model category, model $\infty$-category
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Related concepts
Given a topological group $G$, the Borel model structure is a model category structure on the category of topological G-spaces, hence of topological spaces equipped with continuous group actions.
Analogously, given a simplicial group $G_\bullet$, the Borel model structure is a model category structure on the category of simplicial group actions, hence of simplicial sets equipped with $G$-action
Both of these present the (∞,1)-category of ∞-actions of the ∞-group (see there) presented by $G$.
In the context of equivariant homotopy theory this is also called the “coarse model structure” (e.g. Guillou 2006, section 5), since in general it has more weak equivalences than the fine model structure on topological G-spaces that enters Elmendorf's theorem.
Throughout, write Top for the category of compactly generated weak Hausdorff spaces.
For $G \,\in\, Grp(TopSp)$ a topological group, write
for the Top-enriched category with a single object and $G$ as its unique hom-object.
There is an evident isomorphism of enriched categories
between topological G-spaces and the Top-enriched functor category from $\mathbf{B}G$ (1) to Top (topological presheves).
For $G \in Grp(TopSp)$ a topological group there is a model category-structure
on the category of topological G-spaces whose weak equivalences and fibrations are those morphisms whose underlying continuous functions are so in the classical model structure on topological spaces.
For discrete groups this may be argued as in Guillou 2006, Thm. 5.1. For general topological groups this follows as a special case of the projective model structure on Top-enriched functor (see this Thm.), under the identification $G Act(TopSp) \,\simeq\, TopFun( \mathbf{B}G, TopSp )$ (2).
For $G_\bullet$ a simplicial group write
$\mathbf{B}G_\bullet$ for the one-object sSet-enriched category (here: a simplicial groupoid) whose hom-object is $G_\bullet$.
$G_\bullet Actions(sSet) \;\coloneqq\; sSetCat\big(\mathbf{B}G_\bullet, sSet\big)$ for the sSet-enriched functor category to SimplicialSets.
$G_\bullet Acts\big(sSet_{Qu}\big)_{proj} \coloneqq sSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj}$ for the projective model structure on functors (projective model structure on simplicial presheaves).
This is the $G_\bullet$ Borel model structure, naturally a simplicial model category (DDK 80, Prop. 2.4, Goerss & Jardine 09, Chapter V, Thm. 2.3).
More generally, if $\mathbf{C}$ is an sSet-enriched category which is also tensored over sSet, via an enriched functor
then for $\mathcal{G} \in Grp(sSet)$ the enriched hom-isomorphism of the tensoring
shows that sSet-enriched functors
may equivalently be thought of as simplicial group actions
of $\mathcal{G}$ acting on objects $\mathscr{V} \coloneqq F(\ast)$ via tensoring.
If now $\mathbf{C}$ is in fact a combinatorial simplicial model category, then the respective projective model structure on functors exists (that Prop.), which we may hence think of as the Borel model structure on $\mathcal{G}$-actions on objects of $\mathbf{C}$:
The model category $G Act\big(TopSp_{Qu}\big)_{proj}$ from Prop. is cofibrantly generated with generating cofibrations being (see this Def.) the product with $G$ (regarded with its free left multiplication action) of the generating cofibrations of $TopSp_{Qu}$.
This is a special case of this Thm. about topological functor categories.
Alternatively, the statement is a special case of that for the fine model structure on topological G-spaces for the case of trivial family of closed subgroups (see there).
Since the universal principal space $E G$ (the topological realization $E G \,=\, \big\vert W G\big\vert$ of the universal principal simplicial complex) is
contractible: $E G \simeq_{whe} \ast$;
equipped with a free action by $G$,
Prop. implies that the product with $E G$ in G Act(TopSp) (i.e. the product topological space with the induced diagonal action) serves as cofibrant replacement in $G Act(TopSp)$:
Hm, this is not a proper argument…
There is a Quillen adjunction
between the classical model structure on topological spaces and the projective Borel model structure from Prop. , whose
right Quillen functor, $triv$, assigns trivial actions;
left Quillen functor, $(-)/G$, assigns topological quotient spaces by the relation $G \times X \underoverset{pr_2}{\mathclap{(-)\cdot(-)}}{\rightrightarrows} X$.
