proper geometric morphism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Topos Theory
+-- {: .hide}
[[!include topos theory - contents]]
=--
#### Compact objects
+--{: .hide}
[[!include compact object - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _compact topos_ is the generalization from [[topology]] to [[topos theory]] of the notion of _[[compact topological space]]_.

More generally, over a general [[base topos]], the notion of _proper geometric morphism_ is the generalization to morphisms between toposes of the notion of _[[proper map]]_ between [[topological spaces]].

## Definition
 {#Definition}

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

A [[sheaf topos]] $\mathcal{E}$ is called a **[[compact topos]]** if the [[direct image]] of the [[global section geometric morphism]] $\Gamma : \mathcal{E} \to Set$ preserves [[directed colimit|directed]] [[joins]] of [[subterminal objects]].

A [[geometric morphism]] $f : \mathcal{F} \to \mathcal{E}$ is called **proper** if it exhibits $\mathcal{F}$ as a [[compact topos]] over $\mathcal{E}$. (The [[stack semantics]] of $\mathcal{E}$ can be used to formalize this.)

=--

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

A topos is called **strongly compact** if $\Gamma$ commutes even with all [[filtered colimits]].

A geometric morphism $f : \mathcal{F} \to \mathcal{E}$ is called **tidy** if it exhibits $\mathcal{F}$ as a strongly compact topos over $\mathcal{E}$.

=--

([MV, p. 53](#MoerdijkVermeulen))

This are the first stages of a notion that in [[(∞,1)-topos theory]] continue as follows

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

Let $\kappa$ be a [[regular cardinal]] and $ -1 \leq n \leq \infty$. Then an [[(∞,1)-topos]] is _$\kappa$-compact of height $n$_ if the [[global section geometric morphism]] preserves $\kappa$-[[filtered (∞,1)-category|filtered]] [[(∞,1)-colimits]] of [[n-truncated]] objects.

Accordingly a geometric morphism is _$\kappa$-proper of height $n$_ if it exhibits a $\kappa$-compact of height $n$ $(\infty,1)$-topos over a [[base (∞,1)-topos]].

=--

In this terminology

* a topos _compact of height (-1)_ is the same as a _compact topos_;

* a topos _compact of height 0_ is the same as a _strongly compact topos_;

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

An [[n-coherent (∞,1)-topos]] is compact of height $n$ in the sense of def. \ref{StronglyCompactTopos}, this is ([[Rational and p-adic Homotopy Theory|Lurie XIII, prop. 2.3.9]]).

=--

## Properties

### Stability and closure properties

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

1. Any [[equivalence of categories|equivalence]] is proper and the class of proper maps is closed under composition.

1. If in the [[diagram]]

   $$
     \array{
       A&\xrightarrow{p}&B
       \\
       \downarrow^f&\swarrow^g
       \\
       C
    }
   $$

   $p$ is a [[surjective geometric morphism]] and $f$ is
   proper then so is $g$.

1. If $h$ is proper and $g$ is a [[geometric embedding]] then $p$ is proper.

1. Any [[hyperconnected geometric morphism]] is proper.

1. $f:F\to G$ is proper iff its [[localic reflection]] $Sh_G(X)\to G$ is, i.e. iff $X$ is a [[compact locale|compact]] [[internal locale]] in $G$.

1. If in a pullback square the bottom morphism is open and surjective and the left morphism is proper then so is the right.

=--

([VM, I.1, I.2](#MoerdijkVermeulen))

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

The [[pullback]] of a proper geometric morphism is again proper.

The pullback of a tidy geometric morphism is again tidy.

=--

([VM, theorem 5.8](#MoerdijkVermeulen))

### Properness and Beck-Chevalley conditions

A [[geometric morphism]] $f$ of toposes is said to satisfy the _stable_ (weak) Beck-Chevalley condition if any pullback of $f$ satisfies the (weak) [[Beck-Chevalley condition]] ((weak)BCC).

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

A map satisfies the stable weak BCC iff it is proper.

=--

([MV, Corollary I.5.9](#MoerdijkVermeulen))

### Compact sites
 {#CompactSites}

We discuss classes of [[sites]] such that their [[sheaf topos]] is a compact topos, def. \ref{CompactTopos} ([VM, I.5](#MoerdijkVermeulen)).

(...)

### Strongly compact sites
 {#StronglyCompactSites}

We discuss classes of [[sites]] such that their [[sheaf topos]] is a strongly compact topos, def. \ref{CompactTopos} ([VM, III.4](#MoerdijkVermeulen)).

(...)

## Examples

### Compact toposes

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

Let $\mathbf{H}$ be a [[topos]] and $X \in \mathbf{H}$ an [[object]]. If

*  $X$ is a "[[compact topological space]]-object" in that:

   for every set of morphisms $\{U_i \to X\}_{i \in I}$ such that $\coprod_{i \in I} U_i \to X$ is an [[effective epimorphism]], there is a [[finite set|finite subset]] $J \subset I$ such that $\coprod_{i \in J} U_i \to X$ is still an [[effective epimorphism]];

then

*  The [[slice topos]] $\mathbf{H}_{/X}$ is a compact topos, def. \ref{CompactTopos}.

=--

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

Beware that $X$ being a "compact topological space-object" is different from it being a [[compact object]] (the difference being that between compactness of height (-1) and height 0). For the latter case see prop. \ref{SliceOverCompactIsStronglyCompact} below.

=--

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

The [[terminal object]] of $\mathbf{H}_{/X}$ is the [[identity]] $id_X : X \to X$ in $\mathbf{H}$. A [[subterminal object]] of $\mathbf{H}_{/X}$ is a [[monomorphism]] $U \hookrightarrow X$ in $\mathbf{H}$.

The [[global section geometric morphism]] $\Gamma_X : \mathbf{H}_{/X} \to Set$ sends an object $[E \to X]$ to its set of [[sections]]

$$
  \Gamma_X([E \to X]) = \mathbf{H}(X, E) \times_{\mathbf{H}(X,X)} \{id_X\}
  \,.
$$

Therefore it sends all subterminal object in $\mathbf{H}_{/X}$ to the [[empty set]] except the terminal object $X$ itself, which is sent to the singleton set.

So let $X$ now be a compact-topological-space-object and $U_\bullet : I \to \mathbf{H}_{/X}$ is directed system of subterminals.

If their union $\vee_i U_i$ does not cover $X$, then $\Gamma_X(\vee_i U_i) = \emptyset$. But then also none of the $U_i$ can be $X$ itself, and hence also $\Gamma_X(U_i) = \emptyset$ for all $i \in I$ and so $\vee_i \Gamma_X(U_i) = \emptyset$. On the other hand, if $\vee_i U_i  = X$ then the $\{U_i \to X\}_{i \in I}$ form a cover, hence then by assumption there is a finite subset $\{U_i \to X\}_{i \in J}$ which still covers. By the assumption that the system $U_\bullet$ is a [[directed set]] it also contains the union $X = \vee_{i \in J} U_i$. Therefore $\vee_{i \in I} \Gamma_X(U_i) = \Gamma_X(X) = *$ is the singleton, as is $\Gamma_X(\vee_{i \in I} U_i) = \Gamma_X(X)$. So $\Gamma_X$ preserves directed unions of subterminals and hence $\mathbf{H}_{/X}$ is a compact topos.

=--

### Strongly compact toposes
 {#StronglyCompactToposes}

The following propositions say in summary that

1. the _[[petit topos]]_ over a [[compact topological space]] that is also [[Hausdorff topological space|Hausdorff]] is strongly compact.

1. the _[[gros topos]]_ over a [[compact object]] is strongly compact.

See also ([VM, III.1](#MoerdijkVermeulen)).

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

Examples of strongly compact toposes $\mathcal{E}$, def. \ref{StronglyCompactTopos}, include the following.

1. Every [[coherent topos]] is strongly compact.

1. The [[sheaf topos]] over a [[compact topological space|compact]] [[Hausdorff space|Hausdorff]] [[topological space]] is strongly compact.

=--

([MV, Examples III.1.1](#MoerdijkVermeulen))

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

Let $\mathbf{H}$ be a topos over [[Set]] and $X \in \mathbf{H}$ an object. Then the following are equivalent

1. $X$ is a [[compact object]] (in the sense that the [[hom functor]] $\mathbf{H}(X,-)$ preserves [[filtered colimits]])

1. the [[slice topos]] $\mathbf{H}_{/X}$ is strongly compact, def. \ref{StronglyCompactTopos}.

=--

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

The [[direct image]] $\Gamma_X$ of the [[global section geometric morphism]]

$$
  ((-) \times X \dashv  \Gamma_X) : \mathbf{H}_{/X}
    \stackrel{\overset{(-) \times X}{\leftarrow}}{\underset{\mathbf{\Gamma}_X}{\to}}
  \mathbf{H}
   \stackrel{\overset{\Delta}{\leftarrow}}{\underset{\mathbf{H}(*,-)}{\to}}
  Set
$$

is given by the [[hom functor]] out of the [[terminal object]]. The terminal object in $\mathbf{H}_{/X}$ is the [[identity]] [[morphism]] $id_X : X \to X$. So the terminal geometric morphism  takes any $[E \to X]$ in $\mathbf{H}_{/X}$ to the set of [[sections]], given by the [[pullback]] of the [[hom set]] along the inclusion of the [[identity]]

$$
  \Gamma_X([E \to X]) = \mathbf{H}(X,E) \times_{\mathbf{H}(X,X)} \{id\}
  \,.
$$

By the discussion at _[overcategory -- limits and colimits](overcategory#LimitsAndColimits)_ we have that [[colimits]] in $\mathbf{H}_{/X}$ are computed in $\mathbf{H}$. So if
$[E \to X] \simeq \underset{\longrightarrow_i}{\lim}{[E_i \to X]}$ is a [[filtered colimit]] in $\mathbf{H}_{/X}$, then $E \simeq \underset{\longrightarrow_i}{\lim}{E_i }$ is a filtered colimit in $\mathbf{H}$.

If now $X \in \mathbf{H}$ is a [[compact object]], then this commutes over this colimit and hence

$$
  \begin{aligned}
    \Gamma_X([E \to X])
    &=
    \mathbf{H}(X,\underset{\longrightarrow_i}{\lim} E_i) \times_{\mathbf{H}(X,X)} \{id\}
    \\
    & \simeq
    (\underset{\longrightarrow_i}{\lim}\mathbf{H}(X, E_i)) \times_{\mathbf{H}(X,X)} \{id\}
    \\
    &\simeq
    \underset{\longrightarrow_i}{\lim}
    (\mathbf{H}(X, E_i) \times_{\mathbf{H}(X,X)} \{id\})
    \\
    & \simeq
    \underset{\longrightarrow_i}{\lim}
    \Gamma_X([E_i \to X])
   \end{aligned}
  \,,
$$

where in the second but last step we used that in the [[topos]] [[Set]] [[universal colimits|colimits are preserved by pullback]].

This shows that $\Gamma_X(-) : \mathbf{H}_{/X} \to Set$ commutes over filtered colimits if $X$ is a [[compact object]].

