model category, model -category
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Producing new model structures
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Model structures
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related by the Dold-Kan correspondence
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for rational equivariant -groupoids
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A model category structure for excisive functors on functors from (finite) pointed simplicial sets to pointed simplicial sets (Lydakis 98, theorem 9.2, Biedermann-Chorny-Röndings 06, section 9), hence (see here) a model structure for spectra (Lydakis 98, theorem 11.3).
Special case of a model structure for n-excisive functors.
Write
sSet for the category of simplicial sets;
for the category of pointed simplicial sets;
for the full subcategory of pointed simplicial finite sets.
Write
for the free-forgetful adjunction, where the left adjoint functor freely adjoins a base point.
Write
for the smash product of pointed simplicial sets, similarly for its restriction to :
This gives and the structure of a closed monoidal category and we write
for the corresponding internal hom, the pointed function complex functor.
For , the internal hom is the simplicial set
regarded as pointed by the zero morphism (the one that factors through the base point), and the composition morphism
is given by
We regard all the categories in def. canonically as simplicially enriched categories, and in fact regard and as -enriched categories.
The category that is discussed below to support a model structure for excisive functors is the -enriched functor category
(Lydakis 98, example 3.8, def. 4.4)
In order to compare this to model structures for sequential spectra we consider also the following variant.
Write for the standard minimal pointed simplicial 1-sphere.
Write
for the non-full -enriched subcategory of pointed simplicial finite sets, def. whose
objects are the smash product powers (the standard minimal simplicial n-spheres);
hom-objects are
There is an -enriched functor
(from the category of -enriched copresheaves on the categories of standard simplicial spheres of def. to the category of sequential prespectra in sSet) given on objects by sending to the sequential prespectrum with components
and with structure maps
given by
This is an enriched equivalence of categories.
Consider the -enriched functor category from above.
With we take looping and delooping to mean concretely the operation on smash product and pointed exponential with this particular :
These operations extend objectwise to , where we denote them by the same symbols.
Write
for the functor given on by
Write
for the natural transformation whose component is the -adjunct of the canonical morphism induced from
Write
for the functor given by by the sequential colimit
Write for any Kan fibrant replacement functor.
Say that the stabilization (spectrification) of is
where is the left Kan extension of along the inclusion .
Say that a morphism in is
a stable weak equivalence if its stabilization, def. , takes value on each in weak homotopy equivalences in ;
a stable cofibration if it has the left lifting property against those morphisms whose value on every is a Kan fibration.
(Lydakis 98, def. 9.1, def. 7.1)
The classes of morphisms of def. , define a model category structure
There is a Quillen equivalence between the Bousfield-Friedlander model structure on sequential spectra and the Lydakis model structure from prop. .
There is an -enriched functor
(from the category of -enriched copresheaves on the categories of standard simplicial spheres of def. to the category of sequential prespectra in sSet) given on objects by sending to the sequential prespectrum with components
and with structure maps
given by
This is an enriched equivalence of categories.
The adjunction
(given by restriction along the defining inclusion of def. and by left Kan extension along , and combined with the equivalence of prop. ) is a Quillen adjunction and in fact a Quillen equivalence between the Bousfield-Friedlander model structure on sequential prespectra and Lydakis’ model structure for excisive functors, prop. .
The excisive functors naturally carry a smash product (Lydakis 98, def. 5.1) making the model structure for 1-excisive functors a symmetric monoidal model category (Lydakis 98, section 12). Via the translation to sequential spectra of prop. this is a model for the smash product of spectra (Lydakis 98, theorem 12.5); hence it is a symmetric smash product on spectra.
A monoid with respect to this smash product (hence a ring spectrum) is equivalently a functor with smash products (“FSP”) as earlier considered in (Bökstedt 86).
Since (def. ) is a symmetric monoidal category, canonically becomes symmetric monoidal itself via the induced Day convolution product. We write
for this symmetric monoidal category.
The smash product on considered (Lydakis 98, def. 5.1) coincides with the Day convolution product of def. .
The Day convolution product is characterized (see this proposition) by making a natural isomorphism of the form
where the external smash product on the right is defined by . Now, (Lydakis 98, def. 5.1) sets
where, by (Lydakis 98, prop. 3.23), is the left adjoint to . Hence the adjunction isomorphism gives the above characterization.
Under the Quillen equivalence of prop. the symmetric monoidal Day convolution product on excisive simplicial functors (prop. ) is identified with the proper smash product of spectra realized on sequential spectra by the standard formula.
This implies that any incarnation of the sphere spectrum in , possibly suitably replaced acts as the tensor unit up to stable weak equivalence. The following says that the canonical incarnation of the sphere spectrum actually is the genuine (1-categorical) tensor unit:
Write
for the canonical inclusion .
(the standard incarnation of the sphere spectrum in the model structure for excisive functors).
The object of def. is (up to isomorphism) the tensor unit in .
This is (Lydakis 98, theorem 5.9), but it is immediate with prop. , using that the tensor unit for Day convolution is the functor represented by the tensor unit in the underlying site (this proposition).
Equipped with the Day convolution tensor product (prop. ) the Lydakis model category of prop. becomes a monoidal model category
The pushout product axiom is (Lydakis 98, theorem 12.3). Moreover (Lydakis 98, theorem 12.4), shows that tensoring with cofibrant objects preserves all stable weak equivalences, hence in particular preserves cofibrant resolution of the tensor unit.
This means that (commutative) monoids in the monoidal Lydakis model structure are good models for ring spectra (E-infinity rings/A-infinity rings).
Monoids (commutative monoids) in the Lydakis monoidal model category of prop. are equivalently (symmetric) lax monoidal functors of the form
also known as “functors with smash product” (FSPs).
Since the tensor product is Day convolution of the smash product on , def. , this is a special casse of a general property of Day convolution, see this proposition.
with symmetric monoidal smash product of spectra
excisive functor, model structure on excisive functors
model structure for n-excisive functors
Model structure for excisive functors on simplicial sets (hence also a model structure for spectra) is discussed in:
A similar model structure on functors on topological spaces was given in
and also excisive functors modeled on topological spaces are the -spectra in
Discussion of the restriction from excisive functors to symmetric spectra includes
The functors with smash products (“FSP”s) appearing in (Lydakis 98, remark 5.12) had earlier been considered in
Further generalization of the model structure for excisive functor, in particular to enriched functors and to a model structure for n-excisive functors for is discussed in
Georg Biedermann, Boris Chorny, Oliver Röndigs, Calculus of functors and model categories, Advances in Mathematics 214 (2007) 92-115 [arXiv:math/0601221, doi:10.1016/j.aim.2006.10.009]
Georg Biedermann, Oliver Röndigs, Calculus of functors and model categories II, Algebr. Geom. Topol. 14 (2014) 2853-2913 [arXiv:1305.2834v2, doi:10.2140/agt.2014.14.2853]
Last revised on April 17, 2023 at 09:09:21. See the history of this page for a list of all contributions to it.