By definition of the weak equivalences and fibrations in Prop. , it is immediate that $triv$ preserves these classes of morphisms.
(Borel construction of free action is weak hom. equivalent to plain quotient space)
Let $G$ be a compact Lie group and let $X$ be a G-CW complex whose $G$-action is free. Then the comparison morphism between the Borel construction and the plain quotient space of $X$ is a weak homotopy equivalence:
Since $E G$ is a G-CW complex, the product $X \times E G$ is cofibrant (by the analog of this Prop.) and $X$ is cofibrant by assumption and by Prop. .
Hence, by Prop. , the morphism in question is the image under a left Quillen functor, of a weak equivalence between cofibrant objects. Therefore the claim follows by Ken Brown's lemma (here).
The Borel construction exhibits the left derived functor of the quotient space-left Quillen functor in Prop. :
Since the left derived functor of a left Quillen functor is given by the application of the latter on any cofibrant replacement, the claim follows by Ex. .
or would follow, if that Example were argued properly
(cofibrations of simplicial actions)
The cofibrations $i \colon X \to Y$ in $sSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj}$ (Def. ) are precisely those morphisms such that
the underlying morphism of simplicial sets is a monomorphism;
the $G_\bullet$-action is a relatively free action, i.e. free on all simplices not in the image of $i$.
This is (DDK 80, Prop. 2.2. (ii), Guillou 2006, Prop. 5.3, Goerss & Jardine 09, V Lem. 2.4 & Cor. 2.10).
In particular this means that an object is cofibrant in $sSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj}$ if the $G_\bullet$-action on it is free.
Hence cofibrant replacement is obtained by forming the product with the model $W G_\bullet$ for the total space of the universal principal bundle over $G_\bullet$ (see at simplicial group for notation and more details).
It follows that for $X, A \in sSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj}$ the derived hom space
models the Borel $G$-equivariant cohomology of $X$ with coefficients in $A$.
In particular, if $A$ is fibrant (the underlying simplicial set is a Kan complex) then:
if the $G_\bullet$-action on $A$ is trivial, then
is equivalently maps of simplicial sets out of the Borel construction on $X$;
if $X = \ast$ is the point then
is the homotopy fixed points of $A$.
For $G$ a simplicial group, there is a pair of adjoint functors
which constitute a simplicial Quillen equivalence between the Borel model structure (Def. ) and the slice model structure of the classical model structure on simplicial sets, sliced over the simplicial classifying space $\overline{W}G$.
(this is essentially the statement of DDK 80, Prop. 2.3, Prop. 2.4)
Here:
the right adjoint is the Borel construction
understood as forming associated bundles to universal principal bundles;
the left adjoint forms homotopy fibers.
(This may also be understood as an instance of the “fundamental theorem of $\infty$-topos theory”, see there.)
Consider any morphism in $sSet_{/\overline{W}G}$:
Its image under the left adjoint functor is, by definition, the top left arrow in the following commuting diagram:
Here the right square and the total rectangle are Cartesian squares (pullback squares), by defnition of the functor. It follows by the pasting law that also the square on the left is cartesian. Specifically, since fibrations are preserved under pullback, as shown, the top left morphism in question is the pullback of $f$ along a fibration.
It follows that $(f) \underset{\overline{W}G}{\times} W G$ is:
a weak equivalence if $f$ is a weak equivalence, because the classical model structure on simplicial sets is a right proper model category (see here);
a monomorphism if $f$ is a monomorphism, since monomorphisms are preserved by pullback (see here).
Moreover, since $W G \to \overline{W}G$ is the universal principal bundle, it follows that $c_Y^\ast(W G) \to Y$ is a simplicial principal bundle, so that, in particular, the action of $G$ on $c_Y^\ast(W G)$ is free.
By Prop. this means that $(f) \underset{\overline{W}G}{\times} W G$ is a cofibration if $f$ is a cofibration.
In summary, the left adjoint functor in (4) preserves the classes of weak equivalences and of cofibrations, hence also that of acyclic cofibrations, and so it is a left Quillen functor.
