, B^{\Delta[n]}) = Alg_k^\Delta(A, B \otimes C(\Delta[n])) \simeq Alg_k(A_n,B) \,.$
Proof

Let $f : A \to B \otimes C(\Delta[n])$ be a morphism of cosimplicial algebras and write

$f_n : A_n \to B$

for the component of $f$ in degree $n$ with values in the copy $B = B \otimes k$ of functions $k$ on the unique non-degenerate $n$-simplex of $\Delta[n]$. The $n+1$ coface maps $C(\Delta[n])_n \leftarrow C(\Delta[n])_{n-1}$ obtained as the pullback of the $(n+1)$ face inclusions $\Delta[n-1] \to \Delta[n]$ restrict on the non-degenerate $(n-1)$-cells to the $n+1$ projections $k \leftarrow k^{n+1} : p_i$.

Accordingly, from the naturality squares for $f$

$\array{ A_n &\stackrel{f_n}{\to}& B \\ \uparrow^{\mathrlap{\delta_i}} && \uparrow^{\mathrlap{p_i}} \\ A_{n-1} &\stackrel{f_{n-1}}{\to}& B^{n+1} }$

the bottom horizontal morphism is fixed to have components

$f_{n-1} = (f_n \circ \delta_0, \cdots, f_n \circ \delta_n)$

in the functions on the non-degenerate simplices.

By analogous reasoning this fixes all the components of $f$ in all lower degrees with values in the functions on degenerate simplices.

The above sSet-enrichment makes $cAlg_k^\Delta$ into a simplicially enriched category which is tensored and cotensored over $sSet$.

And this is compatible with the model category structure:

Proposition

With the definitions as above, $(cAlg_k^\Delta)_{proj}$ is a simplicial model category.

Toën 00, theorem 2.1.2

Relation to the model structure on cochain dgc-algebras

Under the monoidal Dold-Kan correspondence this is related to the model structure on commutative non-negative cochain dg-algebras.

Details are in