Conversely, assume that $\Gamma_X(-)$ commutes over all filtered colimits. For every ([[filtered category|filtered]]) [[diagram]] $F_\bullet : I \to \mathbf{H}$ there is the corresponding filtered diagram $X \times F_\bullet : I \to \mathbf{H}_{/X}$, where $[X \times F_i \to X]$ is the projection. As before, the product with $X$ preserves forming colimits

$$
  \underset{\longrightarrow_i}{\lim} ([X \times F_i \to X])
  \simeq
  [X \times  (\underset{\longrightarrow_i}{\lim} F_i) \to X]
  \,.
$$

Moreover, sections of a trivial [[bundle]] are maps into the fiber

$$
  \Gamma_X([X \times F_i \to X]) \simeq \mathbf{H}(X,F_i)
  \,.
$$

So it follows that $X$ is a compact object:

$$
  \begin{aligned}
    \mathbf{H}(X, \underset{\longrightarrow_i}{\lim} F_i)
    &
   \simeq
   \Gamma_X( [X \times (\underset{\longrightarrow_i}{\lim} F_i) \to X])
    \\
    & \simeq
   \Gamma_X(\underset{\longrightarrow_i}{\lim} [X \times  F_i \to X])
   \\
   & \simeq
   \underset{\longrightarrow_i}{\lim}
   \Gamma_X( [X \times  F_i \to X])
   \\
   & \simeq
   \underset{\longrightarrow_i}{\lim}
   \mathbf{H}(X,F_i)
  \end{aligned}
  \,.
$$

=--

### Finite objects

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

An object $X \in \mathcal{T}$ in a topos $\mathcal{T}$ is a [[Kuratowski finite object]] precisely if the [[étale geometric morphism]]

$$
  \mathcal{T}_{/X} \to \mathcal{T}
$$

out of the [[slice topos]] is a proper geometric morphism. And precisely if $X$ is even _decidable_ is this a tidy geometric morphism.

=--

([Moerdijk-Vermeulen, examples III 1.4](#MoerdijkVermeulen))

### Geometric stacks
 {#GeometricStacks}

A typical condition on a  [[geometric stack]] to qualify as an [[orbifold]]/[[Deligne-Mumford stack]] is that its [[diagonal]] be proper. This is equivalent to the corresponding map of toposes being a proper geometric morphism (e.g. [Carchedi 12, section 2](#Carchedi12), [Lurie Spectral, section 3](#LurieSpectral)).

## Related concepts

* [[separated geometric morphism]], [[Hausdorff topos]]

## References
 {#References}

The theory of proper geometric morphisms is largly due to

* [[Ieke Moerdijk]], Jacob Vermeulen,  _Relative compactness conditions for toposes_ ([pdf](https://dspace.library.uu.nl/handle/1874/2374))
  {#MoerdijkVermeulen}

* [[Ieke Moerdijk]], Jacob Vermeulen, _Proper maps of toposes_ , Memoirs of the American Mathematical Society, no. 705 (2000)

based on the [[locale|localic]] case discussed in

* Jacob Vermeulen, _Proper maps of locales_, J. Pure Applied Alg. 92 (1994)

A textbook account is in section C3.2 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_

Discussion with relation to properness of [[geometric stacks]] includes

* {#Carchedi12} [[David Carchedi]], section 2 of _Étale Stacks as Prolongations_ ([arXiv:1212.2282](http://arxiv.org/abs/1212.2282))

Discussion of higher compactness conditions in [[(∞,1)-topos theory]] is in section 3 of

* {#LurieSpectral} [[Jacob Lurie]], _Spectral Schemes_ ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-VII.pdf))

and in section 2.3 of

* [[Jacob Lurie]], _[[Rational and p-adic Homotopy Theory]]_

and for the special case of [[spectral Deligne-Mumford stacks]] in section 1.4 of

* [[Jacob Lurie]], _[[Quasi-Coherent Sheaves and Tannaka Duality Theorems]]_

More on proper geometric morphisms between [[(infinity,1)-toposes|$(\infty,1)$-toposes]]:

* [[Louis Martini]], [[Sebastian Wolf]], _Proper morphisms of ∞-topoi_ [[arXiv:2311.08051](https://arxiv.org/abs/2311.08051)].

[[!redirects proper map of toposes]]
[[!redirects proper maps of toposes]]

[[!redirects proper geometric morphisms]]

[[!redirects compact topos]]
[[!redirects strongly compact topos]]

[[!redirects compact toposes]]
[[!redirects strongly compact toposes]]

[[!redirects compact topoi]]
[[!redirects strongly compact topoi]]


proper homotopy theory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Homotopy theory
+-- {: .hide}
[[!include homotopy - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea of some problems

**Proper homotopy theory** is both an old and a fairly new area of [[algebraic topology]].
It deals with properties of non-[[compact space|compact]] [[topological space]]s, that cannot be detected using maps from [[simplex|simplices]] in these spaces, such as used in the [[fundamental infinity-groupoid|singular complex]].

Historically the subject is traced back to Kerekjarto's classification of non-compact surfaces in 1923, but its emergence as an important tool in geometric topology came with [[Larry Siebenmann]]'s work in 1965.

Suppose $M$ is a _smooth_ [[manifold]] with boundary, $\partial M$, then $M\setminus \partial M$ is an open manifold. Now suppose someone gives us an open manifold $N$, is it possible to detect if there is a compact manifold $M$, with $M\setminus \partial M \cong N$.  Siebenmann showed that certain conditions on the _ends_ of $N$ were necessary and that there were obstructions if the dimension of $M$ was greater than 5.

Its potential importance of proper homotopy,
for example for physical applications, comes from the fact that the phenomena it studies include the limiting behaviour of the system.

## Intuitions

The basic hypothesis will be that $X$ will be a [[connected space|connected and locally connected]] compact [[Hausdorff space]]. It will usually be [[sigma-compact]], i.e., there will be an increasing sequence, $\{K_n\}$, of compact subspaces with each $K_n$ in the interior of $K_{n+1}$ and such that

$$X = \bigcup^\infty_{n=0}K_n,$$

These spaces will most often than not be locally finite [[simplicial complex]]es.

We will  be interested in the [[homotopy]] of such spaces 'out towards its ends'

### Ends

To illustrate the idea of the [[end of a topological space|ends]] of a space $X$, we note that naively $\mathbb{R}$ has two ends, $\infty$ and $-\infty$, whilst $\mathbb{R}^2$ has only one as it is $S^2\setminus \{\infty\}$, (but that is vague!).

More exactly, consider the system of spaces

$$\varepsilon(X) = \{closure(X\setminus K) : K compact \subset X\},$$

This is an inverse system or [[pro-object]] in the category of spaces. Applying the connected component functor, $\pi_0$, to this system of spaces gives $\pi_0(X)$, and, classically, one takes the limit of this to get

$$e(X) = lim \pi_0\varepsilon(X),$$
the _set of ends_ of $X$. In general, $e(X)$ would be given the [[inverse limit topology]], to preserve more of the information coming from its construction.  This space is the _space of (Freudenthal) ends of $X$_. It is a [[profinite space]].

#### Examples

* Let $X_8$ be the figure eight space, the one-point union of two circles, and let $X$ be its universal cover.  This is an infinite 'thorn bush'. It has infinitely many ends and
$$e(X) \cong 2^{\aleph_0}.$$

* Let $M$ be a compact manifold and $X = M\setminus \partial M$, then
$e(X) \cong \pi_0(\partial M)$.

### Proper maps

The assignment sending $X$ to $e(X)$ cannot be functorial on the category of spaces and _continuous_ maps, since the contracting map $\mathbb{R}\to \{0\}$ is continuous, $e(\mathbb{R})$ is $\{-\infty, \infty\}$, whilst $e(\{0\})$ is empty since $\{0\}$ is compact.
The problem is that continuity is really about inverse images (inverse image of open is open), but the inverse image mapping does not preserve compactness  (as in the example!).
+-- {: .un_defn}
###### Definition (recall)
A map $f:X\to Y$ is a [[proper map]] if for each subset $K$ compact in $Y$ then $f^{-1}(K)$ is compact in $X$.
=--

### End spaces continued

If $f:X\to Y$ is proper, then it induces a pro-morphism
$$\varepsilon(f) : \varepsilon(X)\to \varepsilon(Y)$$
and hence a _continuous map of the end spaces
$$e(f):e(X)\to e(Y),$$
and $e$ becomes a functor from some category $Proper$ of spaces and proper maps to $Stone$, the category of [[profinite space]]s / [[Stone space]]s.

### Proper homotopy

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

For any space $X$, the natural inclusions of $X$ into $X \times I$, $e_i(x) = (x,i)$, $i = 0,1$ together with the projection maps from $X\times I$ to $X$, are proper maps.
=--
+-- {: .un_cor}
###### Corollary
The category $Proper$ has a [[cylinder functor]].
=--

We call the corresponding notion of homotopy, 'proper homotopy'.
We get a 'Proper category' and an associated 'proper homotopy category', which we will denote $Ho(Proper)$. If we are restricting to $\sigma$-compact spaces we may write $Proper_\sigma$, and so on.

### Germs at $\infty$

Although a proper map $f: X\to Y$ will induce a continuous $\varepsilon(f)$, on the end spaces, it is clear that $f$ does not need to be defined on the whole of $X$ for this to work, as $\varepsilon$ encodes behaviour 'out towards $\infty$'.  This leads to the notion of a 'germ at $\infty$'.

Suppose $X$ is locally compact Hausdorff and $A\subset X$. The inclusion $j: A\to X$ is 'cofinal' if the closure of $X\setminus A$ is compact.
Note that a cofinal inclusion is proper and induces an isomorphism $\varepsilon(A)\cong \varepsilon(X)$.  Let $\Sigma$ be the class of all cofinal inclusions in $Proper$ and let $Proper_\infty = Proper[\Sigma^{-1}]$, the category obtained by formally inverting the cofinal inclusions.
+--{: .un_defn}
###### Definition
This category is called the _proper category at $\infty$_.
=--
Note that $(Proper,\Sigma)$ admits a [[calculus of fractions|calculus of right fractions]], so any morphism from $X$ to $Y$ in $Proper_\infty$ can be represented by a diagram
$$X\stackrel{j}{\leftarrow} A \stackrel{f}{\rightarrow}Y,$$
with $j$ a cofinal inclusion, i.e., $f$ is defined on some 'neighbourhood of the end of $X$'.

Two such diagrams
$$X\stackrel{j^{\prime}}{\leftarrow} A^{\prime} \stackrel{f^\prime}{\rightarrow}Y   and       X\stackrel{j^{\prime\prime}}{\leftarrow} A^{\prime\prime} \stackrel{f^{\prime\prime}}{\rightarrow}Y$$
represent the same germ if $f^\prime | A = f^{\prime\prime}|A$ for some cofinal subspace  $A$ with $A^\prime \cup A^{\prime\prime}\subset A$.

There is also a homotopy category $Ho(Proper_\infty)$

+-- {: .query}
The end space $e(X)$ is a Stone space so is $Max(R)$ for some Boolean algebra $R$ (Stone duality) In the 1960s someone (Goldman?) looked at a ring, $R$, of 'almost continuous functions' from $X$ to $\mathbb{Z}/2\mathbb{Z}$, that gave the right $e(X)$.  Can this idea help integrate better the ideas of proper homotopy etc. with modern methods of algebraic geometry?
=--

## Proper analogues of the fundamental group

The end space behaves a bit like a $\pi_0$ and usually spaces will have many ends, so are not 'connected at infinity'.  If we try to do a fundamental group or groupoid analogue, this means life will get more complicated.  We will try with the assumption of a space having a single end for simplicity!  We will also assume $X$ is $\sigma$-compact.

