generalized Reedy category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _[[Reedy category]]_, though useful, is in some contexts inconveniently restrictive, since no Reedy category can contain any nonidentity [[isomorphisms]].  This is problematic for many "[[geometric shape for higher structures|shape]] categories" such as Connes' [[category of cycles]] $\Lambda$, [[Segal's category Gamma|Segal's category]] $\Gamma$, the [[tree category]] $\Omega$, and so on.  The notion of *generalized Reedy category* lifts this restriction, while maintaining the truth of the main theorem about Reedy categories: the existence of the [[Reedy model structure]].

In fact, there are two notions of generalized Reedy category in the literature. Cisinski's "catégories squelletiques" (Ch. 8 in  [_PCMH_](#Cisinski)) provide a natural generalization of the [[simplex category]], so that diagrams based on them behave much like [[simplicial objects]]. They were introduced primarily for the purposes of modeling [[homotopy types]]. The "generalized Reedy categories" of [Berger & Moerdijk 2011](#BergerMoerdijk11) are a strictly broader generalization suitable e.g. for describing [[dendroidal sets]]. They were introduced for the purposes of modelling more general classes of structure, particularly [[operads]].

## Definitions

### Generalized Reedy categories

+-- {: .num_defn #BMGeneralizedReedyCategory}
###### Definition

A **Berger-Moerdijk generalized Reedy category** is a [[category]] $R$ together with two [[wide subcategories]] $R_+$ and $R_-$, and a [[function]] $d\colon ob(R)\to \alpha$ called *degree*, for some [[ordinal]] $\alpha$, such that

1. every non-isomorphism in $R_+$ raises degree,

1. every non-isomorphism in $R_-$ lowers degree,

1. every isomorphism in $R$ preserves degree,

1. $R_+\cap R_-$ is the [[core]] of $R$ (equivalently, every isomorphism is in both $R_+$ and $R_-$, i.e. they are not just wide but [[pseudomonic functor|pseudomonic]] subcategories),

1. every morphism $f$ factors as a map in $R_-$ followed by a map in $R_+$, uniquely up to isomorphism,

1. if $f\in R_-$ and $\theta$ is an isomorphism such that $\theta f = f$, then $\theta = 1$ (isomorphisms see the maps in $R_-$ as [[epimorphism|epis]]).

The last condition implies that the isomorphism in the penultimate condition must be unique.  It is not self-dual, but has an obvious dual version.  A BM generalized Reedy category is said to be **dualizable** if it satisfies both this condition and its dual.

=--

+-- {: .num_defn }
###### Definition

A [[morphism]] of generalized Reedy category $S \to R$ is a [[functor]] whish takes $S_+$ to $R_+$, takes $S_-$ to $R_-$ and preserves the degree.

=--

This appears as ([Berger & Moerdijk 2011, def. 1.1](#BergerMoerdijk11)).

+-- {: .num_remark}
###### Remark

Generalized Reedy category structures (as opposed to ordinary structures!) can always be transported along [[equivalence of categories]].

=--

+-- {: .num_defn}
###### Definition

For a **Cisinski generalized Reedy category**, the final condition in
def. \ref{BMGeneralizedReedyCategory} is replaced by

* every morphism in $R_-$ admits a [[section]], and two parallel morphisms in $R_-$ are equal precisely when they have the same sections.

=--

For clarity, in the context of generalized Reedy categories, an ordinary [[Reedy category]] may be called a *strict Reedy category*.

### Prima Facie comparison between Cisinski and Berger-Moerdijk.

The only difference between the Cisinski notion and the Berger-Moerdijk notion is in the final condition -- let's say, between the _Cisinski condition_ and the _Berger-Moerdijk condition_.

It's easy to see that the Cisinski condition is **strictly stronger** than the Berger-Moerdijk condition.  The Berger-Moerdijk condition asks that $R_-$ arrows be something less than epimorphic in $R$. By comparison, the Cisinski condition asks that $R_-$ arrows be _actually_ epimorphic in $R$, in fact that they be _split_ epimorphic, and more.

### Presheaves on Reedy categories.

For $R$ a generalized Reedy category, and $X$ a [[presheaf]] on $R$, there are the evident analogue notions of $n$-cells in $X$, degenerate $n$-cells, faces, boundaries, horns, etc.

(..)

## Properties

### Normal monomorphisms (Cisinski)
 {#NormalMorphisms}

Let $A$ be a Cisinski generalized Reedy category.

+-- {: .num_defn}
###### Definition

An object $a \in A$ is called _degenerate_ precisely if there is a non-isomorphism out of $a$ in $A_-$.

=--

See ([Cisinski, prop. 8.1.9](#Cisinski)).

+-- {: .num_prop}
###### Proposition

For every object $a \in A$ there exists a morphism $\pi : a \to b$ in $A_-$
with $b$ non-degenerate.

=--

This is ([Cisinski, prop. 8.1.13](#Cisisnki)).

Let $X$ be a [[presheaf]] over $A$.

+-- {: .num_defn}
###### Definition

For $a \in A$,  a cell $v \in X(a)$ is called **degenerate** precisely if
there is a morphism $\alpha : a \to a'$ in $A_-$ and a cell $u \in X(a')$

$$
  X(\alpha) : u \mapsto v
$$

=--

See ([Cisinski, cor. 8.1.10](#Cisinski)).

Write $A/X$ for the [[category of elements]]
of $X$.

+-- {: .num_defn}
###### Definition

For $a \in A$ an $a$-cell $v \in X(a)$ is called **dominant** if $(a,v) \in A/X$ has trivial [[automorphism group]].

An $a$-cell $u \in X(a)$ is called **normal** if there is a morphism $\alpha : a \to b$ in $A_-$ with $b$ non-degenerate, and a dominant $b$-cell $v \in X(b)$, such that
$$
  X(\alpha) : v \mapsto u
  \,.
$$

The presheaf $X$ is called **normal** if all its cells are normal.

=--

See ([Cisinski, 8.1.23](#Cisinski)).

+-- {: .num_remark}
###### Remark

A non-degenerate cell is normal precisely if it is dominant.

=--

+-- {: .num_prop}
###### Proposition

$X$ is normal precisely if all its non-degenerate cells are dominant.

=--

This is ([Cisinski, cor. 8.1.25](#Cisinski)).

Let $f : X \to Y$ be a morphism of [[presheaves]] over $A$.

+-- {: .num_defn}
###### Definition

The morphism $f : X \to Y$ is called **normal** if every cell of $Y$ not in the [[image]] of $f$ is normal.

=--

This is ([Cisinski, 8.1.30](#Cisinski)).

+-- {: .num_example}
###### Example

Every [[monomorphism]] between normal presheaves is normal.

=--

+-- {: .num_prop}
###### Proposition

The class of normal monomorphisms in $PSh(A)$ is closed under

* [[pushouts]];

* [[transfinite composition]];

* [[retracts]].

In fact, the class of normal monomorphisms is that generated under these operations from the boundary inclusions $I := \{\partial a \hookrightarrow a\}$.

=--

This is ([Cisinski, prop. 8.1.31, 8.1.35](#Cisinski)).

### Model category structure on presheaves

(...) [[generalized Reedy model structure]]

## Examples

### Berger-Moerdijk type

#### Specific examples

* Any [[Reedy category]] is a generalized Reedy category, in particular the [[simplex category]].

* Any (finite) [[groupoid]] $G$ is also a generalized Reedy category, with $G_+ = G_- = G$.

* Connes' [[category of cycles]] $\Lambda$.

* [[Segal's category Gamma|Segal's category]] $\Gamma$.

