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generalized Reedy model structure

Context

Model category theory

model category

Definitions

Morphisms

Universal constructions

Refinements

Producing new model structures

Presentation of (,1)(\infty,1)-categories

Model structures

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Contents

Idea

Generalized Reedy model structures are a class of model categories that generalize the Reedy model structures when the underlying site is generalized from a Reedy category to a generalized Reedy category.

So these model structures serve to present (∞,1)-categories of (∞,1)-functors on generalized Reedy categories.

As for the Reedy model structures, the generalized Reedy model structure typically models geometric shapes for higher structures. The crucial generalization is that the basic shapes here may have non-trivial automorphisms.

Definition

Let SS be a (Berger-Moerdijk) generalized Reedy category. Let 𝒞\mathcal{C} be a category with small limits and colimits.

The model structure

Definition

For every object sSs \in S, every functor X:S𝒞X : S \to \mathcal{C} and every natural transformation ϕ:XY\phi : X \to Y

  • S +(s)S^+(s) for the full subcategory of the slice category S +/sS^+/s on the non-invertible morphisms into ss;

  • S (s)S^-(s) for the full subcategory of the under category s/S s/S^- on the non-invertible morphisms out of ss;

  • write

    Latch s(X):=lim (rs)S +(s)X(r)Latch_s(X) := {\lim_{\underset{(r \to s) \in S^+(s)}{\to}}} X(r)

    for the colimit of XX over S +(s)S^+(s), called the latching object of XX at ss;

  • write

    Match s(X):=lim (sr)S (s)X(r)Match_s(X) := {\lim_{\underset{(s \to r) \in S^-(s)}{\leftarrow}}} X(r)

    for the limit of XX over S (s)S^-(s), called the matching object of XX at ss.

  • write

    X s Latch s(X)Latch s(Y)Y sX_s \coprod_{Latch_s(X)} Latch_s(Y) \to Y_s

    for the universal morphism induced from the morphism L s(X)X sL_s(X) \to X_s, called the relative latching map of ϕ\phi at ss;

  • write

    X sMatch s(X) Match s(Y)Y sX_s \to Match_s(X) \prod_{Match_s(Y)} Y_s

    for the dual universal morphism, called the relative matching map of ϕ\phi at ss.

See Joyal-Tierney calculus for more on these kinds of objects and morphisms.

Remark

In the above situation, the automorphism group Aut S(s)Aut_S(s) of ss canonically acts on all objects that appear, and all morphisms that appear respect this action.

Equivalently this means that for all ss the above objects and morphisms take place in the presheaf category [BAut(s),𝒞][B Aut(s), \mathcal{C}].

Proof

Since limits and colimits in presheaf categories are computed objectwise.

Choose now once and for all on all on [ sSBAut(s),𝒞][\coprod_{s \in S} B Aut(s), \mathcal{C}] either the projective or the injective model structure on functors (if they exist). We will use the subscript “ proj/inj{}_{proj/inj}”, as in [ sSBAut(s),𝒞] proj/inj[\coprod_{s \in S} B Aut(s), \mathcal{C}]_{proj/inj}, to indicate some fixed choice.

Definition

Let SS be a (Berger-Moerdijk)-generalized Reedy category.

And 𝒞\mathcal{C} be a model category such that the model structure on functors [ sSBAut(s),𝒞] proj/inj[\coprod_{s \in S} B Aut(s), \mathcal{C}]_{proj/inj} exists (sufficient for the projective structure is that 𝒞\mathcal{C} is a cofibrantly generated model category and sufficient for the injective structure is that it is a combinatorial model category).

Write [S,𝒞][S, \mathcal{C}] for the category of presheaves on S opS^{op} with values in 𝒞\mathcal{C}.

Call a morphism f:XYf : X \to Y in [S,𝒞][S, \mathcal{C}]

  • a Reedy cofibration if for each sSs \in S the relative latching map

    X s L s(X)L s(Y)Y s X_s \coprod_{L_s(X)} L_s(Y) \to Y_s

    is a cofibration in [BAut(s),𝒞] proj/inj[B Aut(s), \mathcal{C}]_{proj/inj};

  • a Reedy fibration if for each sSs \in S the relative matching map

    X sM s(X) M s(Y)Y s X_s \to M_s(X) \prod_{M_s(Y)} Y_s

    is a fibration in [S,𝒞] proj/inj[S, \mathcal{C}]_{proj/inj};

  • a Reedy weak equivalence if for each sSs \in S the morphism

    f s:X sY s f_s : X_s \to Y_s

    is a weak equivalence in 𝒞\mathcal{C}.

Degreewise latching and matching objects

We discuss here an alternative way of speaking about the latching and matching objects, one where all indices at a given degree nn \in \mathbb{N} are collected.

Recall from def. 1 that for sSs \in S we write S +(s)S^+(s) for the category of non-invertible degree-increasing morphisms into ss. We introduce the union of these categories over all objects of a fixed degree.

Definition

For nn \in \mathbb{N} write

  • S +(n)= sSd(s)=nS +(s)S^+(n) = \coprod_{s \in S \atop d(s) = n} S^+(s);

  • d n:S +((n))Sd_ n : S^+((n)) \to S for the restriction of the domain opfibration to objects that are non-invertible morphisms in S +S^+ with codomain in degree nn and to morphisms whose codomain is invertible, i.e. to diagrams of the form

    a S + b S + S + notinvertible s s deg=n; \array{ a &\stackrel{\in S^+}{\to}& b \\ \downarrow^{\mathrlap{\in S^+}} && \downarrow^{\mathrlap{\in S^+}} & not\, invertible \\ s & \stackrel{\simeq}{\to} & s' & deg = n } \,;
  • i n:S +(n)S +((n))i_n : S^+(n) \hookrightarrow S^+((n)) for the full subcategory inclusion on constant codomains;

  • G n(S)Core(S)G_n(S) \subset Core(S) for the groupoid of objects of degree nn and isomorphisms between them.