### First attempt

We could try defining $\pi_1(\varepsilon(X))$ as a progroup, then taking its limit.  For this we would take $\{K_n\}$ an exhausting increasing sequence of compact subsets and setting $U_i = X\setminus K_i$, pick a base point $x_i$ in each $U_i$, and we will get groups $\pi_1(U_i,x_i)$.  We however need induced homomorphisms $\pi_1(U_{i+1},x_{i+1})\to \pi_1(U_i,x_i)$, and for this we have to choose an arc in $U_i$ from $x_{i+1}$ to $x_i$.  We can combine these to get a _base ray_, rather than a _base point_,  that is, we need a proper map, $\alpha : [0,\infty)\to X$.  With that we do get an inverse sequence of groups, but there are problems.  What is the dependence of the inverse system on the choice of $\alpha$?

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Let $X$ be an infinite cylinder with an infinite string of circles attached via a proper ray $\alpha: [0,\infty) \to X$.
The space has just one 'end' but you can choose different ways of going from $\pi_1(U_{i+1},x_{i+1})$ to $\pi_1(U_i,x_i)$ for fairly obvious choices of  base points such that the limit groups of the resulting two inverse systems are non-isomorphic! (In the survey listed below, this example is examined in detail, and one of the limits is a free group on one element, the other is trivial! Definitely non-isomorphic!)

This means that $lim \pi_1(\varepsilon(X))$ is not an invariant of the end. This phenomenon is linked to the fact that $\pi_1(\varepsilon(X))$ does not satisfy the [[Mittag-Leffler condition]] for either choice of the base rays.

### Waldhausen boundary

If $X$ and $Y$ are locally compact Hausdorff spaces, there is no obvious candidate for a space of proper maps from $X$ to $Y$, but one can form a simplicial set $\mathbb{P}(X,Y)$ with $\mathbb{P}(X,Y)_n = Proper(X\times \Delta^n,Y)$, which acts as if it was the _singular complex of the mythical space of proper maps from $X$ to $Y$_.
+-- {: .un_defn}
###### Definition
The **Waldhausen boundary** of $X$ is the simplicial set $\mathbb{P}([0,\infty),X)$.
=--
There is an epimorphism from $\pi_0(\mathbb{P}([0,\infty),X))$ to $e(X)$.

*  In the example above of the cylinder with the string of circles attached, $\pi_0(\mathbb{P}([0,\infty),X))$,  is uncountable, and $\pi_1(\mathbb{P}([0,\infty),X))$ maps onto $lim S$.

*  When $X$ has a single end and  $\pi_0(\varepsilon(X))$ is [[Mittag-Leffler condition|Mittag-Leffler]], then $\pi_0(\mathbb{P}([0,\infty),X))$, is a single point, i.e. all possible base rays are properly homotopic.

### Second attempt

Even if we  did not have the above difficulty with the limit groups, we would still have the problem that, as the limit functor is not exact, the resulting limiting homotopy groups would not be that well behaved.  There would not be  any general long exact sequence results (just as with [[Čech homology]]). There is at least one possible replacement for those limiting homotopy groups, but first we note that it is not appropriate to base any such things at a point, rather we should be using a base ray as was discussed above.

One fairly obvious attempt to define a 'fundamental group' for $X$, based at a proper ray $\alpha: [0,\infty) \to X$, would be to note that $\alpha$ gives $\mathbb{P}([0,\infty),X)$ a base point so we could look at  $\pi_1(\mathbb{P}([0,\infty),X),\alpha)$ and more generally at $\pi_n(\mathbb{P}([0,\infty),X),\alpha)$, and we will denote these groups by $\underline{\underline{\pi}}_n(X,\alpha)$.

There are variants 'at infinity' of both the Waldhausen boundary and these groups, obtained using germs instead of proper maps. These will be denoted with a $\infty$ as a super- or suffix on the above notation.

### Brown–Grossman homotopy groups

(Once over lightly here, more details at [[Brown-Grossman homotopy group]].)
In fact there is another different way of looking at these groups, which has a more geometric feel to it. Historically these groups were not the first successful attempt.  This was due to Ed Brown and uses strings of spheres.  These are examples of [[spherical objects]] and the resulting 'groups' (better thought of as '$\Pi_\mathcal{A}$-[[spherical object and Pi(A)-algebra|algebras]]') have a rich structure.  They are discussed at [[Brown-Grossman homotopy group]]s.  They do link with the above homotopy groups of the Waldhausen boundary, which are called the [[Steenrod homotopy group]]s, (for reasons that will be explained there).

## To do!

* Add stuff on Steenrod homotopy groups both for here and for strong shape theory entries.

## References

Survey article:

* [[Tim Porter]], _Proper Homotopy Theory_,  in the _[[Handbook of Algebraic Topology]]_, Ed. I.M.James, Elsevier, 1995, p. 127-167,

Books:

* [[Hans-Joachim Baues|H. J. Baues]] and A. Quintero, _Infinite homotopy theory_, Volume 6 of K-monographs in mathematics,	Springer, 2001

* Bruce Hughes and Andrew Ranicki, _Ends of Complexes_, Cambridge Tracts in Mathematics (No. 123), C.U.P.

Lecture Notes:

*   [[D. A. Edwards]] and [[H. M. Hastings]], _Čech and Steenrod Homotopy Theory with Applications,_ SLNM 542, Springer (1976).

[[!redirects proper homotopy theory]]
[[!redirects Proper Homotopy Theory]]

[[!redirects proper homotopy]]
[[!redirects proper homotopies]]


proper homotopy theory > history

[[!redirects proper homotopy theory (unbroken copy)]]

See [[proper homotopy theory]].

(This page was created when there was a rendering problem with the original at revision 31.  It was identical to revision 32 of the original.)


proper Lie groupoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
#### Topology
+--{: .hide}
[[!include topology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

A [[Lie groupoid]] $(C_1 \stackrel{\overset{s}{\to}}{\underset{t}{\to}} C_0)$ is **proper** if its underlying [[topological groupoid]] is a [[proper topological groupoid]], hence if

$$
  (s,t) : C_1 \to C_0 \times C_0
$$

is a [[proper map]].

So in particular the [[automorphism group]] of any [[object]] in a proper Lie groupoid is a [[compact space|compact]] [[Lie group]]. In this sense proper Lie groupoids generalize [[compact Lie groups]].

## Examples

* An [[orbifold]] is (presented by) a proper [[étale groupoid]].

## Related concepts

* [[effective Lie groupoid]]

## References

* M.J. Pflaum, H. Posthuma, X. Tang, _Geometry of orbit spaces of proper Lie groupoids_ ([arXiv:1101.0180](http://arxiv.org/abs/1101.0180))

[[!redirects proper Lie groupoids]]


proper map

> This entry is about the concept in  [[topology]]. For variants see at _[[proper morphism]]_.

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

#Contents#
* table of contents
{:toc}

## Idea

In [[topology]], the notion of *properness* is that of [[compact topological space|compactness]], but generalized from [[topological spaces]] to [[continuous functions]] between them.

A continuous map $X \to Y$ is proper when “$X$ is compact, relative to $Y$”.

## Definition

Just like there are various equivalent ways of characterizing [[compact topological spaces]], there are also various equivalent ways of characterizing proper functions.

In the following we discuss:

* [characterization via nets](#UsingNets);

* [characterization via intersection of closed sets](#UsingClosedSets);

* [characterization via closedness](#UsingClosedness);

* [characterization as a continuous family of compact fibres](#AsAContinousFamilyOfCompactSpaces);

* [further characterizations](#FurtherCharacterizations).

### Via nets
 {#UsingNets}

Recall that for a [[topological space]] $X$ to be [[compact topological space|compact]] means for every [[ net ]] $x_\bullet$ in $X$ to have a [[cluster point]] in $X$; hence for every net to admits a [[convergence of a net|convergent]] subnet.

A generalisation of this characterisation to a [[continuous map]] $f \colon X \to Y$ is the following:

\begin{definition}
\label{PropernessViaNets}

A [[continuous map]] $f \,\colon\, X \to Y$ is *proper* iff for every [[net]] $x_\bullet$ in $X$ and for every [[cluster point]] $y$ of $f(x_\bullet)$ in $Y$, the net $x_\bullet$ has a [[cluster point]] $x \in X$ with $f(x) = y$.

Equivalently:

Such $f$ is proper iff for every net $x_\bullet \in X$ and every $y \in Y$, if $f(x_\bullet)$ [[convergence of a net|converges]] to $y$ then $x_\bullet$ admits a subnet [[convergence of a net|converging]] to $x \in X$ with $f(x) = y$.

\end{definition}

### Via intersection of closed sets
 {#UsingClosedSets}

Recall that, equivalently, a [[topological space]] $X$ is [[compact topological space|compact]] if and only if for every cofiltered set of non-empty closed subsets $F_i \subset X$, their intersection $\cap_i F_i$ is also non-empty.

\begin{definition}
\label{PropernessViaClosedSets}

A [[continuous map]]  $f \,\colon\, X \to Y$  is *proper* iff

1. $f$ is closed;

1. for every cofiltered set of closed subsets $\{F_i\}_{i \in I}$ of $X$, one has

$$
  f(\cap_{i \in I} F_i) = \cap_{i \in I}f(F_i)
$$

\end{definition}

### Via closedness
 {#UsingClosedness}

Recall that, equivalently, a [[topological space]] $X$ is [[compact topological space|compact]] if and only if the [[projection map]]

$$
  X \times Z \longrightarrow Z
$$

out of the [[product topological space]] with some $Z$
is a [[closed map]], for every [[topological space]] $Z$.

\begin{definition}
\label{PropernessViaClosedness}

A [[continuous map]]  $f \,\colon\, X \to Y$  is *proper* iff its image under the [[Cartesian product]]-[[functor]]

$$
  f \times \mathrm{Id}_Z
   \;\colon\;
  X \times Z
   \;\longrightarrow\;
  Y \times Z
$$

is a [[closed map]] for every [[topological space]] $Z$.

\end{definition}

### As a continuous family of compact spaces
 {#AsAContinousFamilyOfCompactSpaces}

With every function $f \colon X \to Y$, the space $X$ can be described as the union of the fibres $f^{-1}(y)$ with $y \in Y$. The map $f$ is proper when each fibre is compact and the family is continuous in the sense that: for every net $y_\bullet$ converging to $y \in Y$, we ask that the net of sets $f^{-1}(y_\bullet)$ converges to $f^{-1}(y)$. This is equivalent to ask that $f$ be closed.

\begin{definition}
\label{PropernessAsAContinuousFamilyOfCompactFibres}

A [[continuous map]]  $f \,\colon\, X \to Y$  is *proper* if:

1. $f^{-1}(\{y\})$ is compact for every $y \in Y$;

1. $f$ is closed.