Next…
In fact, these functors (4) are sSet-enriched functors which induce an equivalence of $(\infty,1)$-categories between the simplicial localizations $L_W sSetCat\big(\mathbf{B}G_\bullet, sSet\big)_{proj} \simeq L_W sSet_{/\overline{W}H}$ (DDK 80, Prop. 2.5).
This kind of relation is discussed in more detail at ∞-action.
(sSet-enrichement of the adjunction)
The statement that (4) is an sSet-enriched adjunction is not made explicit in DDK 80; there it only says that the functors form a plain adjunction (DDK 80, Prop. 2.3) and that they are each sSet-enriched functors (DDK 80, Prop. 2.4).
The remaining observation that we have a natural isomorphism of sSet-hom-objects
hence
follows from the plain adjunction and the natural isomorphism
which, in turn, follows, for instance, via the pasting law:
For $\mathcal{G} \,\in\, Groups(sSets)$ a simplicial group, write $\mathcal{G}Actions(sSets)$ for the category of $\mathcal{G}$-actions on simplicial sets.
(underlying simplicial sets and cofree simplicial action)
The forgetful functor $undrl$ from $\mathcal{G}Actions$ to underlying simplicial sets is a left Quillen functor from the Borel model structure (Def. ) to the classical model structure on simplicial sets.
Its right adjoint
sends $\mathcal{X} \in sSet$ to
the simplicial set
equipped with the $\mathcal{G}$-action
which in degree $n \in \mathbb{N}$ is the function
that sends
Here and in the following proof we make free use of the Yoneda lemma natural bijection
for any simplicial set $S$ and for $\Delta[n] \in \Delta \overset{y}{\hookrightarrow} sSet$ the simplicial n-simplex.
We already know from Def. that $underl$ preserves all weak equivalences and from Prop. that it preserves all cofibrations. Therefore it is a left Quillen functor as soon as it is a left adjoint at all.
The idea of the existence of the cofree right adjoint to $undrl$ is familiar from topological G-spaces (see the section on coinduced actions there), where it can be easily expressed point-wise in point-set topology. The formula (6) adapts this idea to simplicial sets. Its form makes manifest that this gives a simplicial homomorphism, and with this the adjointness follows the usual logic by focusing on the image of the non-degenerate top-degree cell in $\Delta[n]$:
To check that (6) really gives the right adjoint, it is sufficient to check the corresponding hom-isomorphism, hence to check for $\mathcal{P} \in \mathcal{G}Actions(sSet)$, and $\mathcal{X} \in sSet$, that we have a natural bijection of hom-sets of the form
So given
on the left, define
where $e_n \in \mathcal{G}_n$ denotes the neutral element in degree $n \in \mathbb{N}$ and where $\sigma_n \in (\Delta[n])_n$ denotes the unique non-degenerate element $n$-cell in the n-simplex.
It is clear that this is a natural transformation in $P$ and $X$. We need to show that ${\widetilde \phi}_{(-)} \colon undrl(P) \to X$ uniquely determines all of $\phi_{(-)}$.
To that end, observe for any $g_n \in \mathcal{G}_n$ the following sequence of identifications:
Here:
the first step is the unit law in the component group $\mathcal{G}_n$;
the second step uses the definition (6) of the cofree action;
the third step is the assumption that $\phi_{(-)}$ is a homomorphism of $\mathcal{G}$-actions (equivariance);
the fourth step is the definition (7).
These identifications show that $\phi_{(-)}$ is uniquely determined by ${\widetilde \phi_{(-)}}$, and vice versa.
($\mathbf{B}\mathbb{Z}$-2-action on inertia groupoid)
Let
$G \in Groups(Sets)$
be a discrete group,
$X \in G Actions(Sets)$
be a $G$-action,
$\mathcal{X} \;\coloneqq\; X \sslash G \;\coloneqq\; N( X \times G \rightrightarrows X ) \,=\, X \times G^{\times^\bullet} \in sSet$
the simplicial set which is the nerve of its action groupoid (a model for its homotopy quotient),
$\mathcal{G} \,\coloneqq\, \mathbf{B}\mathbb{Z} \,\coloneqq\, N(\mathbb{Z} \rightrightarrows \ast) \,\coloneqq\, \mathbb{Z}^{\times^\bullet} \,\in\, Groups(sSet)$
the simplicial group which is the nerve of the 2-group that is the delooping groupoid of the additive group of integers.