Proposition

The above categories and functors arrange into a diagram

S dom n S +((n)) cod n G n(S) j n S i n i n S +(n) cod n Obj(S) n, \array{ S &\stackrel{dom_n}{\leftarrow}& S^+((n)) &\stackrel{cod_n}{\to}& G_n(S) &\stackrel{j_n}{\hookrightarrow}& S \\ && {}^{\mathllap{i_n}}\uparrow && \uparrow^{\mathrlap{i_n}} \\ && S^+(n) &\stackrel{cod_n}{\to}& Obj(S)_n } \,,

where the vertical morphisms are (non-full) inclusions and the square is a pullback (in the 1-category Cat) of an opfibration. Therefore it satisfies the Beck-Chevalley condition (see the discussion there) so that we have a natural isomorphism

(cod n) !i n *i n *(cod n) !, (cod_n)_! i_n^* \simeq i_n^* (cod_n)_! \,,

where (cod n) !(cod_n)_! denotes left Kan extension along cod ncod_n.

Remark

The restricted codomain opfibration cod n:S +((n))G ncod_n : S^+((n)) \to G_n is indeed still an opfibration: it is the Grothendieck construction of the pseudofunctor

S +():G n(S)Cat S^+(-) : G_n(S) \to Cat
sS +(s). s \mapsto S^+(s) \,.
Definition

For nn \in \mathbb{N}, let X[S,𝒞]X \in [S, \mathcal{C}]. Write

  • X n:=j n *X=X| G n(S)[G n(S),𝒞]X_n := j_n^* X = X|_{G_n(S)} \in [G_n(S), \mathcal{C}];

  • the nnth latching object is

    Latch n(X):=(cod n)!dom n *X[G n(S),𝒞]. Latch_n(X) := (cod_n)! dom_n^* X \in [G_n(S), \mathcal{C}] \,.
  • the nnth latching morphism

    Latch n(X)X n Latch_n(X) \to X_n

    is the adjunct to the canonical functor

    dom n *X(j ncod n) *X. dom_n^* X \to (j_n cod_n)^* X \,.

The following proposition says that the “global” latching objects indeed contain all the ordinary latching objects in the given degree.

Proposition

For sObj(S) ns \in Obj(S)_n we have

Latch n(X):sLatch s(X) Latch_n(X) : s \mapsto Latch_s(X)

and the component of the nnth latching morphism on ss is the canonical Latch s(X)X(s)Latch_s(X) \to X(s).

Proof

By remark 2 (cod n) !(cod_n)_! is the left Kan extension along an opfibration. By a standard fact (see here at Kan extension) these are computed at any object by the colimit over the fiber over that object.

By definition, that fiber is

cod n 1(s)={a S + b S + S + noninvertible s}. cod_n^{-1}(s) = \left\{ \array{ a &&\stackrel{\in S^+}{\to}&& b \\ & \searrow^{\mathrlap{\in S^+}} && \swarrow_{\mathrlap{\in S^+}} && non\; invertible \\ && s } \right\} \,.

This is indeed S +(s)S^+(s) (by the essential uniqueness of the S +S S^+\circ S^--factorization, this necessarily has the morphisms aba \to b in S +S^+, too.)

So

((cod n) !dom n *X)(s) lim S +(s)(Xdom) =:Latch s(X). \begin{aligned} ((cod_n)_! dom_n^* X)(s) & \simeq \lim_{\underset{S^+(s)}{\to}} (X \circ dom) \\ & \simeq =: Latch_s(X) \end{aligned} \,.

An entirely dual discussion gives the degreewise matching objects: we have a diagram of categories

S cod n S ((n)) dom n G n(S) j n S i n i n S +(n) dom n Obj(S) n, \array{ S &\stackrel{cod_n}{\leftarrow}& S^-((n)) &\stackrel{dom_n}{\to}& G_n(S) &\stackrel{j_n}{\hookrightarrow}& S \\ && {}^{\mathllap{i_n}}\uparrow && \uparrow^{\mathrlap{i_n}} \\ && S^+(n) &\stackrel{dom_n}{\to}& Obj(S)_n } \,,

and

Match nX:=(dom n) *cod n *X, Match_n X := (dom_n)_* cod_n^* X \,,

where (dom n) *(dom_n)_* is the right Kan extension along dom ndom_n.

Skeleta and coskeleta

Over any generalized Reedy category there is an anlog of the notion of simplicial skeleton and simplicial coskeleton.

For nn \in \mathbb{N}, write

t n:S nS t_n : S_{\leq n} \hookrightarrow S

for the full subcategory on the objects of degree n\leq n.

Definition

Left and right Kan extension along t nt_n defines an adjoint triple

((t n) !t n *(t n) *):[S n,𝒞](t n) *t n *(t n) ![S,𝒞]. ((t_n)_! \dashv t_n^* \dashv (t_n)_*) : [S_{\leq n},\mathcal{C}] \stackrel{\overset{(t_n)_!}{\hookrightarrow}}{\stackrel{\overset{t_n^*}{\leftarrow}}{\underset{(t_n)_*}{\hookrightarrow}}} [S,\mathcal{C}] \,.

The induced monads

(sk ncosk n):[S,𝒞]t n *(t n) ![S n,𝒞](t n) *t n *[S,𝒞] (sk_n \dashv cosk_n) : [S,\mathcal{C}] \stackrel{\overset{(t_n)_!}{\leftarrow}}{\underset{t_n^*}{\to}} [S_{\leq n},\mathcal{C}] \stackrel{\overset{t_n^*}{\leftarrow}}{\underset{(t_n)_*}{\to}} [S,\mathcal{C}]

are the nn-skeleton and nn-coskeleton functors, respectively.