\end{definition}

### Further characterizations
 {#FurtherCharacterizations}

\begin{proposition}
\label{EquivalenceOfCharacterization}
Given a [[continuous map]] $f \colon X \to Y$, the following properties are all equivalent:

1. (Def. \ref{PropernessViaNets})

   For every [[net]] $x_\bullet$ in $X$ and every [[cluster point]] $y$ of $f(x_\bullet)$, the net $x_\bullet$ admits a cluster point $x$ with $f(x) = y$;

1. If $\mathcal{F}$ is a [[filter]] on $X$ and if $y$ is a [[cluster point]] of $f(\mathcal{F})$, then $\mathcal{F}$ has a cluster point $x \in X$ with $f(x) = y$;

1. (Def. \ref{PropernessViaClosedSets})

   $f$ is a [[closed map]] and for every cofiltered family $\{F_i\}_{i \in I}$ of
closed subsets of $X$, one has $f(\cap_{i \in I} F_i) = \cap_{i \in I}f(F_i)$;

1. (Def. \ref{PropernessViaClosedness})

   The image $f \times \mathrm{Id}_Z \,\colon\, X \times Z \to Y \times Z$ under the [[Cartesian product]]-[[functor]] is a [[closed map]] for every topological space $Z$;

1. ([[universally closed morphism|universally closed]])

   For every [[continuous map]] $g \colon Z \to Y$, the resulting [[pullback]] map

   \[
      g^\ast(f)
        \;\colon\;
      X \times_Y Z \to Z
   \]

   is a [[closed map]];

1. (Def. \ref{PropernessAsAContinuousFamilyOfCompactFibres})

   $f$ is a [[closed map]] and the [[inverse image]] $f^{-1}\big(\{y\}\big)$ of every $y \in Y$ is [[compact topological space|compact]];

1. $f$ is a [[closed map]] and the [[inverse image]] $f^{-1}(K)$ of every [[compact subspace]] $K \subset Y$ is [[compact topological space|compact]].

\end{proposition}

\begin{definition}
\label{ProperContinuous}
**([[proper maps]])**
\linebreak
A [[continuous function]] $f  \colon X \to Y$
is called _[[proper map|proper]]_ if it satisfies one of the equivalent properties listed in Prop. \ref{EquivalenceOfCharacterization}.

\end{definition}

\begin{remark}
**(Ambiguous terminology)**
\linebreak
The notion of compact space is subject to naming ambiguity. For the same notion, some authors will use the term _quasi-compact_, using _compact_ only when the space is also separated.

For _properness_ the situation is _worse_ as there are three competing definitions. We have defined the one similar to _quasi-compact_ spaces.

In addition one could require $f$ to be separated, that is to require that if a net $x_\bullet$ converges to both $x_1$ and $x_2$ with $f(x_1) = f(x_2)$, then $x_1 = x_2$. This definition of properness resembles the one used in algebraic geometry: see [[ proper morphism ]]. It is also the one to be used in the proper base change theorem.

Finally, some authors use a weaker version of properness, where $f \colon X \to Y$ is proper when $f^{-1}(K)$ is compact for every compact $K \subset Y$. But as explained below, this definition is usually used in situations where these maps are always closed.
\end{remark}

## Further criteria

A continuous map $f \colon X \to Y$ such that $f^{-1}(K)$ is compact for every compact $K \subset Y$ may not be closed; even when both $X$ and $Y$ are very nice spaces like $\mathbf{T}_5$ spaces.

\begin{example}
Let $X$ be an uncountable set and let $p \in X$. Let's still write $X$ for the discrete topological space associated to it. Let $X_p$ denote the topological space whose underlying set is $X$ but whose opens sets are either all the sets not containing $p$ or the sets containing $p$ with countable complement in $X$.

Then $X_p$ is a $\mathbf{T}_5$ topological space and its compact subsets are all finite.
But the identity map
$$ X \longrightarrow X_p$$
 is not closed.
\end{example}

However this is very often the case in practice: for example when $Y$ is a metric space or a locally compact separated space.

\begin{proposition}
Let $f \colon X \to Y$ be a continuous map such that

1. $f^{-1}(K)$ is compact for every compact $K \subset Y$;

1. $Y$ is a [[ compactly generated topological space | $k$-space ]]

then $f$ is closed and thus proper.
\end{proposition}

Also,

+-- {: .num_prop #MapsFromCompactSpacesToHausdorffSpacesAreClosed}
###### Proposition
**(maps from compact spaces to separated spaces are proper)**

Let $f \colon X \longrightarrow Y$ be a [[continuous function]] between [[topological spaces]] such that

1. $X$ is [[compact topological space|compact]];

1. $Y$ is [[Hausdorff topological space | separated]],

then $f$ is proper.

=--

## Properties

Proper maps enjoy analogous properties as [[compact topological spaces]] do, for example the product of proper maps is again proper:

\begin{theorem}
Let $\{f_i \colon X_i \to Y_i\}_{i \in I}$ be a small [[indexed family]] of continuous proper maps (Def. \ref{ProperContinuous}), then their [[functor|functorial]] [[product topological space|product]]

$$
  \prod_{i \in I} \,f_i
    \;\colon\;
  \prod_{i \in I} \, X_i
  \longrightarrow
  \prod_{i \in I} Y_i
$$
is also a proper map.
\end{theorem}

## Comparison with proper maps of locales

\begin{proposition}
Every proper map $f\,:\, X \to Y$ between two topological spaces induces a
proper map $\mathrm{Loc}(f)\,:\, \mathrm{Loc}(X) \to \mathrm{Loc}(Y)$ between the associated locales.

Conversely, if $\mathrm{Loc}(f)\,:\, \mathrm{Loc}(X) \to \mathrm{Loc}(Y)$ is proper and $Y$ is a $\mathrm{T}_\mathrm{D}$-space, then $f\,:\, X \to Y$ is proper.
\end{proposition}

## Related concepts

* [[locally proper map]]

* [[proper morphism]]

* [[open map]], [[closed map]]

* [[proper group action]]

* [[proper homotopy]]

* [[proper equivariant homotopy theory]]

## References

One of the early reference on proper maps is

* [[Nicolas Bourbaki]], _Topologie Générale_ (Éléments de Mathématique, Livre III), Chapitres 1 à 2.  Second Edition, 1951.  Actualites Sci. Ind. 1142, Hermann, Paris.

A considerable expanded treatment is given in the third edition, see Chapter I, Section 10:

* [[Nicolas Bourbaki]], _Topologie Générale_ (Éléments de Mathématique, Livre III), Chapitres 1 à 2.  Third Edition, 1961.  Actualites Sci. Ind. 1142, Hermann, Paris.

English translation of the 1971 edition:

* {#Bourbaki71} [[Nicolas Bourbaki]], *General topology*, Elements of Mathematics III, Springer (1971, 1990, 1995) [[doi:10.1007/978-3-642-61701-0](https://doi.org/10.1007/978-3-642-61701-0)]

{#LocaliVersionReferences} The [[locale|localic]] version of proper maps was introduced in:

* [[Peter Johnstone]], §1 of: _The Gleason cover of a topos, II_, Journal of Pure and Applied Algebra **22** 3 (1981) 229–247 [doi:10.1016/0022-4049(81)90100-6]

where it is attributed to this "preliminary" (and apparently unpublished) account:

* [[Peter Johnstone]], _Factorization and pullback theorems for localic geometric morphisms_, Univ. Cath. de Louvain, Sem. de math, pure, Rapport no. 79 (1979)

[[!redirects proper maps]]
[[!redirects proper continuous map]]
[[!redirects proper continuous maps]]

[[!redirects proper continuous function]]
[[!redirects proper continuous functions]]


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

#Contents#
* table of contents
{:toc}

## Background

+-- {: .num_defn #LocallyCompactSpace}
###### Definition
**([[locally compact topological space]])**

A [[topological space]] $X$ is called _[[locally compact topological space|locally compact]]_
if for every point $x \in X$ and every [[open neighbourhood]] $U_x \supset \{x\}$
there exists a smaller open neighbourhood $V_x \subset U_x$ whose [[topological closure]]
is [[compact topological space|compact]] and still contained in $U$:

$$
  \{x\} \subset V_x \subset \underset{\text{compact}}{Cl(V_x)} \subset U_x
  \,.
$$

=--

## Statement and proof

+-- {: .num_prop}
###### Proposition
**(proper maps to locally compact spaces are closed)**

Let

1. $(X,\tau_X)$ be a [[topological space]],

1. $(Y,\tau_Y)$ a [[locally compact topological space]] according to def. \ref{LocallyCompactSpace},

1. $f \colon X \to Y$ a [[continuous function]].

Then:

If $f$ is a [[proper map]], then it is a [[closed map]].

=--

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

Let $C \subset X$ be a [[closed subset]]. We need to show that every $y \in Y \backslash f(C)$ has an open neighbourhood $U_y \supset \{y\}$ not intersecting $f(C)$ (by [this prop.](Introduction+to+topology+--+1#UnionOfOpensGivesClosure)).

By local compactness of $(Y,\tau_Y)$ (def. \ref{LocallyCompactSpace}), $y$ has an open neighbourhood $V_y$ whose topological closure $Cl(V_y)$ is compact. Hence since $f$ is proper, also $f^{-1}(Cl(V_y)) \subset X$ is compact. Then also the intersection $C \cap f^{-1}(Cl(V_y))$ is compact, and hence so is

$$
  f(C \cap f^{-1}(Cl(V_y)))
  =
  f(C) \cap (Cl(V))
  \;
  \subset Y
  \,.
$$

This is also a [[closed subset]], since [[compact subspaces of Hausdorff spaces are closed]]. Therefore

$$
  U_y \coloneqq V_y \backslash ( f(C) \cap (Cl(V_y)) ) = V_y \backslash f(C)
$$

is an open neighbourhod of $y$ not intersecting $f(C)$.

=--

## Related statements

* [[injective proper maps to locally compact spaces are equivalently the closed embeddings]]

* [[closed injections are embeddings]]

* [[maps from compact spaces to Hausdorff spaces are closed and proper]]

* [[CW-complexes are Hausdorff spaces]]

* [[Hausdorff spaces are sober]], [[schemes are sober]]

* [[continuous image of a compact space is compact]]

* [[closed subsets of compact spaces are compact]]

* [[compact subspaces of Hausdorff spaces are closed]]

* [[closed subspaces of compact Hausdorff spaces are equivalently compact subspaces]]

* [[quotient projections out of compact Hausdorff spaces are closed precisely if the codomain is Hausdorff]]

* [[paracompact Hausdorff spaces are normal]]

* [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]]


proper model category

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

#Contents#
* table of contents
{:toc}

## Idea ##

In a [[model category]] fibrations enjoy [[pullback stability]] and cofibrations are stable under [[pushout]], but weak equivalences need not have either property. In a proper model category weak equivalences are also preserved under certain pullbacks and/or certain pushouts.

Put differently, in a proper model category, [[homotopy pullbacks]] and/or pushouts can be computed with less need for [[fibrant replacement|fibrant and/or cofibrant replacement]].

## Definition ##

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

A [[model category]] is called

* **right proper** if [[weak equivalences]] are preserved by [[pullback]] along [[fibrations]],

* **left proper** if weak equivalence are preserved by [[pushout]] along cofibrations,

* **proper** if it is both left and right proper.

=--

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

More in detail this means the following.
A model category is right proper if for every weak equivalence $f : A \to B$ in $W\subset Mor(C)$ and every fibration $h : C \to B$ the pullback $h^* f : A \times_B C \to C$ in
$$
  \array{
    A \times_C B &\longrightarrow& A
    \\
    \;\;\big\downarrow{}^{\mathrlap{\Rightarrow h^* f \in W}}
    &&
    \big\downarrow{}^{\mathrlap{f \in W}}
    \\
    C &\underset{h \in F}{\longrightarrow}& B
  }
$$
is a weak equivalence.

