model category

for ∞-groupoids

# Contents

## Idea

The standard model category structure on cosimplicial objects in unital, commutative algebras over $k$.

## Definition

Write ${\mathrm{Alg}}_{k}^{\Delta }$ for the category of cosimplicial objects in the category of unital, commutative $k$-algebras. Sending algebras to their underlying $k$-modules yields a forgetful functor

$U:{\mathrm{Alg}}_{k}^{\Delta }\to k{\mathrm{Mod}}^{\Delta }\phantom{\rule{thinmathspace}{0ex}}.$U : Alg_k^\Delta \to k Mod^\Delta \,.

### The model category structure

The Dold-Kan correspondence provides the normalized cochain complex (Moore complex) functor

$N:k{\mathrm{Mod}}_{k}^{\Delta }\to {\mathrm{Ch}}^{•}\left(k{\right)}_{+}\phantom{\rule{thinmathspace}{0ex}}.$N : k Mod_k^\Delta \to Ch^\bullet(k)_+ \,.

Define a morphism $f:A\to B$ of cosimplicial algebras is a morphism is a weak equivalence if

$N\left(U\left(f\right)\right):N\left(U\left(A\right)\right)\to N\left(U\left(B\right)\right)$N(U(f)) : N(U(A)) \to N(U(B))

is a quasi-isomorphism in ${\mathrm{Ch}}_{+}^{•}\left(k\right)$.

Say a morphism of cosimplicial algebras is a fibration if it is a epimorphism (degreewise surjection).

This defines the projective model category structure on ${\mathrm{Alg}}_{k}^{\Delta }$.

### The simplicial model category structure

There is also the structure of an sSet-enriched category of ${\mathrm{Alg}}_{k}^{\Delta }$.

###### Definition

For $X$ a simplicial set and $A\in {\mathrm{Alg}}_{k}$ let ${A}^{X}\in {\mathrm{Alg}}_{k}^{\Delta }$ be the corresponding $A$-valued cochains on simplicial sets

${A}^{X}:\left[n\right]↦\prod _{{X}_{n}}{A}_{n}\phantom{\rule{thinmathspace}{0ex}}.$A^X : [n] \mapsto \prod_{X_n} A_n \,.
###### Remark

If we write $C\left(X\right):={\mathrm{Hom}}_{\mathrm{Set}}\left({X}_{•},k\right)$ for the cosimplicial algebra of cochains on simplicial sets then for $X$ degreewise finite this may be written as

${A}^{X}=A\otimes C\left(X\right)$A^X = A \otimes C(X)

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

###### Definition

For $A,B\in {\mathrm{Alg}}_{k}^{\Delta }$ define the sSet-hom-object ${\mathrm{Alg}}_{k}^{\Delta }\left(A,B\right)$ by

${\mathrm{Alg}}_{k}^{\Delta }\left(A,B\right):={\mathrm{Hom}}_{\mathrm{sSet}}\left(A,{B}^{\Delta \left[•\right]}\right)={\mathrm{Hom}}_{\mathrm{sSet}}\left(A,B\otimes C\left(\Delta \left[•\right]\right)\right)\in \mathrm{sSet}\phantom{\rule{thinmathspace}{0ex}}.$Alg_k^\Delta(A,B) := Hom_{sSet}(A, B^{\Delta[\bullet]}) = Hom_{sSet}(A, B \otimes C(\Delta[\bullet])) \in sSet \,.
###### Remark

For $B\in {\mathrm{Alg}}_{k}$ regarded as a constant cosimplicial object under the canonical embedding ${\mathrm{Alg}}_{k}↪{\mathrm{Alg}}_{k}^{\Delta }$ we have

${\mathrm{Alg}}_{k}^{\Delta }\left(A,{B}^{\Delta \left[n\right]}\right)={\mathrm{Alg}}_{k}^{\Delta }\left(A,B\otimes C\left(\Delta \left[n\right]\right)\right)\simeq {\mathrm{Alg}}_{k}\left({A}_{n},B\right)\phantom{\rule{thinmathspace}{0ex}}.$Alg_k^\Delta(A, B^{\Delta[n]}) = Alg_k^\Delta(A, B \otimes C(\Delta[n])) \simeq Alg_k(A_n,B) \,.
###### Proof

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

${f}_{n}:{A}_{n}\to B$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 $\Delta \left[n\right]$. The $n+1$ coface maps $C\left(\Delta \left[n\right]{\right)}_{n}←C\left(\Delta \left[n\right]{\right)}_{n-1}$ obtained as the pullback of the $\left(n+1\right)$ face inclusions $\Delta \left[n-1\right]\to \Delta \left[n\right]$ restrict on the non-degenerate $\left(n-1\right)$-cells to the $n+1$ projections $k←{k}^{n+1}:{p}_{i}$.

Accordingly, from the naturality squares for $f$

$\begin{array}{ccc}{A}_{n}& \stackrel{{f}_{n}}{\to }& B\\ {↑}^{{\delta }_{i}}& & {↑}^{{p}_{i}}\\ {A}_{n-1}& \stackrel{{f}_{n-1}}{\to }& {B}^{n+1}\end{array}$\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

${f}_{n-1}=\left({f}_{n}\circ {\delta }_{0},\cdots ,{f}_{n}\circ {\delta }_{n}\right)$

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 ${\mathrm{Alg}}_{k}^{\Delta }$ into a simplicially enriched category which is tensored and cotensored over $\mathrm{sSet}$.

And this is compatible with the model category structure:

###### Theorem

With the definitions as above, ${\mathrm{Alg}}_{k}^{\Delta }$ is a simplicial model category.

###### Proof

To06, theorem 2.1.2

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

## References

Details are in section 2.1 of