Define for all X[S,𝒞]X \in [S, \mathcal{C}] the notation sk 1Xsk_{-1}X to denote the initial object and cosk 1Xcosk_{-1}X the terminal object.

Remark

Here (t n) !(t_n)_! and (t n) *(t_n)_* are indeed full and faithful functors, as indicated.

Proof

Since t n1t_{n-1} is a full and faithful functor, so is its left Kan extension (see here at Kan extension). Moreover in an adjoint triple the leftmost functor is full and faithful if and only if the rightmost one is.

The ((t n) !t n *)((t_n)_! \dashv t_n^*)-counit and the (t n *(t n) *)(t_n^* \dashv (t_n)_*)-unit induces natural transformations

sk nid sk_n \to id
idcosk n. id \to cosk_n \,.
Lemma

For all nn \in \mathbb{N}, the nnth latching object, def. 4, is isomorphic to the (n1)(n-1)-skeleton in degree nn, and dually, the degree-nn matching object is isomorphic to the (n1)(n-1)-coskeleton in degree nn. Under these identifications the canonical morphisms on both sides match

Latch n(X)(sk n1(X)) n Latch_n(X) \simeq (sk_{n-1}(X))_n
Match n(X)(cosk n1(X)) n. Match_n(X) \simeq (cosk_{n-1}(X))_n \,.

This is (Ber-Moer, lemma 6.2).

Proof

Observe that for any sSs \in S of degree nn, the canonical inclusion

i s:S +(s)t n1/s i_s : S^+(s) \hookrightarrow t_{n-1}/ s

into the comma category is a cofinal functor:

  • for dsd \to s any object in t n1/st_{n-1}/s it factors essentially uniquely as dS S +sd \stackrel{\in S^-}{\to} \stackrel{\in S^+}{\to} s, and hence the comma category d/i sd/i_s is non-empty;

  • similarly, since every morphism factors essentially uniquely in SS, there is a zig-zag between any two objects in d/i sd / i_s constructed from the isomorphisms that exhibit the essentially unique factorization.

With this the statement follows from the fact that restriction along cofinal functors preserves colimits and the pointwise description of left Kan extension by colimits over comma categories:

sk n1(X) n(s) :=(j n *(t n1) !t n1 *X)(s) ((t n1) !t n1 *X)(s) lim t n1/sXt n1 lim S +(s)Xdom n Latch nX. \begin{aligned} sk_{n-1}(X)_n(s) & := (j_n^* (t_{n-1})_! t_{n-1}^* X )(s) \\ & \simeq ((t_{n-1})_! t_{n-1}^* X )(s) \\ & \simeq \lim_{\underset{t_{n-1}/ s }{\to}} X \circ t_{n-1} \\ & \simeq \lim_{\underset{S^+(s)}{\to}} X \circ dom_n \\ & \simeq Latch_n X \end{aligned} \,.
Lemma

The tower of inclusions

S 0S 1S n1q n1S n S_{0} \hookrightarrow S_{\leq 1} \hookrightarrow \cdots \hookrightarrow S_{\leq n-1} \stackrel{q_{n-1}}{\hookrightarrow} S_{\leq n} \hookrightarrow \cdots

induces towers of natural transformations

sk 0Xsk 1Xsk 2XX, \emptyset \to sk_0 X \to sk_1 X \to sk_2 X \to \cdots \to X \,,

and

Xcosk 2Xcosk 1Xcosk 0X*, X \to \cdots \to cosk_2 X \to cosk_1 X \to cosk_0 X \to * \,,

that exhibit XX as the colimit of its skeleton tower and as the limit of its coskeleton tower.

This is (Ber-Moer, lemma 6.3).

Proof

The morphisms in the tower come from the adjunction units and counits: the morphism

sk nXsk n+1X sk_n X \to sk_{n+1} X

is

(t n+1) !(q n) !q n *t n+1 *X(t n+1) !t n+1 *X. (t_{n+1})_! (q_n)_! q_n^* t_{n+1}^*X \to (t_{n+1})_! t_{n+1}^* X \,.

Therefore a cocone under this morphism

sk n1X sk nX Y \array{ sk_{n-1} X &&\to& sk_n X \\ & \searrow && \swarrow \\ && Y }

is equivalently a diagram

(q n) !q n *t n+1X t n+1 *X t n+1 *Y, \array{ (q_{n})_! q_{n}^* t_{n+1} X &&\to& t_{n+1}^* X \\ & \searrow && \swarrow \\ && t_{n+1}^* Y } \,,

which in turn is equivalently just a morphism t n *Xt n *Yt_n^* X \to t_n^* Y. So a cocone under the whole tower is an object YY equipped for each nn with a morphism t n *Xt n *Yt_n^* X \to t_n^* Y. Clearly XX itself is the inital such object.

Lemma

For every X[S,𝒞]X \in [S, \mathcal{C}] and for all pairs k,lk,l \in \mathbb{N} with klk \leq l, we have natural isomorphisms

sk ksk lsk lsk ksk k sk_k sk_l \simeq sk_l sk_k \simeq sk_k

and

cosk kcosk lcosk lcosk kcosk k. cosk_k cosk_l \simeq cosk_l cosk_k \simeq cosk_k \,.
Proof

With the above notation we have t k=t lq kt_k = t_l \circ q_k. Therefore for instance

sk ksk lX(t l) !(q k) !q k *t l *(t l) !t l *X. sk_k sk_l X \simeq (t_l)_! (q_k)_! q_k^* t_l^* (t_l)_! t_l^* X \,.

Since the left adjoints here are full and faithful functors we have t l *(t l) !idt_l^* (t_l)_! \simeq id and hence

(t l) !(q k) !q k *t l *(t l) !t l *Xsk kX. \cdots \simeq (t_l)_! (q_k)_! q_k^* t_l^* (t_l)_! t_l^* X \simeq sk_k X \,.

