nLab model structure on monoids in a monoidal model category

Contents

Context

Model category theory

model category, model \infty -category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

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

Model structures

for \infty-groupoids

for ∞-groupoids

for equivariant \infty-groupoids

for rational \infty-groupoids

for rational equivariant \infty-groupoids

for nn-groupoids

for \infty-groups

for \infty-algebras

general \infty-algebras

specific \infty-algebras

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

monoidal categories

With braiding

With duals for objects

With duals for morphisms

With traces

Closed structure

Special sorts of products

Semisimplicity

Morphisms

Internal monoids

Examples

Theorems

In higher category theory

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, its 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.

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.

Properties

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

Last revised on May 29, 2022 at 19:22:32. See the history of this page for a list of all contributions to it.