Then the functor groupoid
is known as the inertia groupoid of $X \!\sslash\! G$. Here
denotes, respectively, the set of conjugacy classes of elements of $G$, and the centralizer of $\{g\} \subset G$ – this data serves to express the equivalent skeleton of the inertia groupoid in the last line of (8).
Now, by Prop. the inertia groupoid (8) carries a canonical 2-action of the 2-group $\mathbf{B}\mathbb{Z}$:
By the formula (6), for $n \in \mathbb{Z}$ the 2-group element in degree 1
acts on the morphisms
of the inertia groupoid as follows (recall the nature of products of simplices):
The identity functor gives a Quillen adjunction between the Borel model structure and the final model structure on topological G-spaces? for equivariant homotopy theory (Guillou 2006, section 5).
The left adjoint is
from the Borel model structure to the genuine equivariant homotopy theory.
Because:
First of all, by (Guillou 2006, theorem 3.12, example 4.2) $sSet^{\mathbf{B}G_\bullet}$ does carry a fine model structure. By (Guillou 2006, last line of page 3) the fibrations and weak equivalences here are those maps which are ordinary fibrations and weak equivalences, respectively, on $H$-fixed point simplicial sets, for all subgroups $H$. This includes in particular the trivial subgroup and hence the identity functor
is right Quillen.
Since the universal simplicial principal complex-construction is functorial
the pair of adjoint functors (4) extends to presheaves:
For $\mathcal{C}$ a small sSet-category with
denoting its category of simplicial presheaves, and for
a group object internal to SimplicialPresheaves with
denoting its category of action objects internal to SimplicialPresheaves
we have an adjoint pair
The required hom-isomorphism is the composite of the following sequence of natural bijections:
Here:
the first step is the characterization of hom-sets of a slice category as a fiber of the hom-sets of the underlying category;
the second step is the description of the hom-set of a functor category as an end of object-wise hom-sets;
the third step uses that ends are limits and hence commute with the fiber product;
the fourth step recognizes again, now object-wise, the hom-set in a slice category;
the fifth step is objectwise the hom-isomorphism of (4);
the sixth step recognizes again the end as computing the hom-set in (a subcategory of) a functor category:
Given
$\mathcal{G} \,\in\, Grp(sSet)$ a simplicial group,
$\mathbf{C}$ a combinatorial simplicial model category with
classes of (acyclic) generating cofibrations denoted
$I_{\mathbf{C}},J_{\mathbf{C}} \,\subset\, Mor(\mathbf{C})$,
then its Borel model category (3)
has, by this Prop., (acyclic) generating cofibrations given by tensoring these with $\mathcal{G} = (\mathbf{B}\mathcal{G})(\ast, \ast)$ and understood as equipped with the $\mathcal{G}$-action induced by the regular action of $\mathcal{G}$ on itself:
where we noticed that these are just the images under the (acyclic) generating cofibrations of $\mathbf{C}$ under the left-induced action
But since
the underlying simplicial set of $\mathcal{G}$ is necessarily a cofibrant object of $sSet_{Qu}$
$\mathbf{C}$ is an $sSet$-enriched model category
the pushout-product axiom satisfied by the tensoring implies that the underlying morphisms of these (acyclic) generating cofibration in the Borel structure, i.e. forgetting their equivariance under the simplicial group action, are still (acyclic) cofibrations in $\mathbf{C}$ itself, hence that we also have a Quillen adjunction of the form
whose right adjoint may be thought of as the right induced action or equivalently as right Kan extension along the sSet-enriched functor $\ast \to \mathbf{B}\mathcal{G}$.
We discuss (Prop. below) induced monoidal model category-structure on the Borel model structure, in the generality of coefficients any (cofibrantly generated) simplicial model structure (as above), which in addition carries compatible monoidal simplicial model structure.