* The Moerdijk-Weiss [[tree category]] $\Omega$ is generalized Reedy. The degree is given by the number of vertices in a tree.

  See also the discussion at _[[dendroidal set]]_ and _[[model structure on dendroidal sets]]_.

* Any generalized [[direct category]] or generalized [[inverse category]] is also a generalized Reedy category, in which either $R_-$ or $R_+$ consists only of the isomorphisms.

#### The class of crossed group sites

+-- {: .num_defn }
###### Definition

For $S$ a [[small category]], a **[[crossed group|crossed $S$-group]]** is a [[presheaf]] $G : S^{op} \to Set$ equipped with

1. for each object $s \in S$ a group structure on $G_s$;

1. for all $s, r\in S$ a left $G_r$-[[action]] on the [[hom-set]] $S(s,r)$ ;

such that for all morphisms $\alpha : s \to r$ and $\beta : t \to s$ in $S$ and $g,h \in G_r$ we have

1. $g_*( \alpha \circ \beta)  = g_*(\alpha) \circ \alpha^*(g)_* \beta$;

1. $g_* (id_r) = id_r$;

1. $\alpha^* (g \cdot h) = h_*(\alpha)^*(g)\cdot \alpha^*(h)$;

1. $\alpha^*(e_r) = e_s$;

where $g_*$, $h_*$ denotes the group action and $\alpha^* : G_r \to G_s$ the presheaf map.

The **total category** $S G$ of an [[crossed group|crossed $S$-group]] $G$ is the category with the same objects as $S$, and with morphisms $r \to s$ being pairs $(\alpha, g) \in S(s,r)\times G_r$ and with composition defined by

$$
  (\alpha, g) \circ (\beta, h)
  =
  (\alpha \cdot g_*(\beta), \beta^*(g) \cdot h)
  \,.
$$

=--

([BM11, def. 2.1](#BergerMoerdijk11)).

+-- {: .num_defn }
###### Definition

If $S$ is equipped with a generalized Reedy structure, then an [[crossed group|crossed $S$-group]] $G$ is called **compatible** with that generalized Reedy structure if

1. the $G$-action respects $S^+$ and $S^-$;

1. if $\alpha : r \to s$ is in $S^-$ and $g \in G_s$ such that $\alpha^* (g) = e_r$ and $g_*(\alpha) = \alpha$, then $g = e_s$.

=--

+-- {: .num_example }
###### Example

The category $\Omega_{pl}$ of planar finite rooted [[trees]] is a strict [[Reedy category]]. The category $\Omega$ of non-planar finite rooted [[trees]] is the total category of an $\Omega_{pl}$-[[crossed group]] which to a planar tree $T$ assigns its group of non-planar automorphisms.

=--

+-- {: .num_prop }
###### Proposition

Let $S$ be a strict [[Reedy category]] and let $G$ be a compatible [[crossed group|crossed $S$-group]]. Then there exists a unique dualizabe generalized Reedy structure on $S G$ for which the embedding $S \hookrightarrow S G$ is a morphism of generalized Reedy categories.

=--

([BM11, prop. 2.10](#BergerMoerdijk11)).

## Related concepts

* A "more generalized" notion is a [[c-Reedy category]].
* A more specialized notion is an [[Eilenberg-Zilber category]].

## References

Cisinski's notion of generalized Reedy category appears as def 8.1.1 in

* [[Denis-Charles Cisinski]], _[[joyalscatlab:Les préfaisceaux comme type d'homotopie|Les préfaisceaux comme modèles des types d'homotopie]]_, Astérisque, Volume 308, Soc. Math. France (2006), 392 pages ([pdf](http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf))
 {#Cisinski}

The Berger-Moerdijk definition of generalized Reedy category appears in

* {#BergerMoerdijk11} [[Clemens Berger]], [[Ieke Moerdijk]], *On an extension of the notion of Reedy category*, Mathematische Zeitschrift **269** (2011) 977–1004 [[arXiv:0809.3341](http://arxiv.org/abs/0809.3341), [doi:10.1007/s00209-010-0770-x](https://doi.org/10.1007/s00209-010-0770-x)]

[[!redirects generalized Reedy categories]]
[[!redirects Berger-Moerdijk generalized Reedy category]]
[[!redirects Berger-Moerdijk generalized Reedy categories]]
[[!redirects Cisinski generalized Reedy category]]
[[!redirects Cisinski generalized Reedy categories]]


generalized Reedy model structure

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Model category theory
+--{: .hide}
[[!include model category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
 {:toc}

## 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 structure|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 $S$ be a (Berger-Moerdijk) [[generalized Reedy category]].
Let $\mathcal{C}$ be a category with small [[limits]] and [[colimits]].

### The model structure

+-- {: .num_defn #FirstNotation}
###### Definition

For every [[object]] $s \in S$, every [[functor]] $X : S \to \mathcal{C}$ and every [[natural transformation]] $\phi : X \to Y$

* $S^+(s)$ for the [[full subcategory]] of the [[slice category]] $S^+/s$ on the non-invertible morphisms into $s$;

* $S^-(s)$ for the [[full subcategory]] of the [[under category]] $s/S^-$ on the non-invertible morphisms out of $s$;

* write

  $$Latch_s(X) \coloneqq  {\lim_{\underset{(r \to s) \in S^+(s)}{\to}}} X(r)$$

  for the [[colimit]] of $X$ over $S^+(s)$, called the **latching object** of $X$ at $s$;

* write

  $$Match_s(X) \coloneqq {\lim_{\underset{(s \to r) \in S^-(s)}{\leftarrow}}} X(r)$$

  for the [[limit]] of $X$ over $S^-(s)$, called the **matching object** of $X$ at $s$.

* write

  $$X_s \coprod_{Latch_s(X)} Latch_s(Y) \to Y_s$$

  for the universal morphism induced from the morphism $L_s(X) \to X_s$, called the **relative latching map** of $\phi$ at $s$;

* write

  $$X_s \to Match_s(X) \prod_{Match_s(Y)} Y_s$$

  for the dual universal morphism, called the **relative matching map** of $\phi$ at $s$.

=--

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

+-- {: .num_remark}
###### Remark

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

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

=--

+-- {: .proof}
###### Proof

Since limits and colimits in [[presheaf categories]] are computed objectwise.

=--

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

+-- {: .num_defn #ReedyModelStructure}
###### Definition

Let $S$ be a (Berger-Moerdijk)-[[generalized Reedy category]].

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

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

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

* a **Reedy cofibration** if for each $s \in S$ the relative latching map

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

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

* a **Reedy fibration** if for each $s \in S$ the relative matching map

  $$
    X_s \to M_s(X) \prod_{M_s(Y)} Y_s
  $$

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

* a **Reedy weak equivalence** if for each $s \in S$ the morphism

  $$
    f_s : X_s \to Y_s
  $$

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

=--

### Degreewise latching and matching objects
 {#GlobalLatchingObjects}

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

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

+-- {: .num_defn #SecondNotation}
###### Definition

For $n \in \mathbb{N}$ write

* $S^+(n) = \coprod_{s \in S \atop d(s) = n} S^+(s)$;

* $d_ n : S^+((n)) \to S$ for the restriction of the [[domain opfibration]] to objects that are non-invertible morphisms in $S^+$ with codomain in degree $n$ and to morphisms whose codomain is invertible, i.e. to diagrams of the form

  $$
    \array{
       a &\stackrel{\in S^+}{\to}& b
       \\
       \downarrow^{\mathrlap{\in S^+}} && \downarrow^{\mathrlap{\in S^+}} & not\, invertible
       \\
       s & \stackrel{\simeq}{\to} & s' & deg = n
    }
    \,;
  $$

* $i_n : S^+(n) \hookrightarrow S^+((n))$ for the [[full subcategory]] inclusion on constant codomains;

* $G_n(S) \subset Core(S)$ for the [[groupoid]] of objects of degree $n$ and [[isomorphisms]] between them.

