weak homotopy equivalence

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### Context
#### Homotopy theory
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[[!include homotopy - contents]]
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#### Equality and Equivalence
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[[!include equality and equivalence - contents]]
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# Weak homotopy equivalences
* table of contents
{: toc}

## Idea

A _[[weak homotopy equivalence]]_ is a map between [[topological spaces]] or [[simplicial sets]] or similar which induces [[isomorphisms]] on all [[homotopy groups]].  (The analogous concept in [[homological algebra]] is called a _[[quasi-isomorphism]]_.)

The [[localization]] or [[simplicial localization]] of the categories [[Top]] and [[sSet]] at the weak homotopy equivalences used as [[weak equivalences]] yields the standard [[homotopy category]] [[Ho(Top)]] and [[Ho(sSet)]] or the [[(∞,1)-category]] of [[∞-groupoids]]/[[homotopy types]], respectively.

Weak homotopy equivalences are named after _[[homotopy equivalences]]_. They can be identified with homotopy equivalences after one allows to replace the [[domains]] by a [[resolution]]. The corresponding notions in [[homological algebra]] are [[quasi-isomorphisms]] and [[chain homotopy]]-equivalences.

From another perspective,  the notion of _weak homotopy equivalence_ is 'observational', in that a map is a weak homotopy equivalence if when we look at it through the observations that we can make of it using [[homotopy groups]] or even the [[fundamental infinity-groupoid]], it looks like an equivalence.   In contrast, _[[homotopy equivalence]]_ is more 'constructive'; in that $f$ is a homotopy equivalence if there exists an inverse for it (up to homotopy, of course).  Note that both of these notions are weaker than mere [[isomorphism]] of topological spaces (homeomorphism) and so can be considered examples of [[weak equivalence]]s.

There are actually two related concepts here: whether two spaces are weakly homotopy equivalent and whether a map between spaces is a weak homotopy equivalence.  The former is usually defined in terms of the latter.

## Definition (for topological spaces and simplicial sets)

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

For $X, Y \in $ [[Top]] or $\in $ [[sSet]] two [[topological spaces]] or [[simplicial sets]], a [[continuous function]] or [[simplicial set|simplicial map]] $f : X \to Y$ between them is called a **weak homotopy equivalence** if

1. $f$ induces an [[isomorphism]] of [[connected components]] (path components in the case of topological spaces)

   $$
     \Pi_0(f) \colon \Pi_0(X) \stackrel{\simeq}{\to} \Pi_0(Y)
   $$

   in [[Set]];

1. for all [[points]] $x \in X$ and for all $(1 \leq n) \in \mathbb{N}$ $f$ induces an isomorphism on [[homotopy groups]]

   $$
     \pi_n(f,x) \colon \pi_n(X,x) \stackrel{\simeq}{\to} \pi_n(Y,f(x))
   $$

   in [[Grp]].

=--

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

If $X$ and $Y$ are [[connected space|path-connected]], then (1) is trivial, and it suffices to require (2) for a single (arbitrary) $x$, but in general one must require it for at least one $x$ in each path [[connected component]].

=--

\begin{remark}
It is not enough to require that each pair of homotopy groups are isomorphic. For example, let $Y = S^1 \vee S^2$, and let $X$ be its double cover, which glues two spheres at the north and south poles of a circle. Then there are isomorphisms on the fundamental group $\pi_1(X) \simeq \mathbb{Z} \simeq \pi_1(Y)$, although the map $\pi_1(X) \to \pi_1(Y)$ induced by $f$ is not an isomorphism. And by the properties of covering spaces, all higher homotopy groups are isomorphic. But these two are not weakly homotopy equivalent.
\end{remark}

\begin{remark}
There are many alternative definitions of weak homotopy equivalences.
A simplicial map $f$ is a weak equivalence of simplicial sets
if and only if $Ex^\infty f$ is a simplicial homotopy equivalence
if and only if $Hom(f,A)$ is a simplicial homotopy equivalence for any [[Kan complex]] $A$
if and only if $f$ has a right relative-homotopy-lifting property with respect
to the maps $\partial\Delta^n\to\Delta^n$
if and only if $f$ is a composition of an acyclic cofibration
(i.e., a map with a left lifting property with respect to all maps with a right lifting
property with respect to horn inclusions)
and an acyclic fibration (i.e., a map
with a right lifting property with respect to inclusions $\partial\Delta^n\to\Delta^n)$.
A continuous map $f$ is a weak equivalence of topological spaces
if and only if $|Sing(f)|$ is a homotopy equivalence of topological spaces
if and only if $Hom(A,f)$ is a homotopy equivalence for any [[CW-complex]] $A$
if and only if $f$ has a right relative-homotopy-lifting property with respect
to the maps $S^{n-1}\to D^n$
if and only if $f$ is a composition of an acyclic Serre cofibration
(a retract of a relative CW-complex) and an acyclic [[Serre fibration]].
Both functors $|-|$ and $Sing$ preserve and reflect weak equivalences,
so any of the two classes defines the other.
\end{remark}