=--

\begin{remark}
The above definition is the way it is usually phrased, but in fact it is equivalent to a seemingly weaker condition that is sometimes easier to check: for right properness it suffices to assume that weak equivalences are preserved by pullback along fibrations *between fibrant objects*.  That is, in the more explicit version above, we are free to assume that $B$ (hence also $C$) is fibrant; this then implies the more general version without this hypothesis.  See Proposition \ref{FibrationsBetweenFibrantObjectsSuffice} below.
\end{remark}

## The Rezk criterion for properness

The following criterion shows that the notion of left or right properness only depends on the underlying [[relative category]] of a [[model category]],
i.e., does not depend on fibrations or cofibrations.
This is clear once we observe that the notion of a [[Quillen equivalence]] in the statement below can be replaced by the notion of a [[Dwyer–Kan equivalence]] of underlying [[relative categories]],
or just ordinary equivalences of underlying [[homotopy categories]].

\begin{theorem}
([Rezk](#Rezk02), Proposition 2.7 (arXiv), Proposition 2.5 (journal).)
A model category $M$ is left proper if and only if for every weak equivalence $f\colon X\to Y$
the induced [[Quillen adjunction]]
$$X/M\leftrightarrows Y/M$$
is a [[Quillen equivalence]].
A model category $M$ is right proper if and only if for every weak equivalence $f\colon X\to Y$
the induced [[Quillen adjunction]]
$$M/X\leftrightarrows M/Y$$
is a [[Quillen equivalence]].
\end{theorem}

## Examples
 {#Examples}

### Left proper model categories

* by cor. \ref{AllObjectsFibrantImpliesRightProper}, every model category in which all objects are cofibrant is left proper;

  this includes notably

  * the [[classical model structure on simplicial sets]]

  * the [[model structure for quasi-categories]]

  and many model structures derived from these, such as

  * the injective global [[model structure on simplicial presheaves]] over any (simplicial) category;

* the left [[Bousfield localization]] of every left proper [[combinatorial model category]] at a set of morphisms is again left proper.

  So in particular also the _local_ injective model structures on
  simplicial presheaves over a [[site]] are left proper.

### Non-left proper model categories

A class of model structures which tends to be _not_ left proper are model structures on categories of not-necessarily commutative algebras.

For instance

* the standard model structure on [[simplicial object|simplicial]] [[associative unital algebra]]s (weak equivalences and fibrations are those of the underlying simplicial sets) is  _not_  left proper (see [Rezk](#Rezk02), Example 2.11 (arXiv), Example 2.7 (journal)).

But it is [[Quillen equivalence|Quillen equivalent]] to a model structure that _is_ left proper. This is discussed [below](#ProperEquivModels).

### Right proper model categories ###

* by cor. \ref{AllObjectsFibrantImpliesRightProper} every model category in which each object is fibrant is right proper.

  This includes for instance the standard [[Quillen model structure on topological spaces]].

* The [[classical model structure on simplicial sets]] is (automatically left proper, since all objects are cofibrant, but also) right proper, even though not all objects are fibrant.  See [there](classical+model+structure+on+simplicial+sets#Properness).

### Non-right proper model categories

* "Non-algebraic" models for higher categories (other than higher groupoids) are generally not right proper.  For example, the [[model structure for complete Segal spaces]] and the [[model structure for quasi-categories]] are not right proper (see [mathoverflow](http://mathoverflow.net/questions/40938/is-the-model-category-of-complete-segal-spaces-right-proper)).

### Proper model categories ###

Model categories which are both left and right proper include

* [[Top]]: [[classical model structure on topological spaces]] ([prop.](classical+model+structure+on+topological+spaces#Properness))

* [[sSet]]: [[classical model structure on simplicial sets]] ([here](classical+model+structure+on+simplicial+sets#Properness))

* The _global_ [[model structure on simplicial presheaves]] and any local such model structure over a site with [[point of a topos|enough points]] and weak equivalences the [[stalk]]wise weak equivalences.

* {#ModelStructureOnChainComplexesIsProper} The standard [[model structures on chain complexes]] (see [there](model+structure+on+chain+complexes#Properness)).

* The projective [[model structure on differential graded-commutative algebras]] (unbounded). See [this MO discussion](http://mathoverflow.net/q/204414/381).

### Reedy model structures

\begin{proposition}\label{ReedyStructureInheritsProperness}
  For $\mathcal{R}$ a [[Reedy category]] and $\mathcal{C}$ a [[model category]] which is left or right [[proper model category|proper]], then also the [[Reedy model structure]] on $Func(\mathcal{R}, \mathcal{C})$ is left or right [[proper model category|proper]], respectively.
\end{proposition}
This appears as [Hirschorn (2002), Thm. 15.3.4 (2)](#Hirschorn02), there attributed to [[Daniel Kan]].

### Proper Quillen equivalent model structures {#ProperEquivModels}

While some model categories fail to be proper, often there is a [[Quillen equivalence|Quillen equivalent]] one that does enjoy properness.

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

Every model category whose acyclic cofibrations are [[monomorphism]]s is Quillen equivalent to its [[model structure on algebraic fibrant objects]].
In this all objects are fibrant, so that it is right proper.

=--

([Nikolaus 10, theorem 2.18](#Nikolaus10))

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

Let $T$ be a simplicial (possibly multi-colored) [[theory]], and let $T Alg$ be the corresponding category of simplicial T-algebras. This carries a model category structure where the fibrations and weak equivalences are those of the underlying simplicial sets in the [[classical model structure on simplicial sets]].

Then there exists a morphism of simplicial theories $T \to S$ such that

1. the induced [[adjunction]] $S Alg \stackrel{\to}{\leftarrow} T Alg$ is a [[Quillen equivalence]];

1. $S Alg$ is a proper [[simplicial model category]].

=--

This is the content of ([Rezk 02](#Rezk02))

## Properties

### General

The following says that left/right properness holds _locally_ in every model category, namely between cofibrant/fibrant objects.

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

Given a [[model category]],

1. every [[pushout]] of a [[weak equivalence]] between [[cofibrant objects]] along a [[cofibration]] is again a weak equivalence;

1. every [[pullback]] of a [[weak equivalence]] between [[fibrant objects]] along a [[fibration]] is again a weak equivalence.

=--

A proof is spelled out in [Hirschhorn, prop. 13.1.2](#Hirschhorn), there attributed to [Reedy](#Reedy).

This gives a large class of examples of left/right proper model categories:

\begin{corollary}
\label{AllObjectsFibrantImpliesRightProper}

* A model category in which all objects are cofibrant is left proper.

* A model category in which all objects are fibrant is right proper.

\end{corollary}

See in the list of [Examples](#Examples) below for concrete examples.

Notice that the prop. \ref{GoodPullbacksAndPushouts} applies only (in the right proper case, for concreteness) to pullbacks of fibrations along weak equivalences in which *all three* objects are fibrant, since a fibration with fibrant codomain also has fibrant domain.  The definition of right proper, on the other hand, states this property in the case when *none* of the objects are assumed to be fibrant.

One might consider as an "in-between" assumption the situation when only the common codomain of the fibration and the weak equivalence (hence also the domain of the fibration) are fibrant; but it turns out that this apparently-weaker assumption is sufficient to imply full right properness.  This can be found, for instance, as Lemma 9.4 of [Bousfield 2001](#Bousfield01).

+-- {: .num_prop #FibrationsBetweenFibrantObjectsSuffice}
###### Proposition
Suppose that in some model category, if $X\to Y$ is a fibration and $Z\to Y$ a weak equivalence, with $Y$ (hence also $X$) fibrant, then the pullback $X\times_Y Z \to X$ is a weak equivalence.  Then the model category is right proper, i.e. the same statement is true without the assumption that $Y$ is fibrant.
=--
+-- {: .proof}
###### Proof
Suppose given $X\to Y\leftarrow Z$ where $X\to Y$ is a fibration and $Z\to Y$ a weak equivalence.  Choose a fibrant replacement $Y\to R Y$, and factor $X\to Y \to R Y$ as a weak equivalence $X\to R X$ followed by a fibration $R X \to R Y$.  The assumption now applies to the cospan $R X \to R Y \leftarrow Y$, so that the map $R X \times_{R Y} Y \to R X$ is a weak equivalence.  By 2-out-of-3, the induced map $X \to R X \times_{R Y} Y$ is also a weak equivalence.

Now by [[Ken Brown's lemma]], the pullback functor along $Z\to Y$ preserves weak equivalences between fibrant objects, and in particular preserves this weak equivalence $X \to R X \times_{R Y} Y$.  Thus, the induced map $X\times_Y Z \to R X \times_{R Y} Z$ is a weak equivalence.  However, $R X \times_{R Y} Z \to R X$ is a weak equivalence by the assumption, so by 2-out-of-3, the map $X\times_Y Z \to X$ is also a weak equivalence, as desired.
=--

### Homotopy (co)limits in proper model categories ###
 {#HomotopyLimits}

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

In a left proper model category, ordinary [[pushouts]] along cofibrations are [[homotopy pushouts]].

Dually, in a right proper model category, ordinary [[pullbacks]] along fibrations are [[homotopy pullbacks]].

=--

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

This is stated for instance in [[Higher Topos Theory|HTT, prop A.2.4.4]] or in prop. 1.19 in
[Bar](http://www.math.harvard.edu/~clarkbar/complete.pdf). We follow the proof given in this latter reference.

We demonstrate the first statement, the second is its direct formal dual.

So consider a [[pushout]] diagram

$$
  \array{
    K &\longrightarrow& Y
    \\
    \big\downarrow^{\mathrlap{\in cof}} && \big\downarrow
    \\
    L &\longrightarrow& X
  }
$$

in a left proper model category, where the morphism $K \to L$ is a cofibration, as indicated. We need to exhibit a weak equivalence $X' \stackrel{}{\to} X$ from an object $X'$ that is manifestly a [[homotopy pushout]] of $L \leftarrow K \to Y$.

The standard procedure to produce this $X'$ is to pass to a weakly equivalent diagram with the property that all objects are cofibrant and one of the morphisms is a cofibration. The ordinary pushout of that diagram is well known to be the [[homotopy pushout]], as described there.

So pick a cofibrant replacement $\emptyset \hookrightarrow K' \stackrel{\simeq}{\to}$ of $K$ and factor $K' \to K \to Y$ as a cofibration followed by a weak equivalence
$K' \hookrightarrow Y' \stackrel{\simeq}{\to} Y$ and similarly factor $K' \to K \to L$ as $K' \hookrightarrow L' \stackrel{\simeq}{\to} L$

This yields a weak equivalence of  diagrams

$$
  \array{
    Y &\stackrel{\simeq}{\leftarrow}& Y'
    \\
    \uparrow && \uparrow^{\mathrlap{\in cof}}
    \\
    K &\stackrel{\simeq}{\leftarrow}& K'
    \\
    \downarrow^{\mathrlap{\in cof}}
     && \downarrow^{\mathrlap{\in cof}}
    \\
    L &\stackrel{\simeq}{\leftarrow}& L'
  }
  \,,
$$

where now the diagram on the right is cofibrant as a diagram, so that its ordinary pushout

$$
  X' \coloneqq L' \coprod_{K'} Y'
$$

is a homotopy colimit of the original diagram. To obtain the weak equivalence from there to $X$, first form the further pushouts

$$
  \array{
    K &&&\to&&& Y
    \\
    & \nwarrow^{\mathrlap{\in W}}
     &&&& \nearrow_{\mathrlap{\simeq}} &
    \\
    && K' &\to& Y' &&
    \\
    \downarrow^{\mathrlap{\in cof}}
    && \downarrow^{\mathrlap{\in cof}}
     && \downarrow^{\mathrlap{\in cof}} && \downarrow
    \\
    && L' &\to& X' &&
    \\
    & {}^{\mathllap{\in W}}
    \swarrow &&&& \searrow^{\mathrlap{\simeq}} &
    \\
    L'' \coloneqq K \coprod_{K'} L
    &&&\to&&&
    L'' \coprod_{K} Y
    \\
    \downarrow^{\mathrlap{\in W}} &&&&&&  \downarrow
    \\
    L &&&\to&&& X
  }
  \,,
$$

where the total outer diagram is the original pushout diagram. Here the cofibrations are as indicated by the above factorization and by their stability under pushouts, and the weak equivalences are as indicated by the above factorization and by the left properness of the model category. The weak equivalence $L'' \stackrel{\simeq}{\to} L$ is by the [[category with weak equivalences|2-out-of-3 property]].