=--

+-- {: .num_prop #DiagramOfRestrictions}
###### Proposition

The above categories and functors arrange into a [[diagram]]

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

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]]

$$
  (cod_n)_! i_n^* \simeq i_n^* (cod_n)_!
  \,,
$$

where $(cod_n)_!$ denotes left [[Kan extension]] along $cod_n$.

=--

+-- {: .num_remark #RestrictedCodomainIsOpfibration}
###### Remark

The restricted [[codomain fibration|codomain opfibration]] $cod_n : S^+((n)) \to G_n$ is indeed still an [[opfibration]]: it is the [[Grothendieck construction]] of the [[pseudofunctor]]

$$
  S^+(-) : G_n(S) \to Cat
$$

$$
  s \mapsto S^+(s)
  \,.
$$

=--

+-- {: .num_defn #GlobalLatching}
###### Definition

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

* $X_n \coloneqq j_n^* X = X|_{G_n(S)} \in [G_n(S), \mathcal{C}]$;

* the **$n$th latching object** is

  $$
    Latch_n(X) \coloneqq (cod_n)! dom_n^* X \in [G_n(S), \mathcal{C}]
    \,.
  $$

* the **$n$th latching morphism**

  $$
    Latch_n(X) \to X_n
  $$

  is the [[adjunct]] to the canonical functor

  $$
    dom_n^* X \to (j_n cod_n)^* X
    \,.
  $$

=--

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

+-- {: .num_prop #GlobalLatchingContainsLocalLatchings}
###### Proposition

For $s \in Obj(S)_n$ we have

$$
  Latch_n(X) : s \mapsto Latch_s(X)
$$

and the component of the $n$th latching morphism on $s$ is the canonical $Latch_s(X) \to X(s)$.

=--

+-- {: .proof}
###### Proof

By remark \ref{RestrictedCodomainIsOpfibration}
$(cod_n)_!$ is the left [[Kan extension]] along an [[opfibration]]. By a standard fact (see [here](http://ncatlab.org/nlab/show/Kan+extension#AlongFibrations) at _[[Kan extension]]_) these are computed at any object by the colimit over the fiber over that object.

By definition, that fiber is

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

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

So

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

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

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

and

$$
  Match_n X \coloneqq (dom_n)_* cod_n^* X
  \,,
$$

where $(dom_n)_*$ is the right [[Kan extension]] along $dom_n$.

### Skeleta and coskeleta
 {#SkeletaAndCoskeleta}

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

For $n \in \mathbb{N}$, write

$$
  t_n : S_{\leq n}  \hookrightarrow S
$$

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

+-- {: .num_defn #SkeletaByAdjunction}
###### Definition

Left and right [[Kan extension]] along $t_n$ defines an [[adjoint triple]]

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

The induced [[monads]]

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

are the **$n$-[[skeleton]]** and **$n$-[[coskeleton]]** functors, respectively.

Define for all $X \in [S, \mathcal{C}]$ the notation $sk_{-1}X$ to denote the [[initial object]] and $cosk_{-1}X$ the [[terminal object]].

=--

+-- {: .num_remark }
###### Remark

Here $(t_n)_!$ and $(t_n)_*$ are indeed [[full and faithful functors]], as indicated.

=--

+-- {: .proof}
###### Proof

Since $t_{n-1}$ is a [[full and faithful functor]], so is its left Kan extension
(see [here](Kan+extension#LeftKanOnRepresentables) at _[[Kan extension]]_). Moreover in an [[adjoint triple]] the leftmost functor is full and faithful if and only if the rightmost one is.

=--

The $((t_n)_! \dashv t_n^*)$-[[unit of an adjunction|counit]] and the $(t_n^* \dashv (t_n)_*)$-unit induces [[natural transformations]]

$$
  sk_n \to id
$$

$$
  id \to cosk_n
  \,.
$$

+-- {: .num_lemma #LatchingIsSkeleton}
###### Lemma

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

$$
  Latch_n(X) \simeq (sk_{n-1}(X))_n
$$

$$
  Match_n(X) \simeq (cosk_{n-1}(X))_n
  \,.
$$

=--

This is ([Ber-Moer, lemma 6.2](#BergerMoerdijk)).

+-- {: .proof}
###### Proof

Observe that for any $s \in S$ of degree $n$, the canonical inclusion

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

into the [[comma category]] is a [[cofinal functor]]:

* for $d \to s$ any object in $t_{n-1}/s$ it factors essentially uniquely as $d \stackrel{\in S^-}{\to} \stackrel{\in S^+}{\to} s$, and hence the [[comma category ]] $d/i_s$ is non-empty;

* similarly, since every morphism factors essentially uniquely in $S$, there is a zig-zag between any two objects in $d / i_s$ 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:

$$
  \begin{aligned}
    sk_{n-1}(X)_n(s)
    & \coloneqq
    (j_n^* (t_{n-1})_! t_{n-1}^* X )(s)
    \\
    & \simeq
    ((t_{n-1})_! t_{n-1}^* X )(s)
    \\
    & \simeq
    \lim_{\underset{t_{n-1}/ s }{\to}} X \circ t_{n-1}
    \\
    & \simeq
    \lim_{\underset{S^+(s)}{\to}} X \circ dom_n
    \\
    & \simeq
    Latch_n X
  \end{aligned}
  \,.
$$

=--

+-- {: .num_lemma #coSkeletonTower}
###### Lemma

The tower of inclusions

$$
  S_{0} \hookrightarrow S_{\leq 1} \hookrightarrow \cdots \hookrightarrow S_{\leq n-1}
  \stackrel{q_{n-1}}{\hookrightarrow} S_{\leq n} \hookrightarrow \cdots
$$

induces towers of [[natural transformations]]

$$
  \emptyset \to sk_0 X \to sk_1 X \to sk_2 X \to \cdots \to X
  \,,
$$

and

$$
  X \to \cdots \to cosk_2 X \to cosk_1 X \to cosk_0 X \to *
  \,,
$$

that exhibit $X$ as the [[colimit]] of its skeleton tower and as the [[limit]] of its coskeleton tower.

=--

This is ([Ber-Moer, lemma 6.3](#BergerMoerdijk)).

+-- {: .proof}
###### Proof

The morphisms in the tower come from the [[unit of an adjunction|adjunction units and counits]]: the morphism

$$
  sk_n X \to sk_{n+1} X
$$

is

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

Therefore a [[cocone]] under this morphism

$$
  \array{
    sk_{n-1} X &&\to& sk_n X
    \\
    & \searrow && \swarrow
    \\
    && Y
  }
$$

is equivalently a diagram

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

which in turn is equivalently just a morphism $t_n^* X \to t_n^* Y$. So
a cocone under the whole tower is an object $Y$ equipped for each $n$ with
a morphism $t_n^* X \to t_n^* Y$. Clearly $X$ itself is the [[initial object|inital]]
such object.