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

The [[homotopy category]] of [[Top]] with respect to weak homotopy equivalences is [[Ho(Top)]]${}_{whe}$.

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

Accordingly, weak homotopy equivalences are the [[weak equivalences]] in the standard [[Quillen model structure on topological spaces]] and the [[Quillen model structure on simplicial sets]], and also in the [[mixed model structure]].

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## Properties

### Equivalent characterizations
 {#EquivalentCharacterizations}

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

A continuous map $f : X \to Y$ is a weak homotopy equivalence
precisely if for all $n \in \mathbb{N}$ and for all [[commuting diagrams]] of continuous maps of the form

$$
  \array{
    S^{n-1} &\to& X
    \\
    \downarrow && \downarrow^{\mathrlap{f}}
    \\
    D^n &\to& Y
  }
  \,,
$$

where the left morphism is the inclusion of the $(n-1)$-[[sphere]] as the [[boundary]] of the $n$-[[ball]],
there exists a continuous map $\sigma : D^n \to X$
that makes the resulting upper triangle commute and such that the lower triangle commutes up to a [[homotopy]]

$$
  \array{
    S^{n-1} &&\to&& X
    \\
    \\
    \downarrow && \nearrow & \swArrow& \downarrow^{\mathrlap{f}}
    \\\\
    D^n &&\to&& Y
  }
$$

which is constant along $S^{n-1} \hookrightarrow D^n$.

=--

In this form the statement and its proof appears in
([Jardine](#Jardine)) (where it is also generalized to weak equivalences in a [[model structure on simplicial presheaves]]).
See also around ([Lurie, prop. 6.5.2.1](#Lurie)). The relevant arguments are spelled out in ([May, section 9.6](#May)). A variant is called the _HELP lemma_ in ([Vogt](#Vogt)).

### Relation to homotopy equivalences
 {#RelationToHomotopyEquivalences}

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

Every [[homotopy equivalence]] is a weak homotopy equivalence.

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

It requires a little bit of thought to prove this, because $f$ and its homotopy inverse $g$ need not preserve any chosen basepoint.  But for any $x\in X$ and any $n\ge 1$, we have a [[commuting diagram]]
$$
\array{
  \pi_n(X,x) & & \to & & \pi_n(X,g(f(x)))
  \\
  &
  \searrow
  &&
  \nearrow
  &&
  \searrow
  \\
  &&
  \pi_n(Y,f(x))
  &&
  \longrightarrow
  &&
  \pi_n\big(
    Y,f(g(f(x)))
  \big)
  }
$$
in which the two horizontal morphisms are [[isomorphisms]], because $g f$ and $f g$ are [[homotopy|homotopic]] to [[identity morphism|identities]].  Hence, by the [[two-out-of-six property]] for isomorphisms, the diagonal morphhisms are also all isomorphisms.

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

Conversely, any weak homotopy equivalence between [[m-cofibrant spaces]] (spaces that are homotopy equivalent to [[CW complexes]]) is a [[homotopy equivalence]].

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### Relation to homotopy types
 {#RelationToHomotopyTypes}

We discuss the [[equivalence relation]] generated by weak homotopy equivalence, called _(weak) [[homotopy type]]_. For the "abelianized" analog of this situation see at [[quasi-isomorphism]] the section _[Relation to homology type](quasi-isomorphism#RelationToChainHomologyType)_.

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

The existence of a weak homotopy equivalence from $X$ to $Y$ is a [[reflexive relation|reflexive]] and [[transitive relation]] on [[Top]], but it is not a  [[symmetric relation]].

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

Reflexivity and transitivity are trivially checked. A counterexample to symmetry
is example \ref{CircleAndPseudoCircle} below.

