nLab model structure on monoids in a monoidal model category

Idea

For $C$ a monoidal model category there is under mild conditions a natural model category structure on its category of monoids.

Definition

For $C$ a monoidal category with all colimits, its category of monoids comes equipped (as discussed there) with a free functor/forgetful functor adjunction

$$(F \dashv U) : Mon(C) \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C \,.$$

Typically one uses on $Mon(C)$ the transferred model structure along this adjunction, if it exists.

Theorem

If $C$ is monoidal model category that

• satisfies the monoid axiom in a monoidal model category;

• is a cofibrantly generated model category;

• all objects are small objects,

then the transferred model structure along the free functor/forgetful functor adjunction $(F \dashv U) : Mon(C) \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C$ exists on its category of monoids.

This is part of (SchwedeShipley, theorem 4.1).

Theorem

If the symmetric monoidal model category $C$

• is a cofibrantly generated model category

• its unit is cofibrant

• it has a suitable interval object

then the transferred model structure on monoids exists.

Proof

Regard monoids a algebras over an operad for the associative operad. Then apply the existence results discussed at model structure on algebras over an operad. See there for more details.

Properties

Homotopy pushouts

Suppose the transferred model structure exists on $Mon(C)$. By the discussion of free monoids at category of monoids we have that then pushouts of the form

$$\array{ F(A) &\stackrel{F(f)}{\to}& F(B) \\ \downarrow && \downarrow \\ X &\to& P }$$

exist in $Mon(C)$, for all $f : A \to B$ in $C$

Proposition

Let the monoidal model category $C$ be

• symmetric monoidal;

• and satisfy the monoid axiom in a monoidal model category.

If $f : A\to B$ is an acyclic cofibration in the model structure on $C$, then the pushout $X \to P$ as above is a weak equivalence in $Mon(C)$.

This is SchwedeShipley, lemma 6.2.

Proof

Use the description of the pushout as a transfinite composite of pushouts as described at category of monoids in the section free and relative free monoids.

One sees that the pushout product axiom implies that all the intermediate pushouts produce acyclic cofibrations and the monoid axiom in a monoidal model category implies then that each $P_{n-1} \to P_n$ is a weak equivalence. Moreover, all these moprhisms are of the kind used in the monoid axioms, so also their transfinite composition is a weak equivalence.

$A_\infty$-Algebras

Under mild conditions on $C$ the model structure on monoids in $C$ is Quillen equivalent to that of A-infinity algebras in $C$. See model structure on algebras over an operad for details.

Related concepts

• model structure on monoids in a monoidal model category

  • model structure for ring spectra

• model structure on modules in a monoidal model category

• model structure on commutative monoids in a symmetric monoidal model category

• model structure on algebras over an operad

• model structure on algebras over a monad

• model structure on simplicial T-algebras

References

• Stefan Schwede, Brooke Shipley, Algebras and modules in monoidal model categories Proc. London Math. Soc. (2000) 80(2): 491-511 (pdf)

• David White, Model Structures on Commutative Monoids in General Model Categories (arXiv:1403.6759)