=--

+-- {: .num_lemma #IdempotenceOfCoSkeleta}
###### Lemma

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

$$
  sk_k sk_l \simeq sk_l sk_k \simeq sk_k
$$

and

$$
  cosk_k cosk_l \simeq cosk_l cosk_k \simeq cosk_k
  \,.
$$

=--

+-- {: .proof}
###### Proof

With the above notation we have $t_k = t_l \circ q_k$. Therefore for instance

$$
  sk_k sk_l X
  \simeq
  (t_l)_! (q_k)_! q_k^* t_l^* (t_l)_! t_l^* X
  \,.
$$

Since the left adjoints here are [[full and faithful functors]] we have $t_l^* (t_l)_! \simeq id$ and hence

$$
  \cdots \simeq
  (t_l)_! (q_k)_! q_k^* t_l^* (t_l)_! t_l^* X
  \simeq
  sk_k X
  \,.
$$

Similarly for all the other cases.

=--

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

+-- {: .num_lemma #SkeletalExtension}
###### Lemma

Given

* an object $X_{\leq (n-1)} \in [S_{\leq (n-1)}, \mathcal{C}]$;

* and an object $X_n \in [G_n(S), \mathcal{C}]$;

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

* $X_{\leq (n-1)} = t^*_{n-1} X_{\leq n}$

* $X_n = j_n^* X$;

are in bijection with choices of morphisms

$$
  Latch_n(X_{\leq (n-1)}) \to X_n \to Match_n(X_{\leq (n-1)})
$$

in $[G_n(S), \mathcal{C}]$.

Accordingly, given morphisms $f_{\leq (n-1)} : X_{\leq (n-1)} \to Y_{\leq (n-1)}$ and $f_n : X_n \to Y_n$, then choice of extensions to a morphism $f_{\leq n} : X_{\leq n} \to Y_{\leq n}$ are in bijection with choices of vertical morphisms in commuting diagrams

$$
  \array{
     Latch_n(X_{\leq (n-1)}) &\stackrel{Latch_n(f_{\leq (n-1)})}{\to}& Latch_n(Y_{\leq (n-1)})
     \\
     \downarrow && \downarrow
     \\
     X_n &\stackrel{f_n}{\to}& Y_n
     \\
     \downarrow && \downarrow
     \\
     Match_n(X_{\leq (n-1)})
     &\stackrel{Match_n(f_{\leq (n-1)})}{\to}&
     Match_n(Y_{\leq (n-1)})
  }
  \,.
$$

=--

+-- {: .proof}
###### Proof

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

=--

## Proof of the model category axioms

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

+-- {: .num_theorem }
###### Theorem

Def. \ref{ReedyModelStructure} indeed defines a [[model category|model structure]].

=--

+-- {: .proof}
###### Proof

It is clear that $[S,\mathcal{C}]$ 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](#Lifting)_ and _[Factorization](#Factorization)_.

=--

### Lifting
 {#Lifting}

We work with the "global" latching objects from [above](#GlobalLatchingObjects).

+-- {: .num_remark }
###### Remark

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

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

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

=--

+-- {: .proof}
###### Proof

The pushout in the presheaf category $[G_n(S), \mathcal{C}]$ is computed objectwise, so that the component of the $n$th relative latching morphism at $s \in S$ is the relative latching morphism at $s$, by prop. \ref{GlobalLatchingContainsLocalLatchings}.

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

=--

+-- {: .num_lemma #LiftingLemma}
###### Lemma

In the generalized Reedy structure, def. \ref{ReedyModelStructure}, the following holds.

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

* Reedy cofibration have the left lifting property against acyclic Reedy fibrations $g$ with the special property that all $Latch_n(g)$ are objectwise acyclic fibrations.

=--

This is ([Ber-Moer, lemma 5.2, lemma 5.4](#BergerMoerdijk)).

+-- {: .proof}
###### Proof

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

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

by stepwise constructing lifts in the skeletal filtration, lemma \ref{coSkeletonTower}.

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

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

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

$$
  \array{
    A_0 &\stackrel{}{\to}& Y_0
    \\
    \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{}}
    \\
    B_0 &\stackrel{}{\to}& X_0
  }
$$

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

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

$$
  \array{
    A_{\leq (n-1)} &\stackrel{}{\to}& Y_{\leq (n-1)}
    \\
    \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{}}
    \\
    B_{\leq (n-1)} &\stackrel{}{\to}& X_{\leq (n-1)}
  }
$$

has been found. By lemma \ref{LatchingIsSkeleton} this induces maps $Latch_n(B) \to Latch_n (Y)$ and $Match_n(B) \to Match_n(Y)$ from which we can build the commuting diagram

\[
  \label{MainLiftingDiagram}
  \array{
    A_n \coprod_{Latch_n A} Latch_n B
    &\to&
    Y_n
    \\
    \downarrow^{\mathrlap{v_n}} && \downarrow^{\mathrlap{w_n}}
    \\
    B_n &\to& X_n \times_{Match_n X} Match_n Y
  }
\]

in $[G_n(S), \mathcal{C}]$.

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

$$
  \array{
    Latch_n A &\to& Latch_n B &\to& Latch_n Y
    \\
    \downarrow && && \downarrow
    \\
    A_n &\to& &\to& Y_n
  }
$$

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

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

$$
  \array{
    A_n \coprod_{Latch_n A} Latch_n B
    &\to&
    Y_n
    \\
    \downarrow &\nearrow_{\gamma_n}&
    \\
    B_n
  }
$$

is

* a lift in

  $$
    \array{
       A_n &\to& Y_n
       \\
       \downarrow & \nearrow_{\mathrlap{\gamma_n}}
       \\
       B_n
    }
  $$

* such that it makes

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

  commute;

and in the bottom right the lift

$$
  \array{
     && Y_n
     \\
     & {}^{\mathllap{\gamma_n}}\nearrow & \downarrow
     \\
     B_n &\to& X_n \prod_{Match_n X} Match_n Y
  }
$$

is

* a lift in

  $$
    \array{
      && Y_n
      \\
      & {}^{\mathllap{\gamma_n}}\nearrow & \downarrow
      \\
      B_n &\to& X_n
    }
  $$

* such that it makes

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

  commute.

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

By assumption, the left vertical morphism in (eq:MainLiftingDiagram) is a cofibration in $[G_n, \mathcal{C}]_{proj/inj}$, 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 $\mathcal{C}$ over each $s \in S$.

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

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

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

=--

+-- {: .num_lemma #LatchingIntertwiner}
###### Lemma

Suppose $\phi : R \to S$ is a morphism of
[[generalized Reedy categories]] such that the induced square (see prop. \ref{DiagramOfRestrictions})

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

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

$$
  Latch_k(\phi^*(X)) \stackrel{\simeq}{\to} \phi_k^*(Latch_k X)
  \,,
$$

Given a Reedy category $S$, examples of $\phi$ satisfying this condition include

1. $\phi \coloneqq dom_n \circ i_n : S^+(n) \to S$;

1. $\phi \colon S_{\leq n} \hookrightarrow S$.

=--

This is ([Ber-Moer, lemma 4.4](#BergerMoerdijk)).

+-- {: .proof}
###### Proof

The pullback square is part of the diagram

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

whose rows define, by prop. \ref{DiagramOfRestrictions}, 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](http://ncatlab.org/nlab/show/Beck-Chevalley+condition#PullbacksOfOpfibrations)), we find the intertwining isomorphism as follows:

$$
  \begin{aligned}
     Latch_k \phi^* X
     & \simeq
     (cod^R_k)_! (dom^R_k)^* \phi^* X
     \\
      & \simeq
      (cod^R_k)_! (\phi^+_k)^* (dom^S_k)^* X
     \\
     & \simeq
     (\phi_k)^* (cod^R_k)_! (dom^S_k)^* X
     \\
     & \simeq
     (\phi_k)^*
     Latch_k X
  \end{aligned}
  \,.
$$

Now concerning the two examples.