Similarly for all the other cases.

The following is a useful tool for inductively creating objects by adding higher degree components.

Lemma

Given

  • an object X (n1)[S (n1),𝒞]X_{\leq (n-1)} \in [S_{\leq (n-1)}, \mathcal{C}];

  • and an object X n[G n(S),𝒞]X_n \in [G_n(S), \mathcal{C}];

the choices of X n[S (n1),𝒞]X_{\leq n} \in [S_{\leq (n-1)}, \mathcal{C}] such that

  • X (n1)=t n1 *X nX_{\leq (n-1)} = t^*_{n-1} X_{\leq n}

  • X n=j n *XX_n = j_n^* X;

are in bijection with choices of morphisms

Latch n(X (n1))X nMatch n(X (n1)) Latch_n(X_{\leq (n-1)}) \to X_n \to Match_n(X_{\leq (n-1)})

in [G n(S),𝒞][G_n(S), \mathcal{C}].

Accordingly, given morphisms f (n1):X (n1)Y (n1)f_{\leq (n-1)} : X_{\leq (n-1)} \to Y_{\leq (n-1)} and f n:X nY nf_n : X_n \to Y_n, then choice of extensions to a morphism f n:X nY nf_{\leq n} : X_{\leq n} \to Y_{\leq n} are in bijection with choices of vertical morphisms in commuting diagrams

Latch n(X (n1)) Latch n(f (n1)) Latch n(Y (n1)) X n f n Y n Match n(X (n1)) Match n(f (n1)) Match n(Y (n1)). \array{ Latch_n(X_{\leq (n-1)}) &\stackrel{Latch_n(f_{\leq (n-1)})}{\to}& Latch_n(Y_{\leq (n-1)}) \\ \downarrow && \downarrow \\ X_n &\stackrel{f_n}{\to}& Y_n \\ \downarrow && \downarrow \\ Match_n(X_{\leq (n-1)}) &\stackrel{Match_n(f_{\leq (n-1)})}{\to}& Match_n(Y_{\leq (n-1)}) } \,.
Proof

If the object exists, then the morphisms do, by the above definitions/discussion. Conversely, given these morphisms, we take X n:S𝒞X_{\leq n} : S \to \mathcal{C} to be given by X (n1)X_{\leq (n-1)} on morphism of degree (n1)\leq (n-1), to be given by X nX_n on morphisms between objects of degree nn, and need to define it on the degree-changing morphisms to and from objects of degree nn. This information is provided precisely by the co-cone components of Latch n(X)X nLatch_n(X) \to X_n and by the cone-components of X nMatch n(X)X_n \to Match_n(X).

Proof of the model category axioms

We discuss that for SS a Berger-Moerdijk-generalized Reedy category and 𝒞\mathcal{C} a cofibrantly generated model category, def. 2 indeed defines a model category structure on the functor category [S,𝒞][S,\mathcal{C}].

Theorem

Def. 2 indeed defines a model structure.

Proof

It is clear that [S,𝒞][S,\mathcal{C}] has all limits and colimits (as for any category of presheaves they are computed objectwise in \mathcal{E}) and that the weak equivalences satisfy two-out-of-three, since the weak equivalences in \mathcal{E} do. Also, all three classes of morphisms are closed under retracts, since, for instance, the relative latching morphism of a retract is the retract of a relative latching morphism and so the property follows with the retract-closure of the classes of morphisms in \mathcal{E}.

It remains to show that the relevant lifting and factorization properties hold. This we discuss in a list of lemmas below in Lifting and Factorization.

Lifting

We work with the “global” latching objects from above.

Remark

A morphism f:XYf : X \to Y in [S,𝒞] gReedy[S, \mathcal{C}]_{gReedy} is a Reedy cofibration precisely if for all nn \in \mathbb{N} the global relative latching morphism, def. 4

X n Latch n(X)Latch n(Y)Y n X_n \coprod_{Latch_n(X)} Latch_n(Y) \to Y_n

is a cofibration in [G n(S),𝒞] proj/inj[G_n(S), \mathcal{C}]_{proj/inj}.

Proof

The pushout in the presheaf category [G n(S),𝒞][G_n(S), \mathcal{C}] is computed objectwise, so that the component of the nnth relative latching morphism at sSs \in S is the relative latching morphism at ss, by prop. 2.

The groupoid G n(S)G_n(S) is equivalent to the disjoint union [r]π 0G n(S)BAut S(s)\coprod_{[r] \in \pi_0 G_n(S)} B Aut_S(s) of the automorphism groupoids of one representative in each isomorphism class. A morphism in [G n(S),𝒞] proj/inj[G_n(S), \mathcal{C}]_{proj/inj} is a cofibration precisely if its restriction to all of the [BAut 𝒞(s),𝒞] proj/inj[B Aut_{\mathcal{C}}(s), \mathcal{C}]_{proj/inj} is.

Lemma

In the generalized Reedy structure, def. 2, the following holds.

  • Acyclic Reedy cofibrations ff such that for all nn \in \mathbb{N} the morphism Latch n(f)Latch_n(f) is an objectwise acyclic cofibration have the left lifting property against fibrations;

  • Reedy cofibration have the left lifting property against acyclic Reedy fibrations gg with the special property that all Latch n(g)Latch_n(g) are objectwise acyclic fibrations.

This is (Ber-Moer, lemma 5.2, lemma 5.4).

Proof

We show the first clause. The second is dual. So let f:XYf : X \to Y be an acyclic cofibration with the above extra property, and let g:YXg : Y \to X be a fibration. We will exhibit a lift in any commuting diagram

A Y B X \array{ A &\stackrel{}{\to}& Y \\ \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{}} \\ B &\stackrel{}{\to}& X }

by stepwise constructing lifts in the skeletal filtration, lemma 2.

