model structure on an over category


Model category theory

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For CC a model category and XCX \in C an object, the over category C/XC/X as well as the undercategory X/CX/C inherit themselves structures of model categories whose fibrations, cofibrations and weak equivalences are precisely the morphism that become fibrations, cofibrations and weak equivalences in CC under the forgetful functor C/XCC/X \to C or X/CCX/C \to C.


Cofibrant generation, properness, combinatoriality


If 𝒞\mathcal{C} is

then so are 𝒞 /X\mathcal{C}_{/X} and 𝒞 X/\mathcal{C}^{X/}.

More in detail, if I,JMor(𝒞)I,J \subset Mor(\mathcal{C}) are the classes of generating cofibrations and of generating acylic cofibrations of 𝒞\mathcal{C}, respectively, then

  • the generating (acyclic) cofibrations of 𝒞 X/\mathcal{C}^{X/} are the image under X()X \sqcup(-) of those of 𝒞\mathcal{C}.

This is spelled out in (Hirschhorn 05).


If 𝒞\mathcal{C} is a combinatorial model category, then so is 𝒞 /X\mathcal{C}_{/X}.


By basic properties of locally presentable categories they are stable under slicing. Hence with 𝒞\mathcal{C} locally presentable also 𝒞 /X\mathcal{C}_{/X} is, and by prop. 2 with 𝒞\mathcal{C} cofibrantly generated also 𝒞 /X\mathcal{C}_{/X} is.


If 𝒞\mathcal{C} is an cartesian enriched model category?, then so is 𝒞 /X\mathcal{C}_{/X}.


By basic properties of cartesian enriched categories? they are stable under slicing, where tensoring is computed in 𝒞\mathcal{C}. Hence with 𝒞\mathcal{C} enriched also 𝒞 /X\mathcal{C}_{/X} is. The pushout product axiom now follows from the fact that in overcategories pushouts can be computed in the underlying category 𝒞\mathcal{C}. The unit axiom? follows from the unit axiom of 𝒞\mathcal{C} using the fact that tensorings are computed in 𝒞\mathcal{C}.

Derived hom-spaces


If CC is a simplicial model category and XCX \in C is fibrant, then the overcategory C/XC/X with the above slice model structure is a presentation of the over-(∞,1)-category C /XC^\circ / X: we have an equivalence of (∞,1)-categories

(C/X) C /X. (C/X)^\circ \simeq C^\circ / X \,.

It is clear that we have an essentially surjective (∞,1)-functor C /X(C/X) C^\circ/X \to (C/X)^\circ. What has to be shown is that this is a full and faithful (∞,1)-functor in that it is an equivalence on all hom-∞-groupoids C /X(a,b)(C/X) (a,b)C^\circ/X(a,b) \simeq (C/X)^\circ(a,b).

To see this, notice that the hom-space in an over-(∞,1)-category C /XC^\circ/X between objects a:AXa : A \to X and b:BXb : B \to X is given (as discussed there) by the (∞,1)-pullback

C /X(AaX,BbX) C (A,B) b * * a C (A,X) \array{ C^\circ/X(A \stackrel{a}{\to} X, B \stackrel{b}{\to} X) &\to& C^\circ(A,B) \\ \downarrow && \downarrow^{\mathrlap{b_*}} \\ {*} &\stackrel{a}{\to}& C^\circ(A,X) }

in ∞Grpd.

Let ACA \in C be a cofibrant representative and b:BXb : B \to X be a fibration representative in CC (which always exists) of the objects of these names in C C^\circ, respectively. In terms of these we have a cofibration

A a X \array{ \emptyset &&\hookrightarrow&& A \\ & \searrow && \swarrow_{\mathrlap{a}} \\ && X }

in C/XC/X, exhibiting aa as a cofibrant object of C/XC/X; and a fibration

B b X b Id X \array{ B &&\stackrel{b}{\to}&& X \\ & {}_{\mathllap{b}}\searrow && \swarrow_{\mathrlap{Id}} \\ && X }

in C/XC/X, exhibiting bb as a fibrant object in C/XC/X.

Moreover, the diagram in sSet given by

C/X(a,b) C(A,B) b * * a C(A,X) \array{ C/X(a, b) &\to& C(A,B) \\ \downarrow && \downarrow^{\mathrlap{b_*}} \\ {*} &\stackrel{a}{\to}& C(A,X) }


  1. a pullback diagram in sSet (by the definition of morphism in an ordinary overcategory);

  2. a homotopy pullback in the model structure on simplicial sets, because by the axioms on the sSet Quillen{}_{Quillen} enriched model category CC and the above (co)fibrancy assumptions, all objects are Kan complexes and the right vertical morphism is a Kan fibration.

  3. has in the top left the correct derived hom-space in C/XC/X (since aa is cofibrant and bb fibrant).

This means that this correct hom-space C/X(a,b)(C/X) (a,b)sSetC/X(a,b) \simeq (C/X)^\circ(a,b) \in sSet is indeed a model for C /X(a,b)GrpdC^\circ/X(a,b) \in \infty Grpd.

Quillen adjunctions between slice categories


Given an adjunction LRL\dashv R with L:ABL\colon A\to B and R:BAR\colon B\to A, the following compositions define two Quillen ajdunctions between associated slice categories. If XAX\in A, then

L:A/XB/L:RXL:A/X\leftrightarrows B/L:RX

is an adjunction, where is the composition R:B/LXA/RLXA/XR\colon B/LX\to A/RLX\to A/X, the second arrow is the base change functor along the unit XRLXX\to RLX. If YBY\in B, then

L:A/RYB/Y:RL:A/RY\leftrightarrows B/Y:R

is an adjunction, where L:A/RYB/LRYB/YL\colon A/RY\to B/LRY\to B/Y. The first adjunction is a Quillen equivalence if XX is cofibrant and LXLX is fibrant. The second adjunction is a Quillen equivalence if YY is fibrant.


These adjunctions are Quillen adjunctions because their left (respectively right) adjoints are left (respectively right) Quillen functors: in the model structures on slice categories (co)fibrations and weak equivalences are created by the forgetful functor to~AA or~BB, see Hirschhorn’s note (Hirschhorn 05). An object in A/XA/X given by an arrow ZXZ\to X is cofibrant if and only if ZZ is cofibrant and fibrant if and only if ZXZ\to X is a fibration. Quillen’s criterion for Quillen equivalences now yields the statements about equivalences.



Revised on July 7, 2017 11:09:07 by Dmitri Pavlov (