By definition we have

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

where on the right the vertical sequences in $S$ indicate objects in $(S^+(m))^+((k))$ 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. \ref{LiftingLemma} is in fact automatic.

+-- {: .num_lemma #ExtraPropertyOfAcyclicCofibrations}
###### Lemma

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

=--

This is ([Ber-Moer, lemma 5.3](#BergerMoerdijk)).

+-- {: .proof}
###### Proof

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

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

$$
  \array{
    i_n^* Latch_n A &\to& Y
    \\
    \downarrow^{\mathrlap{i_n^* Latch_n f}} && \downarrow^{g}
    \\
    i_n^* Latch_n B &\to& X
  }
$$

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

$$
  i^*_n Latch_n \coloneqq i_n^* cod_! dom_n^* \simeq cod_! i_n^* dom_n^*
$$

such a filler is equivalently a filler in

\[
  \label{SomeLiftingDiagram}
  \array{
     i_n^* dom_n^* A &\to& cod_n^* Y
     \\
     {}^{\mathllap{i_n^* dom_n^*}}{}\downarrow && \downarrow
     \\
     i_n^* dom_n^* B &\to& cod_n^* X
  }
\]

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

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

Moreover, $i_n^* dom_n^* f$ is a Reedy weak equivalence in this structure, since all its objects have degree $\lt n$. It is now sufficient to show that the assumptions of lemma \ref{LiftingLemma} are satisfied over $S^+(n)$, to obtain the lift.

By lemma \ref{LatchingIntertwiner} the functor

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

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

$$
  (dom_n i_n)^* A \coprod_{Latch_k (dom_n i_n)^* A} Latch_k (dom_n i_n)^* k \to (dom_n i_n)^* B
$$

and the morphism

$$
  (dom_n i_n)_k^*\left(
    A_k \coprod_{Latch_k(A)} Latch_k(B)
    \to
    B_k
  \right)
  \,.
$$

Since $(dom_n i_n)_k$ is a [[faithful functor]] between groupoids, $(dom_n i_n)_k^*$ 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 $(dom_n i_n)^*(f)$ is a Reedy cofibration.
Similarly, since $Latch_k(f)$ is an acyclic cofibration by induction hypothesis, so is $Latch_k((dom_n i_n)_k^* f)$. This way the assumption of lemma \ref{LiftingLemma} are checked for (eq:SomeLiftingDiagram) and so we do have a lift.

=--

+-- {: .num_lemma #ExtraPropertyOfAcyclicFibrations}
###### Lemma

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

=--

This is ([Ber-Moer, lemma 5.4](#BergerMoerdijk)).

+-- {: .proof}
###### 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 $S_-$ as epimorphisms.)

=--

We can finally conclude:

+-- {: .num_prop }
###### Proposition

In the generalized Reedy model structure, def. \ref{ReedyModelStructure},

* acyclic cofibrations have the [[left lifting property]] against fibrations;

* cofibrations have the [[left lifting property]] against acyclic fibrations.

=--

+-- {: .proof}
###### Proof

By lemma \ref{ExtraPropertyOfAcyclicCofibrations} every acyclic Reedy cofibration induces a weak equivalence under $Latch_n$. By lemma \ref{LiftingLemma} this implies the left lifting property against Reedy fibrations. Dually for the second statement.

=--

### Factorization
 {#Factorization}

We demonstrate the factorization axiom in the Reedy model structure, def. \ref{ReedyModelStructure}.

+-- {: .num_lemma #AcyclicCoFibrationsByLatchingMatching}
###### Lemma

A morphism $f : X \to Y$ in $[S, \mathcal{C}]$ is both a cofibration and a weak equivalence according to def. \ref{ReedyModelStructure} precisely if the $n$th relative latching morphism (def. \ref{GlobalLatching})

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

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

Dually, a morphism $f : X \to Y$ in $[S, \mathcal{C}]$ is both a fibration and a weak equivalence according to def. \ref{ReedyModelStructure} precisely if the $n$th relative matching morphism (def. \ref{GlobalLatching})

$$
  X_n \to Y_n \prod_{Match_n(X)} Match_n(Y)
$$

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

=--

This is ([Ber-Moer, prop. 5.6](#BergerMoerdijk)).

+-- {: .proof}
###### Proof

By definition, the morphisms $f_n : X_n \to Y_n$ factor as

$$
  f_n :
  X_n
   \stackrel{u_n}{\to}
  X_n
  \coprod_{Latch_n(X)} Latch_n (Y)
   \stackrel{v_n}{\to}
  Y_n
  \,.
$$

If now $f$ is an acyclic Reedy cofibration, then by lemma \ref{ExtraPropertyOfAcyclicCofibrations} the morphism $Latch_n(X) \to Latch_n(Y)$ is over each object an acyclic cofibration in $\mathcal{C}$ and then so is $u_n$ above, being the pushout of this morphism. It follows by [[two-out-of-three]] that also $v_n$ is a weak equivalence for all $n$.

Conversely, assume that all $v_n$ here are weak equivalences. We show by induction on $n$ that then also the $u_n$ are weak equivalences, and hence that $f$ is a Reedy weak equivalence.

For $n = 0$ we have $u_0 = id$, and so this case is satisfied. So assume now that all $u_k$ for $k \lt n$ are weak equivalences. Then the assumptions of
lemma \ref{ExtraPropertyOfAcyclicCofibrations} are again satisfied, and it follows that $Latch_n(f) : Latch_n(X) \to Latch_n(Y)$ is over each object an acyclic cofibration. Accordingly, so is $u_n$, being its pushout. Therefore, by induction, all $u_n$ are, in particular, weak equivalences.

The argument for fibrations is dual to this.

=--

+-- {: .num_prop }
###### Proposition

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

=--

This is ([Ber-Moer, page 18](#BergerMoerdijk)).

+-- {: .proof}
###### Proof

Let $f : X \to Y$ be any morphism in $[S, \mathcal{C}]$.
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 $t_n^* : [S, \mathcal{C}] \to [S_{\leq n}, \mathcal{C}]$. The other case (cofibration followed by acyclic fibration) works dually.

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

$$
  f_0 : X_0 \stackrel{\simeq}{\to} A_0 \to Y_0
  \,.
$$

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

$$
  f_{\leq (n-1)}
 :
  X_{\leq (n-1)}
  \stackrel{\simeq}{\to}
  A_{\leq (n-1)}
  \to
  Y_{\leq (n-1)}
$$

has been found. This induces the commutative diagram

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

This diagram in turn induces a morphism

$$
  X_n \coprod_{Latch_n(X)} Latch_n(A)
  \to
  Y_n \prod_{Match_n(Y)} Match_n(A)
$$

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

$$
  X_n \coprod_{Latch_n(X)} Latch_n(A)
   \stackrel{\simeq}{\to}
   A_n
  \to
  Y_n \prod_{Match_n(Y)} Match_n(A)
$$

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

By lemma \ref{SkeletalExtension}, the "righmost component" of this data defines an extension of $A_{\leq (n-1)}$ to $A_{\leq n}$. The "leftmost component" defines a factorization of $f_n$ and the "middle component" says that this consistently extends the previously obtained factorization of $f_{\leq n}$.  With this, finally the two morphisms say that this new factorization is by an acyclic Reedy cofibration (by lemma \ref{AcyclicCoFibrationsByLatchingMatching}) followed by a Reedy fibration (by definition).