At n=0n = 0, observe that since L 0(X)=L_0(X) = \emptyset for all XX, the fact that

A 0 L 0(A)L 0(B)B 0 A_0 \coprod_{L_0(A)} L_0(B) \to B_0

is a cofibration in [G 0,𝒞] proj/inj[G_0, \mathcal{C}]_{proj/inj} by assumption, means that in fact f 0:A 0B 0f_0 : A_0 \to B_0 is an acyclic cofibration here. Similarly Y 0X 0Y_0 \to X_0 is a fibration there. But G 0(S)=S 0G_0(S) = S_{\leq 0} and so the restriction of the lifting problem along t 0t_0

A 0 Y 0 B 0 X 0 \array{ A_0 &\stackrel{}{\to}& Y_0 \\ \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{}} \\ B_0 &\stackrel{}{\to}& X_0 }

is a lifting problem in [G 0(S),𝒞] proj/inj[G_0(S), \mathcal{C}]_{proj/inj} of an acyclic cofibration against a fibration, and hence has a filler γ 0:B 0Y 0\gamma_0 : B_0 \to Y_0 there.

Now assume that a filler γ (n1)\gamma_{\leq (n-1)} in

A (n1) Y (n1) B (n1) X (n1) \array{ A_{\leq (n-1)} &\stackrel{}{\to}& Y_{\leq (n-1)} \\ \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{}} \\ B_{\leq (n-1)} &\stackrel{}{\to}& X_{\leq (n-1)} }

has been found. By lemma 1 this induces maps Latch n(B)Latch n(Y)Latch_n(B) \to Latch_n (Y) and Match n(B)Match n(Y)Match_n(B) \to Match_n(Y) from which we can build the commuting diagram

(1)A n Latch nALatch nB Y n v n w n B n X n× Match nXMatch nY \array{ A_n \coprod_{Latch_n A} Latch_n B &\to& Y_n \\ \downarrow^{\mathrlap{v_n}} && \downarrow^{\mathrlap{w_n}} \\ B_n &\to& X_n \times_{Match_n X} Match_n Y }

in [G n(S),𝒞][G_n(S), \mathcal{C}].

Here for instance the top horizontal morphism comes from the commutativity of the square

Latch nA Latch nB Latch nY A n Y n \array{ Latch_n A &\to& Latch_n B &\to& Latch_n Y \\ \downarrow && && \downarrow \\ A_n &\to& &\to& Y_n }

by naturality of the (sk n1t n1 *)(sk_{n-1} \dashv t_{n-1}^*)-counit.

We observe now that finding a lift in (1) will complete the induction step. To see this in more detail, notice that in the top left the lift

A n Latch nALatch nB Y n γ n B n \array{ A_n \coprod_{Latch_n A} Latch_n B &\to& Y_n \\ \downarrow &\nearrow_{\gamma_n}& \\ B_n }

is

  • a lift in

    A n Y n γ n B n \array{ A_n &\to& Y_n \\ \downarrow & \nearrow_{\mathrlap{\gamma_n}} \\ B_n }
  • such that it makes

    Latch n(B) Latch n(Y) B n γ n Y n \array{ Latch_n(B) &\to& Latch_n(Y) \\ \downarrow && \downarrow \\ B_n &\stackrel{\gamma_n}{\to}& Y_n }

    commute;

and in the bottom right the lift

Y n γ n B n X n Match nXMatch nY \array{ && Y_n \\ & {}^{\mathllap{\gamma_n}}\nearrow & \downarrow \\ B_n &\to& X_n \prod_{Match_n X} Match_n Y }

is

  • a lift in

    Y n γ n B n X n \array{ && Y_n \\ & {}^{\mathllap{\gamma_n}}\nearrow & \downarrow \\ B_n &\to& X_n }
  • such that it makes

    B n γ n Y n Match n(B) Match n(Y) \array{ B_n &\stackrel{\gamma_n}{\to}& Y_n \\ \downarrow && \downarrow \\ Match_n(B) &\to& Match_n(Y) }

    commute.

By lemma 4, this is precisely the data that characterizes an extension of γ (n1)\gamma_{\leq (n-1)} to γ n\gamma_{\leq n}.

By assumption, the left vertical morphism in (1) is a cofibration in [G n,𝒞] proj/inj[G_n, \mathcal{C}]_{proj/inj}, and the right vertical morphism is a fibration there. Therefore to get the lift and hence complete the induction step, it is now sufficient to show that the left morphism is also a weak equivalence, hence is a weak equivalence in 𝒞\mathcal{C} over each sSs \in S.

Also by assumption we have that Latch n(f) sLatch_n(f)_s is an acyclic cofibration in 𝒞\mathcal{C} for all ss. Hence so is its pushout A s(A s Latch n(A) sLatch n(B) s)A_s \to (A_s \coprod_{Latch_n(A)_s} Latch_n(B)_s). The morphism v n(s)v_n(s) finally sits in the diagram

Latch n(A) s A s f s B s v n(s) Latch n(B) s (A s Latch n(A) sLatch n(B) s) \array{ Latch_n(A)_s &\to& A_s &\underoverset{\simeq}{f_s}{\to}& B_s \\ \downarrow^{\mathrlap{\simeq}} && \downarrow^{\mathrlap{\simeq}} & \nearrow_{\mathrlap{v_n(s)}} \\ Latch_n(B)_s &\to& (A_s \coprod_{Latch_n(A)_s} Latch_n(B)_s) }

and so is a weak equivalence by two-out-of-three.