This establishes in particular a weak equivalence

$$
  X' \stackrel{\simeq}{\to} L'' \coprod_K Y
  \,.
$$

It remains to get a weak equivalence further to $X$.
For that, take the two outer squares from the above

$$
  \array{
    K &\to& Y
    \\
    \downarrow^{\mathrlap{\in cof}} && \downarrow
    \\
    L'' &\to& L'' \coprod_{K'} Y
    \\
    \downarrow^{\mathrlap{\in W}} && \downarrow
    \\
    L &\to& X
  }
  \,.
$$

Notice that the top square is a pushout by construction, and the total one by assumption.  Therefore by the [[pasting law]], also the lower square is a pushout.

Then factor $K \to Y$ as a cofibration followed by a weak equivalence $K \hookrightarrow Z \stackrel{\simeq}{\to} Y$
and push that factorization through the double diagram, to obtain

$$
  \array{
    K &\stackrel{\in cof}{\to}& Z &\stackrel{\in W}{\to}& Y
    \\
    \downarrow^{\mathrlap{\in \cof}}
    && \downarrow^{\mathrlap{\in cof}} && \downarrow
    \\
    L'' &\stackrel{\in cof}{\to}&
    L'' \coprod_{K} Z
    &\stackrel{\in W}{\to}&
    L'' \coprod_{K'} Y
    \\
    \downarrow^{\mathrlap{\in W}} &&
    \downarrow^{\mathrlap{\in W}}
    && \downarrow
    \\
    L & \to& L \coprod_K Z &\stackrel{\in W}{\to}& X
  }
  \,.
$$

Again by the behaviour of pushouts under pasting, every single square and composite rectangle in this diagram is a pushout. Using this, the cofibration and weak equivalence properties from before push through the diagram as indicated. This finally yields the desired weak equivalence

$$
  L'' \coprod_{K'} Y \stackrel{\simeq}{\to}
  X
$$

by [[category with weak equivalences|2-out-of-3]].

=--

If we had allowed ourselved to assume in addition that $K$ itself is already cofibrant, then the above statement has a much simpler proof, which we list just for fun, too.

+-- {: .proof}
###### Proof of prop. \ref{PullbackAlongFibrationsAreHomotopyPullbacks} assuming that the domain of the cofibration is cofibrant

Let $A \hookrightarrow B$ be a cofibration with $A$ cofibrant and let $A \to C$ be any other morphism. Factor this morphism as
$A \hookrightarrow C' \stackrel{\simeq}{\to} C$ by a cofibration
followed by an acyclic fibration. This give a weak equivalence
of pushout diagrams

$$
  \array{
    C' &\stackrel{\simeq}{\to}& C
    \\
    \uparrow && \uparrow
    \\
    A &\stackrel{=}{\to}& A
    \\
    \downarrow && \downarrow
    \\
    B &\stackrel{=}{\to}& B
  }
  \,.
$$

In the diagram on the left all objects are cofibrant and one morphism is a cofibration,
hence this is a cofibrant diagram and its ordinary colimit is the homotopy colimit.
Using that [[pushout]] diagrams compose to pushout diagrams, that cofibrations are preserved under pushout and that in a left proper model category weak equivalences
are preserved under pushout along cofibrations, we find a weak equiovalence
$hocolim \stackrel{\simeq}{\to} B \coprod_A C$

$$
  \array{
    A &\stackrel{\in cof}{\to}& C' &\stackrel{\in W \cap fib}{\to}& C
    \\
    \downarrow^{\mathrlap{\in cof}} &&
    \downarrow^{\mathrlap{\in cof}} &&
    \downarrow^{\mathrlap{\in cof}}
    \\
    B &\to& hocolim &\stackrel{\in W}{\to}&
    B \coprod_A C
  }
  \,.
$$

The proof for the second statement is the precise formal dual.

=--

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

A model category is [[right proper model category|right proper]] if and only if every fibration is a [[sharp map]].

=--

([Rezk 98](#Rezk98))

### Slice categories
 {#SliceCategories}

For any model category $M$, and any morphism $f \colon A\to B$, the
[[base change]] of [[slice categories]] along $f$
$$
  \Sigma_f
    \;\colon\;
  M/A
    \rightleftarrows
  M/B
   \;\colon\;
  f^* $$
is a [[Quillen adjunction]] between [[slice model categories]] ([this Prop.](slice+model+structure#LeftBaseChangeQuillenAdjunction)).

If this adjunction is a [[Quillen equivalence]], then $f$ must be a [[weak equivalence]].  In general, the converse can be proven only if $A$ and $B$ are fibrant.

\begin{proposition}\label{ViaSliceCategories}
The following are equivalent:

1. $M$ is right proper.

1. If $f$ is any weak equivalence in $M$, then $\Sigma_f \dashv f^*$ is a Quillen equivalence.

\end{proposition}

This is due to [Rezk 02, Prop. 2.5](#Rezk02).

\begin{remark}
The statement of Prop. \ref{ViaSliceCategories} may be read as saying $M$ is right proper iff all [[slice model categories]] have the "correct" Quillen equivalence type.

Since whether or not a Quillen adjunction is a Quillen equivalence depends only on the classes of weak equivalences, not the fibrations and cofibrations, it follows that being right proper is really a property of a [[homotopical category]].  In particular, if one model structure is right proper, then so is any other model structure on the same category with the same weak equivalences.
\end{remark}

### Local cartesian closure

Since most well-behaved model categories are equivalent to a model category in which all objects are fibrant --- namely, the model category of [[algebraically fibrant objects]] --- they are in particular equivalent to one which is right proper.  Thus, right properness by itself is not a property of an $(\infty,1)$-category, only of a particular presentation of it via a model category.

However, if a [[Cisinski model category]] is right proper, then the $(\infty,1)$-category which it presents must be [[locally cartesian closed (∞,1)-category|locally cartesian closed]].  Conversely, any locally cartesian closed (∞,1)-category has a presentation by a right proper Cisinski model category; see [[locally cartesian closed (∞,1)-category]] for the proof.

## Related pages

* [[semi-left-exact left Bousfield localization]]

## References ##

The concept originates in

* {#BousfieldFriedlander78} [[Aldridge Bousfield]], [[Eric Friedlander]], def. 1.1.6 in _Homotopy theory of $\Gamma$-spaces, spectra, and bisimplicial sets_, Springer Lecture Notes in Math., Vol. 658, Springer, Berlin, 1978, pp. 80-130. ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/bousfield-friedlander.pdf))

Textbook account:

* {#Hirschhorn02} [[Philip Hirschhorn]], Chapter 13 of: _[[Model Categories and Their Localizations]]_, AMS Math. Survey and Monographs Vol 99 (2002) [[ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf)]

The usefulness of right properness for constructions of [[homotopy category|homotopy categories]] is discussed in

* [[Rick Jardine]], _Cocycle categories_ ([pdf](http://arxiv.org/PS_cache/math/pdf/0605/0605198v1.pdf))

The general theory can be found in

* {#Hirschhorn} [[Philip Hirschhorn]], chapter 13 of _Model Categories and Their Localizations_, 2003 ([AMS](http://www.ams.org/bookstore?fn=20&arg1=whatsnew&item=SURV-99), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf))

also in

* {#Reedy} [[Chris Reedy]], _Homotopy theory of model categories_ ([pdf](http://www-math.mit.edu/~psh/reedy.pdf))

Proposition \ref{FibrationsBetweenFibrantObjectsSuffice} can be found in

* {#Bousfield01} [[Aldridge Bousfield]], *On the telescopic homotopy theory of spaces*, Trans. Amer. Math. Soc. 353 (2001), 2391-2426, [web with fulltext](https://www.ams.org/journals/tran/2001-353-06/S0002-9947-00-02649-0/)

See also:

* {#Rezk98} [[Charles Rezk]], _Fibrations and homotopy colimits of simplicial sheaves_ ([arXiv:9811038](http://arxiv.org/abs/math/9811038))

* {#Rezk02} [[Charles Rezk]], _Every homotopy theory of simplicial algebras admits a proper model_, Topology and its Applications, Volume 119, Issue 1, 2002, Pages 65-94, (<a href="https://doi.org/10.1016/S0166-8641(01)00057-8">doi:10.1016/S0166-8641(01)00057-8</a>, [arXiv:math/0003065](http://arxiv.org/abs/math/0003065))

* {#Nikolaus10} [[Thomas Nikolaus]], _Algebraic models for higher categories_ ([arXiv:1003.1342](http://arxiv.org/abs/1003.1342))

Examples of nonproper model structures can be found in:

* [[Simon Henry]], _Counter-example to the existence of left Bousfield localization of combinatorial model category_ [MathOverflow](https://mathoverflow.net/questions/325383/counter-example-to-the-existence-of-left-bousfield-localization-of-combinatorial)

* [[Simon Henry]], _Examples of non-proper model structure_,
[MathOverflow](https://mathoverflow.net/questions/339084/examples-of-non-proper-model-structure)

[[!redirects proper model categories]]

[[!redirects right proper model category]]
[[!redirects right proper model categories]]

[[!redirects left proper model category]]
[[!redirects left proper model categories]]

[[!redirects left proper model structure]]
[[!redirects left proper model structures]]
[[!redirects right proper model structure]]
[[!redirects right proper model structures]]

[[!redirects left proper]]
[[!redirects right proper]]
[[!redirects left properness]]
[[!redirects right properness]]


proper morphism

> For the concept in [[topology]] see at _[[proper map]]_.

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

# Contents
* table of contents
{: toc}

## Definitions

### Between topological spaces

While every [[continuous map]] sends [[compact subsets]] to [[compact subsets]] (see at _[[continuous images of compact spaces are compact]]_), it is not true in general that the [[preimage]] of a compact set along a continuous map is compact.

A [[continuous function]] $f : X \to Y$ between [[topological spaces]] is a **[[proper map]]** if it is [[closed map|closed]] and the [[inverse image]] of any [[compact space|compact]] subset is itself compact.

A notion of proper homotopy between proper maps leads to [[proper homotopy theory]].

Similarly, one can consider the conditions on morphisms in other geometric situations, like [[algebraic geometry]], and properness often either reflects or implies good behaviour with respect to the compact objects (cf. also proper push-forward).

### Between schemes

A __proper morphism of [[schemes]]__ is by definition a morphism $f:X\to Y$ which is

1. [[separated morphism|separated]],

2. of [[finite type morphism|finite type]]

3. [[universally closed morphism|universally closed]] (the latter means that for every $h: Z\to Y$ the pullback $h^*(f): Z\times_Y X\to Z$ is closed).