=--

## Properties

### Homotopy skeletal filtration and coskeleton tower
 {#HomotopySkeletalFiltration}

We discuss conditions on an object $X$ such that the skeletal/coskeletal towers
discussed [above](#SkeletaAndCoskeleta) are "homotopy good" in that they exhibit $X$ not just as the colimit/limit over the tower, but as the [[homotopy colimit]]/[[homotopy limit]].

+-- {: .num_lemma #QuillenPropertyOfCoSkeleta}
###### Lemma

For all $n \in \mathbb{N}$

1. The left Kan extension

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

   is a [[left Quillen functor]].

1. The right Kan extension

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

   is a [[right Quillen functor]].

1. The restriction functor

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

   is both a left and a right Quillen functor.

=--

This is ([Ber-Moer, lemma 6.4](#BergerMoerdijk)).

+-- {: .num_prop #ReedyCoFibrantImpliesCoFibrantTowers}
###### Proposition

1. If $X \in [S, \mathcal{C}]$ is Reedy cofibrant according to def. \ref{ReedyModelStructure}, then all $sk_n X$ are Reedy cofibrant and the canonical morphisms $sk_k X \to sk_l X$ are Reedy cofibrations.

1. If $X \in \mathcal{C}$ is Reedy fibrant according to def. \ref{ReedyModelStructure} then all $cosk_n X$ are Reedy fibrant and the canonical morphisms $cosk_l X \to cosk_k X$ are Reedy fibrations.

=--

This is ([Ber-Moer, prop. 6.5](#BergerMoerdijk)).

+-- {: .proof}
###### Proof

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

By lemma \ref{IdempotenceOfCoSkeleta} it is sufficient to consider the
case $sk_n X \to sk_\infty X = X$.

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

$$
  \array{
    Latch_k(sk_n X) &\to& Latch_k (X)
    \\
    \downarrow && \downarrow
    \\
    sk_n(X)_k &\to& X_k
  }
  \,.
$$

For $k \leq n$,  the horizontal morphisms are both isomorphisms. Because by lemma \ref{LatchingIsSkeleton} the top morphism is

$$
  (sk_{k-1} sk_n(X))_k \to (sk_{k-1} X)_k
$$

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

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 $k \gt n$ the left vertical map is an isomorphism, so that the relative latching morphism in this case is $Latch_k(X) \to X_k$, which is a cofibration by the assumption that $X$ is Reedy cofibrant.

Finally, that $sk_n X$ is cofibrant follows directly from lemma \ref{QuillenPropertyOfCoSkeleta}.

=--

### Relation to other model structures

(...)

## Example

* Every ordinary [[Reedy category]] is a generalized Reedy category, and in this case the above model structure reduces to the traditional [[Reedy model structure]].

* The [[model structure for dendroidal complete Segal spaces]] is a [[Bousfield localization of model categories|left Bousfield localization]] of the generalized Reedy model structure over the [[tree category]].

## Related concepts

* [[Joyal-Tierney calculus]]

## References

* [[Clemens Berger]] and [[Ieke Moerdijk]], _On an extension of the notion of Reedy category_, Math. Z. (2008) ([arXiv:0809.3341](http://arxiv.org/abs/0809.3341))
 {#BergerMoerdijk}


generalized ring

In the terminology of [[Nikolai Durov]], a __generalized ring__ is an [[algebraic monad|algebraic (i.e. finitary) monad]] that is [[commutative monad|commutative]].

Durov introduced [[spectrum (geometry)|spectra]] of generalized rings, generalizing the Grothendieck [[prime spectrum]]. This enabled him to develop _generalized algebraic geometry_, a single framework whose special cases include versions of [[tropical geometry]], [[Arakelov geometry]] and [[absolute geometry]] over $F_1$. The categories of [[quasicoherent sheaf|quasicoherent sheaves]] in that setup are not necessarily [[abelian category|abelian]]; for that reason, Durov develops a version of [[homotopical algebra]] in a [[topos]], to be able to replace the usual [[homological algebra]] for abelian sheaves with nonabelian derived functors following Quillen. In his version of homotopical algebra, [[model categories]] are however replaced by [[pseudomodel stacks]]. The ultimate goal is to set a natural framework for a statement and proof of the arithmetic Riemann--Roch theorem of Bismut--Gillet--Soulé and Faltings in a manner parallel to the Grothendieck--Riemann--Roch theorem.

*  N. Durov, New approach to Arakelov geometry, University of Bonn, PhD thesis, 2007; [arXiv:0704.2030](http://www.arxiv.org/abs/0704.2030).

[[!redirects generalized rings]]


generalized scheme

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A general idea of [[Alexander Grothendieck]] was that to study a geometry more general than [[scheme]]s instead of the gluing of [[affine scheme]]s as [[ringed space]]s, one glues the [[functors of points]]; hence **a [[space]] is simply a [[sheaf]] of sets on some [[site]] $Loc$ of local models with a [[Grothendieck topology]]** $\tau$ on it.

An algebraic [[scheme]] $X$ is a [[ringed space]] that is locally isomorphic to an [[affine scheme]]. Alternatively (see Gabriel-Demazure), it is a presheaf of sets on $Aff=CommRing^{op}$ locally representable in Zariski topology on $Aff$. The second approach [[Alexander Grothendieck]] calls [[functor of points]] approach.

To recall the equivalence between the two points of view, every scheme $X$ gives rise to a representable [[presheaf]] on the formal dual of commutative rings

$$
  X(-) : CRing \to Set
$$

$$
  A \mapsto Hom_{Scheme}(Spec A, X)
$$

and this is a [[sheaf]] with respect to Zariski [[Grothendieck topology]] on $Aff$. Sheaves in any other fixed *subcanonical* topology $\tau$ on $Aff$ are called $\tau$-locally affine spaces. The usual schemes are obtained for $\tau=Zariski$ and $Loc=Aff$. Algebraic spaces are another example. In other fields like analytic spaces, sheaves on other categories of local models $Loc$ instead of $Aff$ are considered in classical works.

In general various generalizations which do not have exactness properties of Zariski coverings or [[étale (∞,1)-site|étale coverings]], are usually among algebraic geometers called generalized _spaces_ rather than (generalized) _schemes_; thus the terminology almost scheme is OK because though the local objects are more general the exactness properties are basically the same (similarly for derived schemes of Toen et al. noncommutative schemes of Rosenberg etc.).

There are various way to generalize the scope of the functor of points approach.

There are many generalizations of [[schemes]], some are even called by their respective authors generalized schemes (e.g. Lurie, Durov). Deligne in [[Catégories Tannakiennes]] suggested algebraic geometry in arbitrary symmetric monoidal category.
Aspects of toric geometry and the foundations of the geometry over a field of one element (Smirnov-Kapranov, Dietmar, Connes...) can be founded using structure sheaves of monoids, not rings. Another example is tropical geometry. Rings are sometimes noncommutative (e.g. D-schemes of Beilinson); the underlying topological space can be replaced by a site, locale, topos or a non-distributive lattice by localizations. Usual commutative unital rings suffice for manifolds, rigid analytic spaces, schemes, formal schemes and so on. The emphasis in Lurie is to categorify the space and to take the homotopy version of a ring, restating a formalism fitting the [[derived algebraic geometry]], mainly of Simpson's school.

Several different definitions by several authors exist.

## Locally affine structured (∞,1)-toposes

In

* [[Jacob Lurie]], _[[Structured Spaces]]_ is considered the
definition discussed at

* [[locally representable structured (infinity,1)-topos]].