Lemma

Suppose ϕ:RS\phi : R \to S is a morphism of generalized Reedy categories such that the induced square (see prop. 1)

R +((k)) cod k R G k(R) ϕ k + ϕ k S +((k)) cod k S G k(S) \array{ R^+((k)) &\stackrel{cod_k^R}{\to}& G_k(R) \\ \downarrow^{\mathrlap{\phi_k^+}} && \downarrow^{\mathrlap{\phi_k}} \\ S^+((k)) &\stackrel{cod_k^S}{\to}& G_k(S) }

is a pullback (in the 1-category Cat). Then for each X[S,𝒞]X \in [S, \mathcal{C}], there is a natural isomorphism

Latch k(ϕ *(X))ϕ k *(Latch kX), Latch_k(\phi^*(X)) \stackrel{\simeq}{\to} \phi_k^*(Latch_k X) \,,

Given a Reedy category SS, examples of ϕ\phi satisfying this condition include

  1. ϕ:=dom ni n:S +(n)S\phi := dom_n \circ i_n : S^+(n) \to S;

  2. ϕ:=:S nS\phi := : S_{\leq n} \hookrightarrow S.

This is (Ber-Moer, lemma 4.4).

Proof

The pullback square is part of the diagram

R dom k R R +((k)) cod k R G k(R) ϕ ϕ k + ϕ k S dom k S S +((k)) cod k S G k(S) \array{ R &\stackrel{dom_k^R}{\leftarrow}& R^+((k)) &\stackrel{cod_k^R}{\to}& G_k(R) \\ \downarrow^\phi && \downarrow^{\mathrlap{\phi_k^+}} && \downarrow^{\mathrlap{\phi_k}} \\ S &\stackrel{dom_k^S}{\leftarrow}& S^+((k)) &\stackrel{cod_k^S}{\to}& G_k(S) }

whose rows define, by prop. 1, the latching objects by pull-push. Since the pullback square, being the pullback of an opfibration (the codomain opfibration), satisfies the Beck-Chevalley condition (by the fact discussed here), we find the intertwining isomorphism as follows:

Latch kϕ *X (cod k R) !(dom k R) *ϕ *X (cod k R) !(ϕ k +) *(dom k S) *X (ϕ k) *(cod k R) !(dom k S) *X (ϕ k) *Latch kX. \begin{aligned} Latch_k \phi^* X & \simeq (cod^R_k)_! (dom^R_k)^* \phi^* X \\ & \simeq (cod^R_k)_! (\phi^+_k)^* (dom^S_k)^* X \\ & \simeq (\phi_k)^* (cod^R_k)_! (dom^S_k)^* X \\ & \simeq (\phi_k)^* Latch_k X \end{aligned} \,.

Now concerning the two examples.

By definition we have

(S +(n)) +((k)){t S + t S + S + noninvertible s s deg=k S + S + noninvertible r id r deg=n}, (S^+(n))^+((k)) \simeq \left\{ \array{ t &\stackrel{\in S^+}{\to}& t' \\ \downarrow^{\mathrlap{\in S^+}} && \downarrow^{\mathrlap{\in S^+}} && non\; invertible \\ s &\stackrel{\simeq}{\to}& s' && deg = k \\ \downarrow^{\mathrlap{\in S^+}} && \downarrow^{\mathrlap{\in S^+}} && non \; invertible \\ r &\stackrel{id}{\to}& r && deg = n } \right\} \,,

where on the right the vertical sequences in SS indicate objects in (S +(m)) +((k))(S^+(m))^+((k)) and the whole diagram on the right indicates a morphism there.

One sees that this is indeed the fiber product as claimed.

We now show that the extra condition in prop. 5 is in fact automatic.

Lemma

Let f:XYf : X \to Y in [S,𝒞][S, \mathcal{C}] be a Reedy cofibration, which is a weak equivalence on all objects of degree <n\lt n. Then the morphism Latch n(f):Latch n(X)Latch n(Y) Latch_n(f) : Latch_n(X) \to Latch_n(Y) is over each sSs \in S an acyclic cofibration in 𝒞\mathcal{C}

This is (Ber-Moer, lemma 5.3).

Proof

We show this by induction over nn, using the skeletal filtration def. 5. For n=0n = 0 we have for all XX that Latch nX=sk 1X=Latch_n X = sk_{-1} X = \emptyset, and hence the condition is trivially satisfied.

So assume now that the statement has been shown for all (k<n)(k \lt n), then we need to show that i n *Latch nfi_n^* Latch_n f is an acyclic cofibration in [Obj(S) n,𝒞][Obj(S)_n, \mathcal{C}], hence that every square of the form

i n *Latch nA Y i n *Latch nf g i n *Latch nB X \array{ i_n^* Latch_n A &\to& Y \\ \downarrow^{\mathrlap{i_n^* Latch_n f}} && \downarrow^{g} \\ i_n^* Latch_n B &\to& X }

with gg a fibration in [Obj(S) n,𝒞] proj/inh[Obj(S)_n, \mathcal{C}]_{proj/inh} (hence over every object of degree nn) has a lift. Since by lemma 1 we have

i n *Latch n:=i n *cod !dom n *cod !i n *dom n * i^*_n Latch_n := i_n^* cod_! dom_n^* \simeq cod_! i_n^* dom_n^*

such a filler is equivalently a filler in

(2)i n *dom n *A cod n *Y i n *dom n * i n *dom n *B cod n *X \array{ i_n^* dom_n^* A &\to& cod_n^* Y \\ {}^{\mathllap{i_n^* dom_n^*}}{}\downarrow && \downarrow \\ i_n^* dom_n^* B &\to& cod_n^* X }

being a diagram in [S +(n),𝒞][S^+(n), \mathcal{C}].

We now establish such a filler by using lemma 5 with the Reedy category in question being now S +(n)S^+(n). This has only ++-morphisms and hence Reedy fibrations here are objectwise fibrations, so the morphism on the right is a Reedy fibration over S +(n)S^+(n).