There is a classical and very practical [[valuative criterion of properness]] due Chevalley.

We say that a scheme $X$ is __proper__ if the canonical map $X \to \operatorname{Spec} \mathbb{Z}$ to the [[terminal object]] is proper.

Proper schemes are analogous to [[compact topological spaces]]. This is one reason why one uses the terminology "quasi-compact" when referring to schemes whose underlying topological space is compact.

The [[base change]] formulas for [[cohomology]] for proper and for smooth morphisms of schemes motivated [[Alexander Grothendieck|Grothendieck]] (in [[Pursuing Stacks]]) to define abstract proper and smooth functors in the setting of [[fibered categories]]; this is further expanded on in
([Maltsiniotis](#Maltsiniotis)).

### Between locales

Recall that a [[locale]] $L$ is given by a [[frame]] $O(L)$, its [[frame of opens]], and that a [[continuous map]] $f$ from $K$ to $L$ is given by an [[adjunction]] $f^* \dashv f_* \colon O(K) \rightleftarrows O(L)$ such that the [[inverse image]] function $f^*$ preserves finitary [[meets]] (or equivalently is a frame [[homomorphism]], since it must preserve all [[joins]]).

Such a map $f$ is __proper__ iff the [[direct image]] function $f_*$ preserves [[directed joins]] (or equivalently is [[Scott-continuous function|Scott-continuous]], or equivalently is a morphism of [[preframes]]), and also satisfies the [[Frobenius reciprocity]]-like condition that $f_*(U\cup f^*(V)) = f_*(U) \cup V$ (which by itself states that the map is [[closed map|closed]]).

In particular, the map $L\to 1$ is proper iff $L$ is both [[compact locale|compact]] and [[covert space|covert]].  But in this case the second condition is redundant, since every compact locale is automatically covert; see [[covert space]] for a proof.

Proper maps of locales can also be characterized as those that are universally closed, i.e. every pullback of them (along any map of locales) is closed.

### Between toposes

Proper maps of locales can be generalized to [[geometric morphisms]] of [[Grothendieck toposes]]; see *[[proper geometric morphism]]*.

The topos-theoretic condition refers only to directed unions of subterminal objects, suggesting a stronger condition that it preserve all [[filtered colimits]].  This is a strictly stronger condition even for locales (i.e. [[localic toposes]]), called being *tidy*.  In fact properness and tidiness are the first two rungs on an infinite ladder of higher properness for [[higher toposes]].

\begin{remark}
**(Inconsistent terminology)**
\linebreak
In the world of 1-toposes, the concept of proper map splits into two
subconcepts

$$
  \mathrm{tidy} \Rightarrow \mathrm{proper}
$$

with the stronger concept being called “tidy”.

More generally in $\infty$-toposes, one gets an infinite tower of related
definitions

$$
  \infty-\mathrm{proper} \Rightarrow ... \Rightarrow 1-proper \Rightarrow 0-proper
$$

This is where the inconsistency arises: in $\infty$-topos theory it is customary to call “proper morphism” the strongest notion while in 1-topos theory it is customary to use it for the weaker notion.
Hence, a proper map of spaces $f\,:\, X \to Y$ will induce a proper map of 1-toposes but not a proper map of $\infty$-toposes (without extra assumptions).

\end{remark}

## Related concepts

* [[proper geometric morphism]]

* [[proper map]], [[open map]], [[closed map]]

* [[open morphism]], [[closed morphism]]

* [[proper base change theorem]]

* [[proper homotopy theory]]

* [[maps from compact spaces to Hausdorff spaces are closed and proper]]

## References

* [[Georges Maltsiniotis]], _Structures d'asphéricité, foncteurs lisses, et fibrations_, Ann. Math. Blaise Pascal __12__, pp. 1-39 (2005) ([ps](http://people.math.jussieu.fr/~maltsin/ps/asphbl.ps)).
 {#Maltsiniotis}

(TO ADD: The definition of a proper dg algebra, proper dg category, proper A-inf cat ???)

* Wikipedia, *[Proper morphism](http://en.wikipedia.org/wiki/Proper_morphism)*

* Daniel Halpern-Leistner, Anatoly Preygel, _Mapping stacks and categorical notions of properness_, [arxiv/1402.3204](http://arxiv.org/abs/1402.3204)

[[!redirects proper morphism]]
[[!redirects proper morphisms]]
[[!redirects proper scheme]]
[[!redirects proper schemes]]
[[!redirects proper morphism of schemes]]
[[!redirects proper morphisms of schemes]]

[[!redirects proper functor]]
[[!redirects proper functors]]


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=--
#### Higher algebra
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=--
=--
=--

This page collects links related to the article

* [[Jacob Lurie]],

  _Proper Morphisms, Completions, and the Grothendieck Existence Theorem_

  ([pdf](http://www.math.harvard.edu/~lurie/papers/DAG-XII.pdf))

on basic properties of [[E-∞ geometry]]

#Contents#
* table of contents
{:toc}

## 1. Generalities on spectral Deligne-Mumford stacks/algebraic spaces

### 1.1 Points of spectral Deligne-Mumford stacks

### 1.2 Étale morphisms

* [[étale morphism of schemes]]

### 1.3 Localic spectral Deligne-Mumford stacks

* [[algebraic space]]

### 1.4 Quasi-compactness of spectral Deligne-Mumford stacks

### 1.5 Local properties of spectral Deligne-Mumford stacks

## 2. Noetherian approximation

## 3. Properness

### 3.1 Strongly proper morphisms

### 3.2 The direct image theorem

### 3.3 Proper linear $\infty$-categories

* [[Wirthmüller context]]

### 3.4 Valuative criteria

## 4. Completions of modules

* [[torsion module]], [[torsion approximation]]

* [[completion of a module]]

## 5. Completions of spectral Deligne-Mumford stacks

### 5.1 Formal completions

### 5.2 Truncations in $QCoh(\mathcal{X}_K^{\wedge})$

### 5.3 The Grothendieck existence theorem

* [[Grothendieck existence theorem]]

### 5.4 Algebraizability of formal stacks

## 6. Relationship with formal moduli problems

* [[formal moduli problem]]

## A. Stone duality

* [[Stone duality]]

category: reference


proper subset

# Proper subsets
* table of contents
{: toc}

## Idea

A [[subset]] $A$ of a [[set]] $S$ is __proper__ if it is not the [[improper subset]] of $S$ ($S$ as a subset of itself).  We may also say (in the context of [[concrete sets]]) that $S$ is a proper [[superset]] of $A$ or that $S$ properly [[containment|contains]] $A$.

## Definition

### In material set theory

In [[material set theory]], we may state that $A$ is proper in any of these equivalent ways:

1.  $A$ is not equal (as a subset) to $S$;
2.  There exists an element $x$ of $S$ such that $x \notin A$;
3.  Given any way of expressing $A$ as the [[intersection]] of a [[family of subsets]] of $S$, this family is [[inhabited set|inhabited]].

Actually, these three definitions are equivalent only if we accept the principle of [[excluded middle]]; in [[constructive mathematics]], we usually prefer (3).  (For example, consider the notion of [[proper filter]] on a set $X$, thought of as a subset of the [[power set]] of $X$.)  However, (3) is not [[predicative mathematics|predicative]]; see [[positive element]] for discussion of this in the dual context.  Also, (2) may be strengthened using an [[inequality relation]] other than the [[denial inequality]].

### In structural set theory

In [[structural set theory]], subsets are represented by [[injections]], and so we may state that the injection $m:A \hookrightarrow S$ is proper in any of these equivalent ways:

1.  The [[injection]] $m:A \hookrightarrow S$ is not an [[bijection]];
2.  There exists an element $x$ of $S$ such that for all elements $y \in A$, $m(x) \neq y$;
3.  Given any way of expressing $m:A \hookrightarrow S$ as the [[intersection]] of a [[family]] of [[injections]] into $S$, this family is [[inhabited set|inhabited]].

Similarly as the case in [[material set theory]], these three definitions are equivalent only if we accept the principle of [[excluded middle]].

### More generally

In any category $C$, subsets become [[subobjects]] and are thus represented by the [[monomorphisms]] in $C$. Then a **proper subobject** is defined in any of the following ways:

1.  The [[monomorphism]] $m:A \hookrightarrow S$ is not an [[isomorphism]];
2.  If $C$ is a [[well-pointed category]] with [[terminal object]] $1$, then there exists a global element $x:1 \to S$ such that for all global elements $y:1 \to A$, $m \circ x \neq y$;
2.  If the [[subobject preorder]] of every object in $C$ is a [[preorder|pre]]-[[inflattice]], then given any way of expressing $m:A \hookrightarrow S$ as the [[intersection]] of a [[family]] of [[monomorphisms]] into $S$, this family is [[inhabited set|inhabited]].

## See also

* [[subset]], [[proper subset]], [[improper subset]]

* [[subobject]], [[proper subobject]], [[improper subobject]]

* [[filter]], [[proper filter]], [[improper filter]]

[[!redirects proper subset]]
[[!redirects proper subsets]]
[[!redirects proper superset]]
[[!redirects proper superset]]
[[!redirects proper containment]]

[[!redirects proper subobject]]
[[!redirects proper subobjects]]


proper topological groupoid

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#Contents#
* table of contents
{:toc}

## Definition

A [[topological groupoid]] $X_1 \stackrel{\overset{s}{\to}}{\underset{t}{\to}} X_0$ is called **proper** if the [[continuous map]]

$$
  (s,t):X_1 \to X_0\times X_0
$$

is a [[proper map]].

## Properties

In particular the [[automorphism group]] of any [[object]] in a proper topological groupoid is a [[compact group]]. In this sense proper topological groupoids generalize [[compact groups]].

## Examples

A [[Lie groupoid]] is called a _[[proper Lie groupoid]]_ if its underlying topological groupoid is proper.

An [[orbifold]] is a [[proper Lie groupoid]] which is also an [[étale groupoid]].

## Related concepts

* [[locally compact groupoid]]

[[!redirects proper groupoid]]

[[!redirects proper groupoids]]
[[!redirects proper topological groupoid]]


proper topology

Let $S$ be a [[scheme]].  The **proper topology** is a [[Grothendieck topology]] on the [[full subcategory]] of [[proper morphism]]s in $Sch/S$ (the category of schemes over $S$).  Namely, it is the topology generated by the [[pretopology]] where covering families are single-element families $\{ p : Y \to X \}$ with $p$ proper and [[surjective]].

## See also

* [[etale topology]]

## References

* [[Shun-Ichi Kimura]], _A cohomological characterization of Alexander schemes_.  Invent. math 137, 575-611 (1999).


properad

# Properads
* table of contents
{: toc}

## Definition

A __properad__ in a [[symmetric monoidal
category]] $C$ is a [[monoid]] in the [[monoidal
category]] of [[symmetric sequence|bisymmetric sequences]] in $C$ (i.e.,
functors $\Sigma\times\Sigma\to C$) equipped
with a version of [[substitution product]]
modeled on connected directed graphs with 2 levels
instead of corollas, which are used for [[operads]].
See §1.2 in [Vallette](#Vallette) for details.
The idea is that operations in a properad can
have multiple inputs and outputs, as opposed
to a single output in an operad.

### Relation to polycategories, dioperads and PROPs

The notion of coloured properads is more general than that of [[polycategories]] in that properads have [[composition]]-operations along more than one object. See at *[polycategory -- Relation to properads](polycategory#RelationToProperads)* for a more detailed explanation.

