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commutative monoid in a symmetric monoidal model category

Contents

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

monoidal categories

With symmetry

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

Let CC be symmetric monoidal category and CC the category of commutative monoids in CC. When CC is further a model category, there are certain conditions under which there is an induced model structure on CMon(C)CMon(C), where the weak equivalences and fibrations are defined as in CC.

Distinction between E-infinity monoids

Recall that CMon(C)CMon(C) can be described as the category of algebras over an operad over the operad Comm. If the operad Comm were cofibrant, then for the existence of the induced model structure on CMon(C)CMon(C) it would be sufficient to require the monoid axiom, and to use the model structure on algebras over an operad as discussed there. However Comm is in general not cofibrant, and this is the distinction between Comm and the E-infinity operad: the latter is a cofibrant replacement of the former. This is why the model structure on commutative monoids may not always exist even when the model structure on E-infinity monoids in a symmetric monoidal model category does.

For example, take CC to be the category of chain complexes over a field of positive characteristic. Then the category of commutative monoids in CC is the category of commutative dg-algebras. This does not have an induced model structure, as explained in MO/23885/2503.

See Rectification for some results on when the model structures on E-infinity monoids? and commutative monoids are Quillen equivalent?, though.

Model structure

Let CC be a symmetric monoidal model category and let CMon(C)CMon(C) denote the category of commutative monoids in the underlying category of CC. Define a weak equivalence (resp. fibration) of commutative monoids to be a weak equivalence (resp. fibration) of the underlying objects of CC. Below we will give sufficient conditions for this to define a model structure on CMon(C)CMon(C).

Theorem

Suppose that

Then the category CMon(C)CMon(C) is a combinatorial model category with weak equivalences and fibrations as defined above.

See (Lurie, Proposition 4.5.4.6).

Next we state a more general version of this result, for which we will require some set-theoretic assumptions. Suppose that CC is cofibrantly generated by a set of cofibrations II and a set of trivial cofibrations JJ. Let ICI \otimes C-cell denote the closure of IC={iid X:iI,XOb(C)}I \otimes C = \{ i \otimes \id_X : i \in I, X \in Ob(C) \} under cobase change and transfinite composition, and similarly for JCJ \otimes C-cell.

Theorem

Suppose that

Then the category CMon(C)CMon(C) is a cofibrantly generated model category with weak equivalences and fibrations as defined above.

If CC is further simplicial (resp. combinatorial, tractable), then so is CMon(C)CMon(C).

See (White 14, Theorem 3.2).

Rectification

Rectification of E E_\infty-monoids is the question of whether the weak equivalence between the operads Comm and the E-infinity operad induces a Quillen equivalence on the model categories of algebras. Since the model category of algebras over an operad over the E-infinity operad is a presentation of the (infinity,1)-category of commutative monoids in a symmetric monoidal (infinity,1)-category, rectification for CC is equivalent to saying that the (infinity,1)-category presented by the model structure on commutative monoids is equivalent to the (infinity,1)-category of commutative monoids in a symmetric monoidal (infinity,1)-category in the symmetric monoidal (infinity,1)-category presented by CC.

The following cases are particularly interesting.

See (Lurie, Theorem 4.5.4.7) for sufficient conditions for rectification to hold. See also (White 14, Paragraph 4.2) for more discussion.

A general rectification criterion for symmetric monoidal model categories is formulated in PS 14, Proposition 10.1.2 and Theorem 9.3.6. It says that given a tractable? symmetric monoidal model category that satisfies a certain compact generatedness assumption with a morphism of admissible operads? ABA\to B (e.g., A=E A=E_\infty, B=CommB=Comm), the Quillen adjunction between AA-monoids and BB-monoids induced by the morphism of operads ABA\to B is a Quillen equivalence if and only if for any cofibration ss and any n0n\ge0 the morphism (A nB n) Σ ns n(A_n\to B_n)\wedge_{\Sigma_n}s^{\wedge n} is a weak equivalence, where \wedge denotes the pushout product with respect to the monoidal structure.

References

Last revised on March 11, 2015 at 15:24:04. See the history of this page for a list of all contributions to it.