* [[E-∞ scheme]]

## Generalized schemes of Durov

N. Durov replaces the commutative rings by commutative [[algebraic monad]]s (aka generalized rings) in sets and defines spectra in that context, and glues them together. This way he defines what he calls __generalized schemes__: in a nutshell generalized schemes are schemes glued from affine spectra of generalized rings. The corresponding category of quasicoherent $\mathcal{O}$-modules is not abelian in general. See also the separate entry [[generalized scheme after Durov]].

## Other generalized schemes

O. Gabber considers replacing rings by *almost rings*, this results in the theory of [[almost schemes]].

One should note that the Grothendieck school has occasionally studied ringed sites where ring is not required to be commutative and considered quasicoherent sheaves and cohomology in that context. D-schemes of Beilinson are an example where this formalism is useful.

Rosenberg considers generalized relative schemes as categories over an arbitrary base category with a relatively affine cover satisfying some exactness conditions. The scheme as a category is in fact abstracting the category of quasicoherent sheaves over some generalized scheme. Rosenberg calls the Zariski version of that formalism [[noncommutative scheme]]; some other versions of locally affine spaces can be also relativized.

[[rigid analytic geometry|Rigid analytic geometry]] is featuring locally affinoid spaces (affinoid spaces are spectra of Banach algebras over complete ultrametric fields which belong to a special class called [[affinoid algebra]]s; Berkovich spectra are most often used) in so-called G-topology.

[[!redirects generalized schemes]]


generalized scheme">nLab:generalized


generalized Scherk-Schwarz reduction

[[!redirects generalised Scherk-Schwarz reduction]]
[[!redirects generalised Scherk-Schwarz reduction]]
+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Duality
+--{: .hide}
[[!include duality - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _generalized Scherk-Schwarz reduction_ is a [[dimensional reduction]] combined with a [[duality in string theory|duality]] twist, such as for [[exceptional field theory]] and [[double field theory]].

Consider an [[exceptional field theory]] on a certain extended space, i.e. a [[smooth manifold]] locally [[isomorphism|isomorphic]] to $U \times R$, where $R$ is the underlying [[vector space]] of the [[fundamental representation]] of some [[duality in string theory|duality]] [[group]] $G$ (e.g. [[U-duality]] or [[T-duality]]). Let us call $x$ the local coordinates of the base manifold $U$ and $(y,\tilde{y})$ the ones of the [[fiber]] $R$, called _internal manifold_.

A _generalized Scherk-Schwarz reduction_ is performed by introducing twisting $G$-valued matrices $E^{A}_{\;\;M}(y,\tilde{y})$ and by encoding all the dependence of the fields on the coordinates of the internal manifold with the following ansatz:

$$ T_{M_1 \dots M_n}(x,y,\tilde{y}) = E^{A_1}_{\;\;M_1}(y,\tilde{y}) \cdots E^{A_n}_{\;\;M_n}(y,\tilde{y}) T_{A_1\dots A_n}(x) $$

on any generalized [[tensor field]] $T_{M_1 \dots M_n}$ on the extended space. The $G$-valued matrices $E^{A}_{\;\;M}(y,\tilde{y})$ can be seen as generalized [[frame fields]] for the internal manifold.

## Examples

### Double field theory on the torus

Consider a [[double field theory]] on the torus $T^{2n}$, where the [[T-duality]] group $O(n,n)$ acts in the [[fundamental representation]]. We are in the case where the _external space_ is just a point. Let the doubled metric be

$$ \mathcal{H}_{M N} = \begin{pmatrix} g -B g^{-1}B & B g^{-1} \\ -g^{-1}B  &  g^{-1}\end{pmatrix} $$

Now, if we define $O(n,n)$-valued matrices

$$ E^A_{\;\;M} = \begin{pmatrix}e^a_{\mu} & 0 \\ e^\rho_a B_{\rho\mu} & e^\mu_a \end{pmatrix} $$

we can immediately write the generalized Scherk-Schwarz reduction over the point by

$$ \mathcal{H}_{M N}(y,\tilde{y}) = E^{A_1}_{\;\;M_1}(y,\tilde{y}) \cdots E^{A_n}_{\;\;M_n}(y,\tilde{y}) \mathcal{H}_{A B} $$

where $\mathcal{H}_{A B} = \mathrm{diag}(\delta_{a b},\,\delta^{a b})$. This simple example captures the analogy with ordinary [[frame fields]].

## Related concepts

* [[doubled geometry]], [[double field theory]]

* [[exceptional field theory]]

* [[gauged supergravity]]

## References

### General

The original [[Scherk-Schwarz mechanism]]:

* {#ScherkSchwarz89} [[Joël Scherk]], [[John Schwarz]], _How to get masses from extra dimensions_, in: Supergravities in Diverse Dimensions, pp. 1282-1309 (1989)  ([doi:10.1142/9789814542340_0083](https://doi.org/10.1142/9789814542340_0083))

The [[duality in string theory|duality]]-twisted generalized version:

* [[Atish Dabholkar]], [[Chris Hull]], _Duality Twists, Orbifolds, and Fluxes_, JHEP 0309:054, 2003 ([arXiv:hep-th/0210209](https://arxiv.org/abs/hep-th/0210209))

* [[Chris Hull]], _A Geometry for Non-Geometric String Backgrounds_, JHEP 0510:065, 2005 ([arXiv:hep-th/0406102](https://arxiv.org/abs/hep-th/0406102))

### In M-theory

A lift of [[D8-branes]] to [[M-theory]] [[M-branes]] by [[generalized Scherk-Schwarz reduction]], relating to [[D7-branes]] in [[F-theory]], is proposed in

* {#Hull98} [[Chris Hull]], _Massive String Theories From M-Theory and F-Theory_, JHEP 9811:027, 1998 ([arXiv:hep-th/9811021](https://arxiv.org/abs/hep-th/9811021))

[[!redirects generalized Scherk-Schwarz mechanism]]


[[!redirects generalized second law of theormodynamics]]

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The standard [[second law of theormodynamics]] states that in a closed system of [[classical mechanics]] or [[quantum mechanics]] the (coarse-grained) [[entropy]] is non-decreasing with time.

This is statement is to be tacitly understood as applying to dynamics taking place on an a-priori fixed [[spacetime]] (a fixed "background"). If however one takes also [[gravity]] into account as part of the "system", then the statement of the second law becomes much more subtle.

The _generalized second law of theoremodynamics_ is an attempt to generalize the naive second law to [[general relativity]]. A central idea is that the [[Bekenstein-Hawking entropy]] of [[horizon]]s (such as of [[black hole]]s) has to be added to the naive entropy of the matter in the "universe".

## Related concepts

* gravitational entropy

  * [[Bekenstein-Hawking entropy]]

  * [[generalized second law of thermodynamics]]

  * [[black holes in string theory]]

  * [[holographic entanglement entropy]]

## References

A detailed discussion is in

* Aron Wall, _Ten proofs of the generalized second law_ ([arXiv:0901.3865](http://arxiv.org/abs/0901.3865))

* Aron Wall, _A proof of the generalized second law for rapidly-evolving Rindler horizons_ ([arXiv:1007.1493](http://arxiv.org/abs/1007.1493))

* Aron Wall, _A proof of the generalized second law for rapidly changing fields and arbitrary horizon slices_ ([arXiv:1105.3445](http://arxiv.org/abs/1105.3445))

The first paper critiques previous work.  The second proves the generalized 2nd law in a semiclassical framework on a background [[spacetime]] that has both boost and null translation symmetries.  The argument makes heavy use of the concept of [[relative entropy]].  The third paper generalizes the second one to a much wider class of spacetimes, assuming some axioms about the behavior of [[quantum field theory|quantum fields]] on these spacetimes.

