model category, model -category
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Universal constructions
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Producing new model structures
Presentation of -categories
Model structures
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on chain complexes/model structure on cosimplicial abelian groups
related by the Dold-Kan correspondence
for equivariant -groupoids
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for rational equivariant -groupoids
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For a model category, there is canonically induced a model category structure on the corresponding category of pointed objects , namely the coslice model structure under “the point”, i.e. under the terminal object .
As a special case of the general discussion at coslice model structures, we have:
Let be a model category with terminal object dented . Then there is a model category structure on its category of pointed objects , hence on the category under , whose classes of morphisms (weak equivalences, fibrations, cofibrations) are those created by the forgetful functor .
(e.g. Hovey 99, Prop. 1.1.8)
(classical model structure on pointed topological spaces)
For , the classical model structure on topological spaces, then the model structure on pointed topological spaces induced via prop. we call the classical model structure on pointed topological spaces . Its homotopy category of a model category is the classical pointed homotopy theory .
The fibrant objects in the pointed model structure , prop. , are those that are fibrant as objects of .
But the cofibrant objects in are those for which the basepoint inclusion is a cofibration in .
For (Ex. ), the cofibrant pointed topological spaces are typically referred to as spaces with non-degenerate basepoints. Notice that the point itself is cofibrant in , so that cofibrant pointed topological spaces are in particular cofibrant topological spaces.
For any model category, with its pointed model structure according to Prop. , then corresponding homotopy category is, (by this Remark), canonically enriched in pointed sets, in that its hom-functor is of the form
If is a monoidal model category with cofibrant tensor unit, then the pointed model structure on (Prop. ) is also a monoidal model category, and the smash productmapping space adjunction of prop. is a Quillen adjunction
(induced Quillen adjunction on model categories of pointed objects)
Given a Quillen adjunction between model categories
there is induced a Quillen adjunction between the corresponding model categories of pointed objects
where
the right adjoint acts directly as on the triangular commuting diagrams in that define the morphisms in ;
the left adjoint is the composite of the corresponding direct application of followed by pushout along the adjunction counit (using that since right adjoints preserve limits and hence terminal objects):
It is fairly straightforward to check this directly (e.g. Hovey 1999, Prop. 1.3.5), but it is also a special case of this general prop. about slice model categories — to make this explicit, notice that passing to opposite categories with their opposite model structures turns the original Quillen adjunction into the opposite Quillen adjunction:
Now the passage to pointed objects corresponds to slicing (instead of co-slicing), since
whence item (1) in that Prop. says that there is a Quillen adjunction of the form
hence with opposite Quillen adjunction of the required form
with acting directly as on underlying diagrams, and with acting as the composite of following by pullback – in – along the adjunction unit of . Since the component morphism of the unit of the opposite adjunction is that of the adjunction unit of , and since pullback in an opposite category is pushout in the original category, this implies the claim.
Textbook accounts:
Last revised on July 20, 2021 at 15:11:10. See the history of this page for a list of all contributions to it.