nLab
model structure on monoids in a monoidal model category

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

for \infty-groupoids

for ∞-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general

specific

for stable/spectrum objects

for (,1)(\infty,1)-categories

for stable (,1)(\infty,1)-categories

for (,1)(\infty,1)-operads

for (n,r)(n,r)-categories

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

Monoidal categories

Higher algebra

Contents

Idea

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

Definition

For CC a monoidal category with all colimits the category of monoids comes equipped (as discussed there) with a free functor/forgetful functor adjunction

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

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

Properties

Existence

Theorem

If CC is monoidal model category that

then the transferred model structure along the free functor/forgetful functor adjunction (FU):Mon(C)UFC(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 CC

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)Mon(C). By the discussion of free monoids at category of monoids we have that then pushouts of the form

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

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

Proposition

Let the monoidal model category CC be

If f:ABf : A\to B is an acyclic cofibration in the model structure on CC, then the pushout XPX \to P as above is a weak equivalence in Mon(C)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 n1P nP_{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 A_\infty-Algebras

Under mild conditions on CC the model structure on monoids in CC is Quillen equivalent to that of A-infinity algebras in CC. 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)