model category, model $\infty$-category
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For $\mathcal{C}$ a model category, there is canonically induced a model category structure on the corresponding category of pointed objects $\mathcal{C}^{\ast/}$, namely the coslice model structure under “the point”, i.e. under the terminal object $\ast \in \mathcal{C}$.
As a special case of the general discussion at coslice model structures, we have:
Let $\mathcal{C}$ be a model category with terminal object dented $\ast \in \mathcal{C}$. Then there is a model category structure on its category of pointed objects $\mathcal{C}^{\ast/}$, hence on the category under $\ast$, whose classes of morphisms (weak equivalences, fibrations, cofibrations) are those created by the forgetful functor $\mathcal{C}^{\ast/} \to \mathcal{C}$.
(e.g. Hovey 99, Prop. 1.1.8)
(classical model structure on pointed topological spaces)
For $\mathcal{C} = Top_{Qu}$, 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 $Top_{Qu}^{\ast/}$. Its homotopy category of a model category is the classical pointed homotopy theory $Ho(Top^{\ast/})$.
The fibrant objects in the pointed model structure $\mathcal{C}^{\ast/}$, prop. , are those that are fibrant as objects of $\mathcal{C}$.
But the cofibrant objects in $\mathcal{C}^{\ast/}$ are those for which the basepoint inclusion is a cofibration in $X$.
For $\mathcal{C}^{\ast/} = Top^{\ast/}_{Qu}$ (Ex. ), the cofibrant pointed topological spaces are typically referred to as spaces with non-degenerate basepoints. Notice that the point itself is cofibrant in $Top_{Qu}$, so that cofibrant pointed topological spaces are in particular cofibrant topological spaces.
For $\mathcal{C}$ any model category, with $\mathcal{C}^{\ast/}$ 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 $\mathcal{C}$ is a monoidal model category with cofibrant tensor unit, then the pointed model structure on $\mathcal{C}^{\ast/}$ (Prop. ) is also a monoidal model category, and the smash product$\dashv$mapping 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 $R$ on the triangular commuting diagrams in $\mathcal{C}$ that define the morphisms in $\mathcal{C}^{\ast\!/}$;
the left adjoint is the composite of the corresponding direct application of $L$ followed by pushout along the adjunction counit $L(\ast) \simeq L \circ R(\ast) \xrightarrow{ \;\epsilon_\ast \; } \ast$ (using that $R(\ast) \simeq \ast$ 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 $R^{op}$ acting directly as $R$ on underlying diagrams, and with $L^{op}$ acting as the composite of $L$ following by pullback – in $\mathcal{C}^{op}$ – along the adjunction unit of $(R^{op} \dashv L^{op})$. Since the component morphism of the unit of the opposite adjunction $(R^{op} \dashv L^{op})$ is that of the adjunction unit of $(L \dashv R)$, 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.