=--

But we can consider the genuine equivalence relation _generated_ by weak homotopy equivalence:

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

We say two spaces $X$ and $Y$ have the same **(weak) [[homotopy type]]** if they are equivalent under the [[equivalence relation]] _generated_ by weak homotopy equivalence.

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

Equivalently this means that $X$ and $Y$ have the same (weak) homotopy type
if there exists a [[zigzag]] of weak homotopy equivalences

$$
  X \leftarrow \to\leftarrow \dots \to Y
  \,.
$$

This in turn is equivalent to saying that $X$ and $Y$ become [[isomorphism|isomorphic]] in the [[homotopy category]] [[Ho(Top)]]/[[Ho(sSet)]] with the weak homotopy equivalences [[localization|inverted]].

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

Two spaces $X$ and $Y$ may have isomorphic homotopy groups without being weak homotopy equivalence: for this all the isomorphisms must be induced by an actual map $f : X \to Y$, as in the above definition.

However, if, roughly, one remembers,  how all the homotopy groups [[nLab:action|act]] on each other, then this is enough information to exhibit the full homotopy type. This collection of data is called the _[[Postnikov tower]]_ decomposition of a homotopy type.

=--

### Relation to free homotopy sets
 {#RelationToFreeHomotopySets}

For $K, X \,\in\, TopSp$, write $Map(K, X)$ for their [[mapping space]], i.e. not considering or respecting any [[basepoint]]. For example,  $Map(S^1, Y)$ is the [[free loop space]] of $Y$, in contrast to the [[based loop space]]

$$
  \Omega_x X
    \xhookrightarrow{\; fib_x(ev_\ast) \;}
  Map(S^1, X)
    \overset{\; ev_x \;}{\twoheadrightarrow}
  X
$$

for any base point $x \,\in\, X$.

Moreover, when $K$ is a [[CW complex]], write

\[
  \label{FreeHomotopySets}
  {[K,X]} \,\coloneqq\, \tau_0 \,Map(K,X)\,
\]

for the *free homotopy set* of maps from $K$ to $X$, hence the set of [[homotopy classes]] of map $K \to X$, hence the set of [[connected components]] of the mapping space.

\begin{proposition}
**(weak homotopy equivalences detected on free homotopy sets)**
\label{WeakHomotopyEquivalencesDetectedOnTermsOfFreeHomotopySet}
\linebreak
  For $f \,\colon\, X \to Y$, a [[continuous function]] between [[connected topological spaces]], the following are equivalent:

1. $f$ is a [[weak homotopy equivalence]]

   $$
     \underset{n \in \mathbb{N}}{\forall}
     \;
     \;\;
     p_n(X)
     \underoverset
       {\simeq}
       {f_\ast}
       {\longrightarrow}
     \pi_n(Y)
     \,;
   $$

1. $f$ induces an [[isomorphism]] on all free homotopy sets (eq:FreeHomotopySets) out of [[CW-complexes]]:

   $$
     \underset{ K \in CWCplx}{\forall}
     \;\;
     [K, X]
     \underoverset
       {\simeq}
       {f_\ast}
       {\longrightarrow}
     [K, Y]
     \,;
   $$

* $f$ induces

  1. an [[isomorphism]] on all free homotopy sets (eq:FreeHomotopySets) out of $K = $any [[n-sphere]] of [[positive number|positive]] [[dimension]],

     $$
       \underset{
         n \in \mathbb{N}_+
       }{\forall}
       \;\;
       [S^n, X]
       \underoverset
         {\simeq}
         {f_\ast}
         {\longrightarrow}
       [S^n, Y]
       \,.
     $$

  1. a [[surjection]] on the free homotopy set out of the [[wedge sum]] of [[circles]] [[index set|indexed]] by (the set [[underlying]]) the [[fundamental group]] of $Y$:

     $$
        \big[
          \underset{\pi_1(Y)}{\vee} S^1
          ,\,
          X
        \big]
        \overset{\; f_\ast \;}{\twoheadrightarrow}
        \big[
          \underset{\pi_1(Y)}{\vee} S^1
          ,\,
          Y
        \big]
     $$

\end{proposition}
([Matumoto, Minami and Sugawara 1984, Thm. 2](#MatumotoMinamiSugawara84))
\begin{proof}
  The implication $(1) \Rightarrow (2)$ is a standard/classical conclusion in homotopy theory, for example it is a small special case of the fact that  $Map(K,-)$ out of any cell complex $K$ preserves weak homotopy equivalences ([this Prop.](classical+model+structure+on+topological+spaces#HomProductAdjunctionForCofibrantObjectInTopCGIsQuillen)).