Properads are less general than [[PROPs]], which also allow a "composition" of nonconnected graphs, by tensoring morphisms with $\otimes$.

## Related concepts

* [[PROP]]
* [[polycategory]]
* [[dioperad]]
* [[multicategory]]
* [[operad]]

The **pluricategories** of [Kavanagh](#Kavanagh) (Definition 2.1.11) are similar to properads, except that they have identity morphisms for each list of objects, rather than a unary morphism for each object.

## References

* {#Vallette} [[Bruno Vallette]], A Koszul duality for props.
[arXiv:math/0411542](http://arxiv.org/abs/math/0411542)

Properads are called **compact polycategories** in:

* Ross Duncan. "Types for quantum computing." Phd thesis (2006).

The notion of **pluricategory** is defined in:

* {#Kavanagh} Ryan Kavanagh. _Communication-Based Semantics for Recursive Session-Typed Processes_. Thesis. Carnegie Mellon University, 2021 ([pdf](https://rak.ac/publication/2021-communication-based-semantics-for-recursive-session-typed-processes/thesis.pdf))

About the [[(∞,1)-category]] version

* Shaul Barkan, Jan Steinebrunner, _The equifibered approach to ∞-properads_ ([arXiv:2211.02576](https://arxiv.org/abs/2211.02576)).

[[!redirects properads]]
[[!redirects Properads]]
[[!redirects pluricategory]]
[[!redirects pluricategories]]
[[!redirects compact polycategory]]
[[!redirects compact polycategories]]


properly discontinuous action

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#Contents#
* table of contents
{:toc}

## Definition

The [[action]] of a [[topological group]] $G$ on a [[topological space]] $X$ is called _properly discontinuous_ if every point $x \in X$ has a [[neighbourhood]] $U_x$ such that the [[intersection]] $g(U_x) \cap U_x$ with its translate under the group action via some element $g \in G$ is [[inhabited set|non-empty]] only for the [[neutral element]] $e \in G$:

$$
  g(U_x) \cap U_x \neq \emptyset
  \phantom{AA}
  \Rightarrow
  \phantom{AA}
  g = e
$$

This is equivalent to the condition that the [[quotient space]] [[coprojection]] $X \longrightarrow X/G$ is a [[covering space]]-projection.

Therefore properly discontinuous actions are also called _covering space actions_ ([Hatcher](#Hatcher)).

## Related concepts

* [[proper action]]

* [[transitive action]], [[free action]], [[semi-free action]], [[regular action]]

* [[Clifford-Klein form]]

## References

* Jack Lee, chapter 12 of _Introduction to Topological Manifolds_

* Jack Lee, chapter 21 of _Introduction to Smooth Manifolds_

* Jack Lee, [MO comment](https://math.stackexchange.com/a/1083696/58526), Dec. 2014

* {#Hatcher} [[Alan Hatcher]], _[Algebraic Topology](https://www.math.cornell.edu/~hatcher/AT/ATpage.html)_

[[!redirects properly discontinuous actions]]

[[!redirects covering space action]]
[[!redirects covering space actions]]


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=--
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#Contents#
* table of contents
{:toc}

## Definition

+-- {: .num_defn #ProperlySupportedPseudoDifferentialOperator}
###### Definition
**([[properly supported peudo-differential operator]])

A [[pseudo-differential operator]] $Q$ on a [[manifold]] $X$ is called _properly supported_ if for each [[compact subset]] $K \subset X$ there exists a compact subset $K' \subset X$ such that for $u$ a [[distribution]] with [[support of a distribution|support]] in $K$ it follows that the [[derivative of distributions]] $Q u$ has support in $K'$

$$
  supp(u)\subset K \phantom{A}\Rightarrow \phantom{A} supp(Q u) \subset K'
$$

and such that

$$
  u\vert_{K'} = 0 \phantom{A} \Rightarrow \phantom{A} (Q u)\vert_{K} = 0
  \,.
$$

=--

([Hörmander 85 (18.1.21)](#Hoermander85) recalled e.g. in [Radzikowski 96. p. 8,9](#Radzikowski96))

## Examples

+-- {: .num_example}
###### Example
**([[differential operators]] are properly supported [[pseudo-differential operators]])**

Every ordinary [[differential operator]] $D$, regarded as a [[pseudo-differential operator]], is properly supported (def. \ref{ProperlySupportedPseudoDifferentialOperator}), since differential operators do not increase the [[support]] of the functions they act on:

$$
  supp(D f) \subset supp(f)
  \,.
$$

=--

## Related concepts

* [[propagation of singularities theorem]]

## References

* {#DuistermaatHoermander72} [[Johann Duistermaat]], [[Lars Hörmander]], _Fourier integral operators II_, Acta Mathematica 128, 183-269, 1972 ([Euclid](https://projecteuclid.org/euclid.acta/1485889724))

* {#Hoermander85} [[Lars Hörmander]], _The analysis of partial differential operators III_, Springer 1985

* {#Radzikowski96} [[Marek Radzikowski]], _Micro-local approach to the Hadamard condition in quantum field theory on curved space-time_, Commun. Math. Phys. 179 (1996), 529–553 ([Euclid](http://projecteuclid.org/euclid.cmp/1104287114))

[[!redirects properly supported pseudo-differential operator]]


properties of functors -- contents

**basic properties of...**

* **[[functors]]**

  * [[faithful functor]]

  * [[full functor]]

  * [[full and faithful functor]]

  * [[injective-on-objects functor]]

  * [[essentially injective functor]]

  * [[essentially surjective functor]]

    * [[surjective-on-objects functor]]

    * [[bijective-on-objects functor]]

    * [[dominant functor]]

  * [[split essentially surjective functor]]

  * [[essentially surjective and full functor]]

  * [[equivalence of categories]]

  * [[conservative functor]]

  * [[pseudomonic functor]]

  * ([[cocontinuous functor|co]])[[continuous functor]]

  * [[final functor]]

* **[[2-functors]]**

  * [[full and faithful 2-functor]]

  * [[locally fully faithful 2-functor]]

  * [[equivalence of 2-categories]]

* **[[(∞,1)-functors]]**:

  * [[fully faithful (∞,1)-functor]]

  * [[essentially surjective (∞,1)-functor]]

  * [[equivalence of (∞,1)-categories]]

  * [[homotopy final functor]], [[final (∞,1)-functor]]


* [[Lukas Buhné]], _Properties of Integral Morava K-Theory and the Asserted Application to the Diaconescu-Moore-Witten Anomaly_, Diploma thesis Hamburg (2011) ([spire:899170](http://inspirehep.net/record/899170))

on the application of [[integral Morava K-theory]] to the [[Diaconescu-Moore-Witten anomaly]], following

* {#SatiKriz04} [[Igor Kriz]], [[Hisham Sati]], _M-theory, type IIA superstrings, and elliptic cohomology_, Adv. Theor. Math. Phys. 8 (2004), no. 2, 345–394 ([arXiv:hep-th/0404013](http://arxiv.org/abs/hep-th/0404013))

For more along these lines see also

* [[Hisham Sati]], _[[Geometric and topological structures related to M-branes]]_ (2010)

category: reference


property sup

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# The property (sup)
* table of contents
{: toc}

## Idea

The property (sup) is an optional property of an [[abelian category]], introduced in ([Gabriel](#Gabriel)).

## Definition

An [[abelian category]] has __property (sup)__ if:

+-- {: .un_defn}
###### Definition (sup)

For any ascending chain $\Omega$ of [[subobjects]] of a fixed object $M$, the [[supremum]] of $\Omega$ exists; and, for any subobject $L\hookrightarrow M$, the canonical morphism
$$
sup\{L\cap P | P\in \Omega\}\to (sup \Omega) \cap L
$$
is an [[isomorphism]].
=--

## Examples

Gabriel's property (sup) is satisfied by any [[Grothendieck category]] (in some expositions it is listed as a part of the definition), e.g. the category of all [[modules]] over a fixed [[ring]] $R$, and the category of [[sheaf|sheaves]] of [[abelian groups]] on a fixed [[topological space]] $X$.

According to the Appendix B of the Thomason--Trobaugh contribution to [[Grothendieck Festschrift]], it is not known whether the category $Qcoh_Y$ of [[quasicoherent sheaf|quasicoherent sheaves]] over an arbitrary scheme $Y$ is a Grothendieck category (although this is known for a large class), but it is an elementary fact that it does satisfy Gabriel's property (sup). If an abelian category is [[noetherian category|noetherian]] it clearly satisfies the property (sup).

## References

* [[Pierre Gabriel]], _[[Des catégories abéliennes]]_
{#Gabriel}

[[!redirects property sup]]
[[!redirects property (sup)]]
[[!redirects Gabriel's property (sup)]]


prophk2015

The quantization is one of the leading problems in physics. Introducing constraints, or factoring the symmetries (reduction) in geometry of phase space leads to generalizations of Poisson structures like Dirac structures,
and from the point of view of quantization the situation is singular and
needs cohomological resolutions. New geometric objects which appear in such constructions like Courant algebroids and higher analogues like $L_\infty$-algebroids (and almost algebroids of Strobl et al.) will be in focus of that research.
More systematically cohomological obstructions are seen in a higher categorical framework called derived geometry. Most notably, work of
Pantev, Toen et. al.
natural notions of derived versions of symplectic structures (motivated by field theory) are systematically studied. We are interested to investigate concrete resolutions appearing in quantization from this point of view.

Another topic is to understand appearance of cyclic cohomology in the case of some simple cases of singularities like orbifold singularities. A result of Hinich and independently Skoda is that inertia orbifold (or homotopically equivalent loop orbifold, that is loop space of the orbifold) which plays the central role in this picture has category of equivariant sheaves which is Drinfeld center of the category of sheaves on the original manifold. The orbifold cohomology is, as shown by Baranovsky in the simplest cases (we propose to further extend his result, say for the case with a gerbe) related to cyclic homology. HKR theorem relates de Rham cohomology to Hochschild but in smooth case only, singular case requires cyclic homology again. Toen and Vezzosi have shown further that derived loop spaces (in algebraic context)
are related to cyclic homology again (similar result with formal loop space is by Kapranov and Vasserot). We want to study similar correspondences in the setting of differential geometry with singularities.

Quantum phase spaces are recently studied by Skoda, Meljanac and Stojic as carrying Hopf algebroid structures; they also enable to use Hopf algebroid generalization of Drinfeld twist in quantization problems. Recently Szabo, Schupp, Jur\v co and others explored nonabelian cocycles on spaces (e.g. coming from gerbes) to study Drinfeld twist operators leading not only to twisted theories but also to nonassociative geometry. We are interested in considering this problem from more general point of view where the phase is
Hopf algebroid/like structure.

A.Kotov, T.Strobl, Generalizing geometry - algebroids and sigma models, arxiv/1004.0632, in Handbook on Pseudo-Riemannian Geometry

T. Pantev, B. Toen, M. Vaquie, G. Vezzosi, Shifted symplectic structures, arxiv/1111.3209, to appear Publ. IHES

B. Toën, G. Vezzosi, A note on Chern character, loop spaces and derived algebraic geometry, arxiv/0804.1274, Proc. of the 2007 Abel Symposium

V. Baranovsky, Orbifold cohomology as periodic cyclic homology, math.AG/0206256

G. E. Barnes, A. Schenkel, R. J. Szabo, Working with nonassociative geometry and field theory,  arXiv:1601.07353