nLab model structure on pointed objects

Idea

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 $\ast \in \mathcal{C}$.

Definition

As a special case of the general discussion at coslice model structures, we have:

Proposition

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)

Examples

Example

(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 $Ho(Top^{\ast/})$.

Properties

Remark

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.

Example

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

$$[-,-]_\ast \;\colon\; Ho(\mathcal{C}^{\ast/})^\op \times Ho(\mathcal{C}^{\ast/}) \longrightarrow Set^{\ast/} \,.$$

Remark

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

$$\big( X \wedge (-) \;\dashv\; (-)^X \big) \;\;\colon\;\; \mathcal{C}^{\ast/} \leftrightarrow \mathcal{C}^{\ast/} \,.$$

Proposition

(induced Quillen adjunction on model categories of pointed objects)
Given a Quillen adjunction between model categories

$$\mathcal{D} \underoverset {\underset{R}{\longrightarrow}} {\overset{L}{\longleftarrow}} {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;} \mathcal{C} \,,$$

there is induced a Quillen adjunction between the corresponding model categories of pointed objects

$$\mathcal{D}^{\ast\!/} \underoverset {\underset{R^{\ast\!/}}{\longrightarrow}} {\overset{L^{\ast\!/}}{\longleftarrow}} {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;} \mathcal{C}^{\ast\!/} \,,$$

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):

  $$L^{\ast\!/} \;\colon\; \mathcal{C}^{\ast\!/} \xrightarrow{ \;\; L \;\; } \mathcal{D}^{L(\ast)\!/} \;\simeq\; \mathcal{D}^{L\circ R(\ast)\!/} \xrightarrow{ \;\; (-) \sqcup \epsilon_\ast \;\; } \mathcal{D}^{\ast\!/} \,.$$

Proof

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:

$$\mathcal{C}^{op} \underoverset {\underset{L^{op}}{\longrightarrow}} {\overset{R^{op}}{\longleftarrow}} {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;} \mathcal{D}^{op} \,.$$

Now the passage to pointed objects corresponds to slicing (instead of co-slicing), since

(1)$$\mathcal{C}^{\ast\!/} \;\; \simeq \big( \mathcal{C}^{op}_{/\ast} \big)^{op} \,, \;\;\;\;\;\;\; \text{similarly} \;\;\; \mathcal{D}^{\ast\!/} \;\; \simeq \big( \mathcal{D}^{op}_{/R(\ast)} \big)^{op} \,,$$

whence item (1) in that Prop. says that there is a Quillen adjunction of the form

$$\mathcal{C}^{op}_{/R(\ast)} \underoverset {\underset{L^{op}_{/\ast}}{\longrightarrow}} {\overset{R^{op}_{/\ast}}{\longleftarrow}} {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;} \mathcal{D}^{op}_{/\ast} \,,$$

hence with opposite Quillen adjunction of the required form

$$\mathcal{D}^{\ast\!/} \simeq \big(\mathcal{D}^{op}_{/R(\ast)}\big)^{op} \underoverset {\underset{ R^{\ast\!/} \,\coloneqq\, \big( R^{op}_{/\ast} \big)^{op} }{\longrightarrow}} {\overset{ L^{\ast\!/} \,\coloneqq\, \big( L^{op}_{/\ast} \big)^{op} }{\longleftarrow}} {\;\;\;\;\;\bot_{\mathrlap{{}_{Qu}}}\;\;\;\;\;} \big(\mathcal{C}^{op}_{/\ast}\big)^{op} \simeq \mathcal{C}^{\ast\!/} \,,$$

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.

References

Textbook accounts:

• Mark Hovey, Prop. 1.1.8 in: Model Categories, Mathematical Surveys and Monographs, Volume 63, AMS (1999) (ISBN:978-0-8218-4361-1, doi:10.1090/surv/063, pdf, Google books)