The implication $(2) \Rightarrow (3)$ is trivial, as the conditions in (3) are just a special case of the condition in (2).

So the point of the statement is that (3) is already sufficient to recover (1). This is the content of [Matumoto, Minami and Sugawara 1984, Thm. 1 & Lem. 1.3](#MatumotoMinamiSugawara84).
\end{proof}

## For other kinds of spaces

A map of [[simplicial sets]] is called a weak homotopy equivalence equivalently if its [[geometric realization]] is a weak homotopy equivalence of topological spaces, as above.  (Since the geometric realization of any simpicial set is a
[[CW complex]], in this case its geometric realization is actually a [[homotopy equivalence]].)

Likewise, a [[functor]] between [[small category|small]] categories is sometimes said to be a weak homotopy equivalence if its [[nerve]] is a weak homotopy equivalence of simplicial sets, hence of topological spaces after [[geometric realization of categories]].  These are the weak equivalences in the
[[Thomason model structure]] on categories (not the [[canonical model structure]]). The statement of [[Quillen's theorem A]] and [[Quillen's theorem B]] in in this contex.

Similarly, one can define weak homotopy equivalences between any sort of object that has a [[geometric realization]], such as a [[cubical set]], a [[globular set]], an [[n-category]], an [[n-fold category]], and so on.

Note that in some of these cases, such as as simplicial sets, symmetric sets, and probably cubical sets, there is also a notion of "homotopy equivalence" from which this notion needs to be distinguished.  A simplicial homotopy equivalence, for instance, is a simplicial map $f:X\to Y$ with an inverse  $g:Y\to X$ and simplicial homotopies $X\times \Delta^1 \to X$ and $Y\times \Delta^1 \to Y$ relating $f g$ and $g f$ to identities.

A different direction of generalization is the notion of a [[homotopy equivalence of toposes]].

## Examples

### Of non-reversible weak homotopy equivalences
 {#ExamplesOfNonReversibleWHEs}

We discuss examples of weak homotopy equivalences that have no weak homotopy equivalence going the other way, according to prop. \ref{ReflexiveAndTransitiveButNotSymmetric} above.

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

Let $S^1 \in $ [[Top]] denote the ordinary [[circle]] and $\mathbb{S}$ the
[[pseudocircle]].

There is a [[continuous function]] $S^1 \to \mathbb{S}$ which is a weak homotopy equivalence, hence in particular $\pi_1(\mathbb{S}) \simeq \mathbb{Z}$. But every continuous map the other way round has to induce the trivial map on $\pi_1$.

=--

This is the simplest in a class of counter-examples discussed in ([McCord](#McCord)).

## Related concepts

* [[equality]]

* [[isomorphism]]

* [[equivalence]]

* [[weak equivalence]]

* [[homotopy equivalence]], **weak homotopy equivalence**

  * [[rational homotopy equivalence]]

  * [[n-equivalence]]

  * [[equivariant weak homotopy equivalence]]

  * [[stable weak homotopy equivalence]]

* [[Bousfield equivalence]]

* [[homotopy equivalence of toposes]]

* [[equivalence in an (∞,1)-category]]

* [[equivalence of (∞,1)-categories]]

## References

A general account is for instance in section 9.6 of

* {#May} [[Peter May]], _[[A Concise Course in Algebraic Topology]]_

The characterization of weak homotopy equivalences by lifts up to homotopy seems is in

* {#Jardine} [[J. F. Jardine]], _Simplicial Presheaves_, Journal of Pure and Applied Algebra 47, 1987, no.1, 35-87.

* {#Vogt} [[Rainer Vogt]], _The HELP-Lemma and its converse in Quillen model categories_ ([arXiv:1004.5249](http://arxiv.org/abs/1004.5249))

For related and general discussion see also section 6.5 of

* {#Lurie} [[Jacob Lurie]], _[[Higher Topos Theory]]_

Examples for the non-symmetry of the weak homotopy equivalence relation are in

* {#McCord} [[Michael McCord]], _Singular homology groups and homotopy groups of finite topological spaces_, Duke Math. J. Volume 33, Number 3 (1966), 465-474. ([euclid.dmj/1077376525](http://projecteuclid.org/euclid.dmj/1077376525))

See also:

* Topospaces-Wiki, _[Weak homotopy equivalence of topological spaces](http://topospaces.subwiki.org/wiki/Weak_homotopy_equivalence_of_topological_spaces)_

Detection in terms of free homotopy sets:

* {#MatumotoMinamiSugawara84} [[Takao Matumoto]], [[Norihiko Minami]], [[Masahiro Sugawara]], *On the set of free homotopy classes and Brown's construction*, Hiroshima Math. J. 14(2): 359-369 (1984) ([doi:10.32917/hmj/1206133043](https://projecteuclid.org/journals/hiroshima-mathematical-journal/volume-14/issue-2/On-the-set-of-free-homotopy-classes-and-Browns-construction/10.32917/hmj/1206133043.full))

[[!redirects weak homotopy equivalences]]


weak inverse

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###Context###
#### Higher category theory
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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A weak inverse or quasi-inverse is like an inverse, but weakened to work in situations where being an inverse on the nose would violate the [[principle of equivalence]].

## Definitions

Given a [[functor]] $F: C \to D$, a __weak inverse__ of $F$ is a functor $G: D \to C$ with [[natural isomorphism]]s
$$ \iota: id_C \to G \circ F,\; \epsilon: F \circ G \to id_D .$$

If it exists, a weak inverse is unique up to natural isomorphism, and furthermore can be improved to form an [[adjoint equivalence]], where $\iota$ and $\epsilon$ sastisfy the [[triangle identities]].

More generally, given a $2$-[[2-category|category]] $\mathcal{B}$ and a morphisms $F: C \to D$ in $\mathcal{B}$, a __weak inverse__ of $F$ is a morphism $G: D \to C$ with $2$-isomorphisms
$$ \iota: id_C \to G \circ F,\; \epsilon: F \circ G \to id_D .$$

Weak inverses give the proper notion of [[equivalence of categories]] and [[equivalence]] in a $2$-category.  Note that you must use [[anafunctor]]s to get the weak notion of equivalence of categories here without using the [[axiom of choice]].

## Links with homotopy theory

Given the [[geometric realization of categories]] functor $ \vert -\vert: Cat \to Top$, weak inverses are sent to [[homotopy]] inverses. This is because the product with the [[interval groupoid]] is sent to the product with the topological interval $[0,1]$. In fact, less is needed for this to be true, because the classifying space of the [[interval category]] is also the topological interval. If we define a _lax inverse_ to be given by the same data as a weak inverse, but with $\iota$ and $\epsilon$ replaced by [[natural transformation|natural transformations]], then the classifying space functor sends lax inverses to homotopy inverses.  An example of a lax inverse is an [[adjunction]], but not all lax inverses arise this way, as we do not require the triangle identities to hold.

([[David Roberts]]: I'm just throwing this up here quickly, it probably needs better layout or even its own page.)

## Related concepts

* [[inverse]]

[[!redirects weak inverses]]


weak Kan complex

#Contents#
* table of contents
{:toc}

## Idea

A **weak Kan complex** [[Boardman & Vogt (1973)](#BoardmanVogt73)] is a [[simplicial set]] for which all _inner_ [[horns]] have a filler.

The notion was later popularized under the name *[[quasi-category]]*. See there for more.

## References

* {#BoardmanVogt73} [[Michael Boardman]], [[Rainer Vogt]], *Homotopy invariant algebraic structures on topological spaces*, Lecture Notes in Mathematics **347** Springer (1973) [[doi:10.1007/BFb0068547](https://doi.org/10.1007/BFb0068547)]

[[!redirects weak Kan complexes]]


weak limit

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### Context
#### Category theory
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[[!include category theory - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A **weak limit** for a [[diagram]] in a [[category]] is a [[cone]] over that diagram which satisfies the existence property of a [[limit]] but not necessarily the uniqueness. The dual concept is a [[weak colimit]].

Beware that "weak" here does **not** correspond to that in "[[weak n-category]]", in particular it does **not** refer to [[homotopy limits]]. Nevertheless, there is a relation, see [below](#RelationToHomotopyLimits). It is due to this relation that weak limits in [[homotopy categories]] play a key role in the [[Brown representability theorem]].