To show that entropy increases, a crucial step in Wall's proof is to use the fact that the [[relative entropy]] obeys

$$
  S(\rho'|\sigma') \leq S(\rho|\sigma)
$$

where the [[state]]s $\rho'$ and $\sigma'$ are obtained by restricting the states $\rho$ and $\sigma$ to a subalgebra of the original algebra of observables.

[[!redirects generalized second law]]


generalized smooth algebras > history

< [[generalized smooth algebras]]
[[!redirects generalized smooth algebras -- history]]


generalized smooth space

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

# Generalised smooth spaces
* table of contents
{: toc}

## Idea

Generalised smooth spaces are, roughly speaking, generalisations of [[smooth manifolds]].  Their _raison d'etre_ is the following

> Manifolds are fantastic spaces.  It's a pity that there aren't more of them.

Many spaces that occur in mathematics aren't manifolds but one would like to be able to treat them as if they were manifolds; in particular, they should have some *smooth* structure that goes beyond mere topology.  By considering examples of these spaces and by considering what specifically one would like to do with or to them, it is possible to generalise the idea of a smooth manifold to encompass the new examples whilst preserving enough structure to retain the old tools.  There have been several such generalisations in recent mathematical history.  A (partial) list is below.

Often the examples of spaces that one would like to consider as manifolds are formed by applying a categorical construction to ordinary manifolds; such as limits, quotients, or function spaces.  This leads one to ask for the categorical properties of each of the resulting categories of generalised smooth spaces.

Another obvious question to ask is what tools and techniques can be extrapolated from smooth manifolds to generalised smooth spaces.  In addition, whilst some techniques have obvious generalisations there may be some hidden twists that are not apparent on just smooth manifolds.

## Examples

According to the general nonsense of [[space and quantity]], generalized smooth spaces may be determined by [[sheaf|sheaves]] on smooth test spaces, in which case we call them [[smooth spaces]] here, or by co-(pre)sheaves on test spaces, in which case we call them [[structured generalized spaces]] here.

* [[smooth space|Smooth spaces]]

  * [[Chen space|Chen spaces]] (called _differentiable spaces_ in Chen's works).
  * [[diffeological space|Diffeological spaces]]

    * [[differentiable stack|Differentiable stacks]]/[[Lie groupoid]]
      * [[orbifold|Orbifolds]]

  * [[∞-Lie groupoid]]

  * [[Kriegl and Michor's cartesian closed category of manifolds]]

* [[CartSp]]-[[structured (∞,1)-topos|Structured (∞,1)-topos]]

  * [[differential module|Differential modules]]
  * [[Smith space (generalized smooth space)|Smith spaces]]
  * [[stratifold|Stratifolds]]
  * [[polyfold|Polyfolds]]
  * [[derived smooth manifold|Derived smooth manifolds]]

* both

  * [[Froelicher space|Frölicher spaces]]

{#DiagramOfAdjointFunctors} The relationships between (some) of the categories can be sumarised by the following diagram of [[adjoint functors]]:

\begin{imagefromfile}
    "file_name": "Stacey-NotionsOfSmoothSpaceRelationships.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -30,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "(from [Stacey 2011](#Stacey11))"
\end{imagefromfile}

## Literature

* John Baez and Alexander Hoffnung, _Convenient Categories of Smooth Spaces_, Trans. Amer. Math. Soc., Vol. 363 No. 11 (2011), 5789-5825, doi:[10.1090/S0002-9947-2011-05107-X](https://doi.org/10.1090/S0002-9947-2011-05107-X), [arXiv:0807.1704](http://arxiv.org/abs/0807.1704), [blog](http://golem.ph.utexas.edu/category/2008/05/convenient_categories_of_smoot.html).

* Patrick Iglesias-Zemmour, _Diffeology_, Mathematical Surveys and Monographs **185**, AMS (2013), [publisher page](https://doi.org/10.1090/surv/185), ([book site](http://math.huji.ac.il/~piz/Site/TheBook-AMS.html)).

* Matthias Kreck, _Differential Algebraic Topology: From Stratifolds to Exotic Spheres_, Graduate Studies in Mathematics Volume **110**, AMS (2010)  ([publisher page](https://bookstore.ams.org/gsm-110)), ([author pdf](https://www.hcm.uni-bonn.de/fileadmin/user_upload/kreck-DA.pdf)).

* William Lawvere, _Taking categories seriously_, Reprints in Theory and Applications of Categories, No. 8 (2005) pp. 1-24 ([journal](http://www.tac.mta.ca/tac/reprints/articles/8/tr8abs.html)).

* David Spivak, _Quasi-smooth derived manifolds_, thesis (2010) ([pdf](https://web.archive.org/web/20221206173621/https://math.mit.edu/~dspivak/files/thesis1.pdf)).

* {#Stacey11} [[Andrew Stacey]], _Comparative Smootheology_, Theory and Applications of Categories, **25** 4 (2011) 64-117, [[tac:25-4](http://www.tac.mta.ca/tac/volumes/25/4/25-04abs.html)), [arXiv:0802.2225](http://arxiv.org/abs/0802.2225)]

* Martin Laubinger, _Differential Geometry in Cartesian Closed Categories of Smooth Spaces_ ([pdf](https://web.archive.org/web/20150920152120/http://etd.lsu.edu/docs/available/etd-02212008-165645/unrestricted/laubingerdiss.pdf))

* Alexander Hoffnung, _Smooth spaces: convenient categories for differential geometry_, ([pdf slides](https://web.archive.org/web/20100629002755/http://math.ucr.edu/~alex/goettingen.pdf))

* Alexander Hoffnung, _From Smooth Spaces to Smooth Categories_, ([pdf slides](https://web.archive.org/web/20110714114128/http://aix1.uottawa.ca/%7Escpsg/Fields09/alex.hoffnung.pdf))

There are also Hofer's [[polyfold|polyfolds]].

Concerning [[smooth ∞-stacks]] there is useful material in

* Daniel Dugger, _Sheaves and homotopy theory_ ([web](http://math.uoregon.edu/~ddugger/cech.html), [pdf](http://ncatlab.org/nlab/files/cech.pdf))

## Remarks

Dual to generalized smooth spaces are [[generalized smooth algebra|generalized smooth algebras]] of functions on them, according to the general lore of [[space and quantity]].

## Further discussion

We had extensive discussion of generalized smooth spaces at the $n$-Café:

 * _Comparative Smootheology_ ([I](http://golem.ph.utexas.edu/category/2008/01/comparative_smootheology.html), [II](http://golem.ph.utexas.edu/category/2008/04/comparative_smootheology_ii.html), [III](http://golem.ph.utexas.edu/category/2008/09/comparative_smootheology_iii.html)), [IV](http://golem.ph.utexas.edu/category/2009/04/comparative_smootheology_iv.html)

 * [_Convenient categories of smooth spaces_](http://golem.ph.utexas.edu/category/2008/05/convenient_categories_of_smoot.html)

 * [_Spivak on derived manifolds_](http://golem.ph.utexas.edu/category/2008/08/david_spivak_on_derived_manifo.html)

 * [_Bär on fiber integration in differential cohomology_](http://golem.ph.utexas.edu/category/2008/11/br_on_fiber_integration_in_dif.html)

[[!redirects generalized smooth space]]
[[!redirects generalized smooth spaces]]
[[!redirects generalised smooth space]]
[[!redirects generalised smooth spaces]]

[[!redirects generalized smooth structure]]
[[!redirects generalized smooth structures]]