nLab model structure on pointed objects



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For 𝒞\mathcal{C} a model category, there is canonically induced a model category structure on the corresponding category of pointed objects 𝒞 */\mathcal{C}^{\ast/}, namely the coslice model structure under “the point”, i.e. under the terminal object *𝒞\ast \in \mathcal{C}.


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


Let 𝒞\mathcal{C} be a model category with terminal object dented *𝒞\ast \in \mathcal{C}. Then there is a model category structure on its category of pointed objects 𝒞 */\mathcal{C}^{\ast/}, hence on the category under *\ast, whose classes of morphisms (weak equivalences, fibrations, cofibrations) are those created by the forgetful functor 𝒞 */𝒞\mathcal{C}^{\ast/} \to \mathcal{C}.

(e.g. Hovey 99, Prop. 1.1.8)



(classical model structure on pointed topological spaces)

For 𝒞=Top Qu\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 */Top_{Qu}^{\ast/}. Its homotopy category of a model category is the classical pointed homotopy theory Ho(Top */)Ho(Top^{\ast/}).



The fibrant objects in the pointed model structure 𝒞 */\mathcal{C}^{\ast/}, prop. , are those that are fibrant as objects of 𝒞\mathcal{C}.

But the cofibrant objects in 𝒞 */\mathcal{C}^{\ast/} are those for which the basepoint inclusion is a cofibration in XX.

For 𝒞 */=Top Qu */\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 QuTop_{Qu}, so that cofibrant pointed topological spaces are in particular cofibrant topological spaces.


For 𝒞\mathcal{C} any model category, with 𝒞 */\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

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

If 𝒞\mathcal{C} is a monoidal model category with cofibrant tensor unit, then the pointed model structure on 𝒞 */\mathcal{C}^{\ast/} (Prop. ) is also a monoidal model category, and the smash product\dashvmapping space adjunction of prop. is a Quillen adjunction

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


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

𝒟 QuRL𝒞, \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

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


  • the right adjoint acts directly as RR on the triangular commuting diagrams in 𝒞\mathcal{C} that define the morphisms in 𝒞 */\mathcal{C}^{\ast\!/};

  • the left adjoint is the composite of the corresponding direct application of LL followed by pushout along the adjunction counit L(*)LR(*)ϵ **L(\ast) \simeq L \circ R(\ast) \xrightarrow{ \;\epsilon_\ast \; } \ast (using that R(*)*R(\ast) \simeq \ast since right adjoints preserve limits and hence terminal objects):

    L */:𝒞 */L𝒟 L(*)/𝒟 LR(*)/()ϵ *𝒟 */. 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\!/} \,.


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:

𝒞 op QuL opR op𝒟 op. \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)𝒞 */(𝒞 /* op) op,similarly𝒟 */(𝒟 /R(*) op) op, \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

𝒞 /R(*) op QuL /* opR /* op𝒟 /* op, \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

𝒟 */(𝒟 /R(*) op) op QuR */(R /* op) opL */(L /* op) op(𝒞 /* op) op𝒞 */, \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 opR^{op} acting directly as RR on underlying diagrams, and with L opL^{op} acting as the composite of LL following by pullback – in 𝒞 op\mathcal{C}^{op} – along the adjunction unit of (R opL op)(R^{op} \dashv L^{op}). Since the component morphism of the unit of the opposite adjunction (R opL op)(R^{op} \dashv L^{op}) is that of the adjunction unit of (LR)(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.