## Weak pullbacks

A **weak pullback** of a  [[cospan]]

$$
  A\overset{f}{\to} C \overset{g}{\leftarrow} B
$$

(in some [[category]]) is a [[commutative diagram|commutative square]]

$$
  \array{
    P & \overset{p}{\to} & A
    \\
    {}^{\mathllap{q}} \downarrow && \downarrow^{\mathrlap{f}}
    \\
    B & \overset{g}{\to} & C
  }
$$

such that for every commuting square

\[\array{ X & \overset{x}{\to} & A\\
   {}^{\mathllap{y}} \downarrow && \downarrow^{\mathrlap{f}}\\
   B & \overset{g}{\to} & C}\]

there exists a morphism $h: \colon X\to P$, not necessarily unique, such that $x = h p$ and $y = h q$;

If the actual [[pullback]] $A \underset{C}{\times}B$ exists, then this condition means equivalently that the universal morphism

$$
  P \longrightarrow A \underset{C}{\times}B
$$

is a [[split epimorphism]].

## Weak terminal objects

Every [[inhabited set]] is a weak [[terminal object]] in [[Set]], since there always exists a [[function]] from any [[set]] to any inhabited set.  But only a [[singleton]] is a terminal object.

## Projective objects and exact completion

In any category with finite limits and [[projective object|enough projectives]], the full [[subcategory]] of [[projective object]]s has weak finite limits.  For example, given a cospan $A\overset{f}{\to} C \overset{g}{\leftarrow} B$ of projective objects, let $P\to A\times_C B$ be a projective cover of the actual pullback; then any square
\[\array{ X & \overset{x}{\to} & A\\
  ^y \downarrow && \downarrow ^f\\
  B & \overset{g}{\to} & C}\]
with $X$ projective induces a morphism $X\to A\times_C B$, which lifts to a morphism $X\to P$ since $X$ is projective.

Conversely, from any category with weak finite limits one can construct an [[exact category|exact completion]] in which the original category sits as the projective objects, and the exact categories constructible in this way are precisely those having enough projectives.

## Relation to homotopy limits
 {#RelationToHomotopyLimits}

Unlike usages of 'weak' in terms like [[weak n-category]], a weak limit is not be like a [[homotopy limit]] or a [[2-limit]], which satisfy uniqueness (as well as existence) albeit only up to higher [[homotopy|homotopies]] or [[weak equivalence|equivalences]].

However, some homotopy limits induce the corresponding type of weak limit in the corresponding [[homotopy category]].  For example, suppose that
$$\array{ P & \overset{p}{\to} & A\\
  ^q \downarrow && \downarrow ^f\\
  B & \overset{g}{\to} & C}
$$
is a homotopy pullback in some category $M$ having a notion of [[homotopy]], such as a [[model category]].  In particular, this square commutes up to homotopy, and thus it commutes in the homotopy category $Ho(M)$.  Then any square
$$\array{ X & \overset{x}{\to} & A\\
  ^y \downarrow && \downarrow ^f\\
  B & \overset{g}{\to} & C}$$
that commutes in $Ho(M)$ commutes up to homotopy in $M$, and thus (by the ("local") universal property of homotopy pullbacks), there is a map $h:X\to P$ and homotopies $p h \simeq x$ and $q h\simeq y$; thus the given square is a weak pullback in $Ho(M)$.  While the universal property of a homotopy pullback means that $h$ is unique up to homotopy, this is only true for a given _choice_ of homotopy $f x \simeq g y$, and different such homotopies can induce inequivalent $h$'s.  Thus in $Ho(M)$, which remembers only the _existence_ of homotopies, we have only a weak pullback.

Note, though, that not _all_ homotopy limits produce weak limits in the homotopy category, because in general it will not be possible to lift a cone that commutes in $Ho(M)$ to a cone that commutes up to _coherent_ homotopy in $M$.  However, in "simple" cases such as pullbacks, products, equalizers, sequential inverse limits, and so on, this is always true (and it will be true whenever the diagram category is a [[quiver]]).  On the other hand, homotopy products in $M$ give actual (not weak) products in $Ho(M)$, since there are no homotopies necessary.

## Related concepts

* [[weak adjoint]]

* [[weak multilimit]]

## References

* [[Peter Freyd]], _Representations in abelian categories_ 1966 Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965) pp. 95–120 Springer, New York

[[!redirects weak limits]]

[[!redirects weak pullback]]
[[!redirects weak pullbacks]]

[[!redirects weak pushout]]
[[!redirects weak pushouts]]

[[!redirects weak finite limit]]
[[!redirects weak finite limits]]

[[!redirects weak terminal object]]
[[!redirects weak terminal objects]]
[[!redirects weakly terminal]]


weak local ring

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### Context
#### Algebra
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[[!include higher algebra - contents]]
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\tableofcontents

## Idea

A weak notion of [[local ring]] in the context of [[constructive mathematics]].

