nLab
model structure on monoids in a monoidal model category

Context

Model category theory

Monoidal categories

Higher algebra

Idea
Definition
Properties
Related concepts
References

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 ⊣ U) : Mon (C) \overset{\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

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

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

\begin{matrix} F (A) & \overset{F (f)}{\to} & F (B) \\ ↓ & ↓ \\ X & \to & P \end{matrix}

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.

$A_{∞}$ -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 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)

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

nLab model structure on monoids in a monoidal model category

Context

Model category theory

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (∞,1)-categories

Model structures

for ∞-groupoids

for n-groupoids

for ∞-groups

for ∞-algebras

general

specific

for stable/spectrum objects

for (∞,1)-categories

for stable (∞,1)-categories

for (∞,1)-operads

for (n,r)-categories

for (∞,1)-sheaves / ∞-stacks

Monoidal categories

With symmetry

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Examples

Theorems

In higher category theory

Higher algebra

Algebraic theories

Algebras and modules

Higher algebras

Model category presentations

Geometry on formal duals of algebras

Theorems

Contents

Idea

Definition

Properties

Existence

Theorem

Theorem

Proof

Homotopy pushouts

Proposition

Proof

A ∞-Algebras

Related concepts

References

nLab
model structure on monoids in a monoidal model category

Presentation of $(∞, 1)$ -categories

for $∞$ -groupoids

for $n$ -groupoids

for $∞$ -groups

for $∞$ -algebras

for $(∞, 1)$ -categories

for stable $(∞, 1)$ -categories

for $(∞, 1)$ -operads

for $(n, r)$ -categories

for $(∞, 1)$ -sheaves / $∞$ -stacks

$A_{∞}$ -Algebras