Moreover, i n *dom n *fi_n^* dom_n^* f is a Reedy weak equivalence in this structure, since all its objects have degree <n\lt n. It is now sufficient to show that the assumptions of lemma 5 are satisfied over S +(n)S^+(n), to obtain the lift.

By lemma 6 the functor

(dom ni n) *:[G k(S),𝒞][G k(S +(n)),𝒞] (dom_n i_n)^* : [G_k(S), \mathcal{C}] \to [G_k(S^+(n)), \mathcal{C}]

intertwines the latching objects on both sides. Therefore we have an isomorphism between the relative latching morphism of interest

(dom ni n) *A Latch k(dom ni n) *ALatch k(dom ni n) *k(dom ni n) *B (dom_n i_n)^* A \coprod_{Latch_k (dom_n i_n)^* A} Latch_k (dom_n i_n)^* k \to (dom_n i_n)^* B

and the morphism

(dom ni n) k *(A k Latch k(A)Latch k(B)B k). (dom_n i_n)_k^*\left( A_k \coprod_{Latch_k(A)} Latch_k(B) \to B_k \right) \,.

Since (dom ni n) k(dom_n i_n)_k is a faithful functor between groupoids, (dom ni n) k *(dom_n i_n)_k^* preserves cofibrations in the projective structure (and trivially does so always in the injective structure), and so the above relative latching morphism is a cofibration, hence (dom ni n) *(f)(dom_n i_n)^*(f) is a Reedy cofibration. Similarly, since Latch k(f)Latch_k(f) is an acyclic cofibration by induction hypothesis, so is Latch k((dom ni n) k *f)Latch_k((dom_n i_n)_k^* f). This way the assumption of lemma 5 are checked for (2) and so we do have a lift.

Lemma

Dually, let f:XYf : X \to Y in [S,𝒞][S, \mathcal{C}] be a Reedy fibration which is a weak equivalence over all objects of degree <n\lt n. Then the morphism Match n(f):Match n(X)Match n(Y)Match_n(f) : Match_n(X) \to Match_n(Y) is over each sSs \in S an acyclic fibration in 𝒞\mathcal{C}.

This is (Ber-Moer, lemma 5.4).

Proof

(Essentially the dual proof to that above. Except for one slight difference in the last part. Here – and only here – do we need the last clause in the definition of generalized Reedy category, the one that says that isomorphisms see the maps in S S_- as epimorphisms.)

We can finally conclude:

Proposition

In the generalized Reedy model structure, def. 2,

Proof

By lemma 7 every acyclic Reedy cofibration induces a weak equivalence under Latch nLatch_n. By lemma 5 this implies the left lifting property against Reedy fibrations. Dually for the second statement.

Factorization

We demonstrate the factorization axiom in the Reedy model structure, def. 2.

Lemma

A morphism f:XYf : X \to Y in [S,𝒞][S, \mathcal{C}] is both a cofibration and a weak equivalence according to def. 2 precisely if the nnth relative latching morphism (def. 4)

X n Latch n(X)Latch n(Y)Y n X_n \coprod_{Latch_n(X)} Latch_n(Y) \to Y_n

is an acyclic cofibration in [G n(S),𝒞] proj/inj[G_n(S), \mathcal{C}]_{proj/inj} for all nn \in \mathbb{N}.

Dually, a morphism f:XYf : X \to Y in [S,𝒞][S, \mathcal{C}] is both a fibration and a weak equivalence according to def. 2 precisely if the nnth relative matching morphism (def. 4)

X nY n Match n(X)Match n(Y) X_n \to Y_n \prod_{Match_n(X)} Match_n(Y)

is an acyclic fibration in [G n(S),𝒞] proj/inj[G_n(S), \mathcal{C}]_{proj/inj} for all nn \in \mathbb{N}.

This is (Ber-Moer, prop. 5.6).

Proof

By definition, the morphisms f n:X nY nf_n : X_n \to Y_n factor as

f n:X nu nX n Latch n(X)Latch n(Y)v nY n. f_n : X_n \stackrel{u_n}{\to} X_n \coprod_{Latch_n(X)} Latch_n (Y) \stackrel{v_n}{\to} Y_n \,.

If now ff is an acyclic Reedy cofibration, then by lemma 7 the morphism Latch n(X)Latch n(Y)Latch_n(X) \to Latch_n(Y) is over each object an acyclic cofibration in 𝒞\mathcal{C} and then so is u nu_n above, being the pushout of this morphism. It follows by two-out-of-three that also v nv_n is a weak equivalence for all nn.

Conversely, assume that all v nv_n here are weak equivalences. We show by induction on nn that then also the u nu_n are weak equivalences, and hence that ff is a Reedy weak equivalence.

For n=0n = 0 we have u 0=idu_0 = id, and so this case is satisfied. So assume now that all u ku_k for k<nk \lt n are weak equivalences. Then the assumptions of lemma 7 are again satisfied, and it follows that Latch n(f):Latch n(X)Latch n(Y)Latch_n(f) : Latch_n(X) \to Latch_n(Y) is over each object an acyclic cofibration. Accordingly, so is u nu_n, being its pushout. Therefore, by induction, all u nu_n are, in particular, weak equivalences.

The argument for fibrations is dual to this.

Proposition

Every morphism in [S,𝒞][S, \mathcal{C}] factors as an acyclic Reedy cofibration (according to def. 2) followed by a Reedy fibration, and it factors as a Reedy cofibration followed by an acyclic Reedy fibration

This is (Ber-Moer, page 18).

Proof

Let f:XYf : X \to Y be any morphism in [S,𝒞][S, \mathcal{C}]. We construct a factorization into an acyclic cofibration followed by a fibration by induction on the degree, i.e. by induction over the restrictions along t n *:[S,𝒞][S n,𝒞]t_n^* : [S, \mathcal{C}] \to [S_{\leq n}, \mathcal{C}]. The other case (cofibration followed by acyclic fibration) works dually.