For $\mathcal{G} \in Grp(sSet)$ and $\mathbf{C}$ a combinatorial simplicial model category with compatible closed monoidal enriched category-structure, i.e. with an sSet-enriched functor $(-)\otimes(-)$, the objectwise tensor product
and the objectwise internal hom
make the Borel model structure (3)
a monoidal model category, at least in that the pushout-product axiom holds. A sufficient condition that also the unit axiom hold is that all objects of $\mathbf{C}$ are cofibrant.
It is sufficient to check the pushout-product axiom on (acyclic) generating cofibrations (by this remark). To that end, given
generating cofibrations$\;A \to B$ and $X \to Y$ in $I_{\mathcal{G}Act(\mathbf{C})}$, we need to check that their pushout product is still a cofibration, which means that for any
acyclic fibration$\;Z \to W$ in $WFib_{\mathcal{G}Act(\mathbf{C})} = undrl^{-1} WFib_{\mathbf{C}}$
we need to find a lift in any commuting square of the form
By Joyal-Tierney calculus, this is equivalent to finding a lift in the internal hom-adjunct diagram:
Now observe that the (acyclic) Borel-cofibration on the left is, by (9), in the image of the left adjoint functor $\mathcal{G}\cdot(-)$ whose right adjoint is the forgetful functor remembering just the underlying morphisms in $\mathbf{C}$. Therefore a lift in the above diagram is equivalently a lift of its adjunct, which is just the underlying diagram in $\mathbf{C}$:
But now using
that the underlying map of the previous acyclic fibration is still an acyclic fibration in $\mathbf{C}$, by definition of the projective/Borel model structure,
that the underlying maps of the previous cofibrations are still cofibrations in $\mathbf{C}$, by (11),
the pullback-power axiom satisfied in the monoidal model category $\mathbf{C}$
it follows that the left map in this diagram is a cofibration and the right map is still an acyclic fibration, whence a lift exists by the model category axioms on $\mathbf{C}$.
A directly analogous argument applies in the cases where $A \to B$ or $X \to Y$ are in addition weak equivalences. Hence the pushout-product axiom is verified.
Finally, to verify the unit axiom:
If $\mathbb{1} \,\in\, \mathbf{C}$ denotes the given tensor unit, then the tensor unit in $\mathcal{G}Act(\mathbf{C})$ is the constant functor $\mathbf{B}\mathcal{G} \to \ast \overset{\mathbb{1}}{\longrightarrow} \mathbf{C}$, corresponding to the trivial action on $\mathbb{1}$. Writing $p_{const \mathbb{1}} \colon Q const(\mathbb{1}) \to const \mathbb{1}$ for a cofibrant resolution, we need to see that given an object $X \in \mathcal{G}Act(\mathbf{C})$ the composite
is a weak equivalence.
Since weak equivalence are just the underlying weak equivalences, for this it is sufficient (with the assumption that all objects in $\mathbf{C}$ are cofibrant), that $(-) \otimes X$ is a left Quillen functor on $\mathbf{C}$, since as such it preserves all weak equivalences between cofibrant objects, by Ken Brown’s lemma.
But the underlying object of $X$ is still cofibrant in $\mathbf{C}$, by (11), therefore $(-) \otimes X$ is left Quillen by the pushout-product axiom satisfied by the monoidal model category $\mathbf{C}$.
The model structure, the characterization of its cofibrations, and its equivalence to the slice model structure of $sSet$ over $\bar W G$ is due to
This Quillen equivalence also mentioned as:
Discussion in relation to the “fine” model structure of equivariant homotopy theory which appears in Elmendorf's theorem is in
Textbook account of (just) the Borel model structure:
Discussion with the model of ∞-groups by simplicial groups replaced by groupal Segal spaces is in
Discussion of monoidal model category-enhancement on the Borel model structure:
Discussion of a the integral model structure for actions of all simplicial groups:
Steffen Sagave, Christian Schlichtkrull, Prop. 6.4 in: Diagram spaces and symmetric spectra, Advances in Mathematics, 231 (2012), 2116-2193 (arXiv:1103.2764, doi:10.1016/j.aim.2012.07.013)
Last revised on November 2, 2023 at 08:25:58. See the history of this page for a list of all contributions to it.