nLab model structure on monoids in a monoidal model category

model category

Model structures

for ∞-groupoids

for $(\infty,1)$-sheaves / $\infty$-stacks

Monoidal categories

monoidal categories

In higher category theory

Higher algebra

higher algebra

universal algebra

Contents

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 the 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.

Properties

Existence

Theorem

If $C$ is monoidal model category that

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 $C$

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.

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

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.

References

Revised on February 4, 2013 17:54:10 by Urs Schreiber (82.113.99.102)