For n=0n = 0 we have S 0=G 0(S)S_{\leq 0} = G_0(S) and we factor f 0f_0 in the model structure [G 0(S),𝒞] proj/inj[G_0(S), \mathcal{C}]_{proj/inj}

f 0:X 0A 0Y 0. f_0 : X_0 \stackrel{\simeq}{\to} A_0 \to Y_0 \,.

Now assume for some nn \in \mathbb{N} that a factorization of

f (n1):X (n1)A (n1)Y (n1) f_{\leq (n-1)} : X_{\leq (n-1)} \stackrel{\simeq}{\to} A_{\leq (n-1)} \to Y_{\leq (n-1)}

has been found. This induces the commutative diagram

Latch n(X) Latch n(A) Latch n(Y) X n Y n Match n(X) Match n(A) Match n(Y). \array{ Latch_n(X) &\to & Latch_n(A) &\to& Latch_n(Y) \\ \downarrow && && \downarrow \\ X_n && && Y_n \\ \downarrow && && \downarrow \\ Match_n(X) &\to & Match_n(A) &\to& Match_n(Y) } \,.

This diagram in turn induces a morphism

X n Latch n(X)Latch n(A)Y n Match n(Y)Match n(A) X_n \coprod_{Latch_n(X)} Latch_n(A) \to Y_n \prod_{Match_n(Y)} Match_n(A)

in [G n(S),𝒞][G_n(S), \mathcal{C}], which we may factor as a trivial cofibration followed by a fibration

X n Latch n(X)Latch n(A)A nY n Match n(Y)Match n(A) X_n \coprod_{Latch_n(X)} Latch_n(A) \stackrel{\simeq}{\to} A_n \to Y_n \prod_{Match_n(Y)} Match_n(A)

in [G n(S),𝒞] proj/inj[G_n(S), \mathcal{C}]_{proj/inj}.

By lemma 4, the “righmost component” of this data defines an extension of A (n1)A_{\leq (n-1)} to A nA_{\leq n}. The “leftmost component” defines a factorization of f nf_n and the “middle component” says that this consistently extends the previously obtained factorization of f nf_{\leq n}. With this, finally the two morphisms say that this new factorization is by an acyclic Reedy cofibration (by lemma 9) followed by a Reedy fibration (by definition).

Properties

Homotopy skeletal filtration and coskeleton tower

We discuss conditions on an object XX such that the skeletal/coskeletal towers discussed above are “homotopy good” in that they exhibit XX not just as the colimit/limit over the tower, but as the homotopy colimit/homotopy limit.

Lemma

For all nn \in \mathbb{N}

  1. The left Kan extension

    (t n) !:[S n,𝒞] gReedy[S,𝒞] gReedy (t_n)_! : [S_{\leq n}, \mathcal{C}]_{gReedy} \to [S,\mathcal{C}]_{gReedy}

    is a left Quillen functor.

  2. The right Kan extension

    (t n) *:[S n,𝒞] gReedy[S,𝒞] gReedy (t_n)_* : [S_{\leq n}, \mathcal{C}]_{gReedy} \to [S,\mathcal{C}]_{gReedy}

    is a right Quillen functor.

  3. The restriction functor

    (t n) *:[S,𝒞] gReedy[S n,𝒞] gReedy (t_n)^* : [S, \mathcal{C}]_{gReedy} \to [S_{\leq n},\mathcal{C}]_{gReedy}

    is both a left and a right Quillen functor.

This is (Ber-Moer, lemma 6.4).

Proposition
  1. If X[S,𝒞]X \in [S, \mathcal{C}] is Reedy cofibrant according to def. 2, then all sk nXsk_n X are Reedy cofibrant and the canonical morphisms sk kXsk lXsk_k X \to sk_l X are Reedy cofibrations.

  2. If X𝒞X \in \mathcal{C} is Reedy fibrant according to def. 2 then all cosk nXcosk_n X are Reedy fibrant and the canonical morphisms cosk lXcosk kXcosk_l X \to cosk_k X are Reedy fibrations.

This is (Ber-Moer, prop. 6.5).

Proof

We discuss the skeleta. The case of coskeleta is dual.

By lemma 3 it is sufficient to consider the case sk nXsk X=Xsk_n X \to sk_\infty X = X.

To check that this is a Reedy cofibration if XX is Reedy cofibrant, consider the diagram that induces the relative latching object for this morphism

Latch k(sk nX) Latch k(X) sk n(X) k X k. \array{ Latch_k(sk_n X) &\to& Latch_k (X) \\ \downarrow && \downarrow \\ sk_n(X)_k &\to& X_k } \,.

For knk \leq n, the horizontal morphisms are both isomorphisms. Because by lemma 1 the top morphism is

(sk k1sk n(X)) k(sk k1X) k (sk_{k-1} sk_n(X))_k \to (sk_{k-1} X)_k

and this is an iso by lemma 3. The bottom morphism is isomorphic to (t k *sk n(X)) k(t k *X) k(t_k^* sk_{n}(X))_k \to (t_k^* X)_k which is isomorphic to the identity by the kind of argument in lemma 3.

Therefore also the relative latching morphism is an isomorphism in this case (use that pushouts of isos are isos and use 2-out-of-3 for isos), hence in particular a cofibration.

Similarly, for k>nk \gt n the left vertical map is an isomorphism, so that the relative latching morphism in this case is Latch k(X)X kLatch_k(X) \to X_k, which is a cofibration by the assumption that XX is Reedy cofibrant.

Finally, that sk nXsk_n X is cofibrant follows directly from lemma 10.

Relation to other model structures

(…)

Example

References

Revised on March 28, 2012 16:59:13 by Urs Schreiber (131.174.41.113)