model structure on monoids in a monoidal model category in nLab
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
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on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
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
With braiding
With duals for objects
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category with duals (list of them)
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dualizable object (what they have)
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ribbon category, a.k.a. tortile category
With duals for morphisms
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monoidal dagger-category?
With traces
Closed structure
Special sorts of products
Semisimplicity
Morphisms
Internal monoids
Examples
Theorems
In higher category theory
Higher algebra
Algebraic theories
Algebras and modules
Higher algebras
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symmetric monoidal (∞,1)-category of spectra
Model category presentations
Geometry on formal duals of algebras
Theorems
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
(F⊣U):Mon(C)→U←FC. (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.
This is part of (SchwedeShipley, theorem 4.1).
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:A→Bf : A \to B in CC
This is SchwedeShipley, lemma 6.2.
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.
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model structure on monoids in a monoidal model category
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model structure on commutative monoids in a symmetric monoidal model category
References
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Stefan Schwede, Brooke Shipley, Algebras and modules in monoidal model categories Proc. London Math. Soc. (2000) 80(2): 491-511 (pdf)
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David White, Model Structures on Commutative Monoids in General Model Categories (arXiv:1403.6759)
Last revised on May 29, 2022 at 19:22:32. See the history of this page for a list of all contributions to it.