## Definition

A **weak local ring** is a [[commutative ring]] such that

* $0 \neq 1$; and

* the sum of two non-invertible elements is non-invertible

These are the same as [[local rings]] in classical mathematics, but are a weaker (and thus more general) notion than local rings in [[constructive mathematics]].

## Properties

The non-invertible elements in a weak local ring form an [[ideal]]. Thus, the quotient of a weak local ring by its ideal of non-invertible elements form a [[weak Heyting field]] (cf. [Richman 2020](#Richman20)) or a [[Johnstone residue field]] (cf. [Johnstone 1977](#Johnstone77)).

Every weak local ring has an [[equivalence relation]] $\approx$, defined as $x \approx y$ if and only if $x - y$ is non-invertible. Then weak Heyting fields are precisely the weak local rings for which $\approx$ implies [[equality]].

## Examples

* Every [[weak Heyting field]] is an weak local ring where every non-invertible element is equal to zero.

* The dual algebra $\mathbb{R}[\epsilon]/\epsilon^2$ of the [[MacNeille real numbers]] $\mathbb{R}$ is a weak local ring where the [[nilpotent]] [[infinitesimal]] $\epsilon \in \mathbb{R}[\epsilon]/\epsilon^2$ is a non-zero non-invertible element.

* For any prime number $p$ and any positive natural number $n$, the [[prime power local ring]] $\mathbb{Z}/p^n\mathbb{Z}$ is an weak local ring, whose ideal of non-invertible elements is the ideal $p(\mathbb{Z}/p^n\mathbb{Z})$. The quotient of $\mathbb{Z}/p^n\mathbb{Z}$ by its ideal of non-invertible elements is the [[finite field]] $\mathbb{Z}/p\mathbb{Z}$.

* Every [[local ring]] is a weak local ring with an [[apartness relation]] $\#$ such that for all $a \in R$ and $b \in R$, $a \# b$ if and only if $a - b$ is invertible. The negation of $a \# b$ is an [[equivalence relation]] which holds if and only if $a - b$ is non-invertible, making every local ring a weak local ring.

## Weakly ordered local rings

A weakly ordered local ring is a weak local ring $R$ with a [[preorder]] $\leq$ such that

* for all $a \in R$ and $b \in R$, $a \approx b$ if and only if $a \leq b$ and $b \leq a$
* for all $a \in R$, $b \in R$, and $c \in R$, if $a \leq b$, then $a + c \leq b + c$
* for all $a \in R$ and $b \in R$, if $0 \leq a$ and $0 \leq b$, then $0 \leq a \cdot b$

If additionally, for all $a \in R$ and $b \in R$, $a \approx b$ implies that $a = b$, then a weakly ordered local ring becomes a weakly ordered Heyting field, and the preorder becomes a [[partial order]].

Every [[ordered local ring]] and thus every [[ordered field]] is a weakly ordered local ring.

## See also

* [[local ring]]

* [[ordered local ring]]

## References

* {#Johnstone77} [[Peter Johnstone]], *Rings, Fields, and Spectra*, Journal of Algebra **49** (1977) pp 238-260. doi:[10.1016/0021-8693(77)90284-8](https://doi.org/10.1016/0021-8693%2877%2990284-8)

* {#Richman20} [[Fred Richman]], *Laurent series over $\mathbb{R}$*. Communications in Algebra, Volume 48, Issue 5, 11 Jan 2020 Pages 1982-1984 [[doi:10.1080/00927872.2019.1710166](https://doi.org/10.1080/00927872.2019.1710166)]

[[!redirects weak local ring]]
[[!redirects weak local rings]]


weak model category

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#Contents#
* table of contents
{:toc}

## Idea

The concept of _weak model categories_ is a relaxation of that of [[model categories]], even weaker than the concept of [[semimodel categories]], but such that it still allows for a rich theory largely analogous to that of actual [[model categories]]:

The weak analogue of the construction of the [[homotopy category of a model category]] still exists, as do notions of [[Quillen adjunction]] and [[Quillen equivalence]].

Also, for example, an analogue of left or right _[[Bousfield localization of model categories]]_ still makes sense for weak model categories; and, as a bonus in contrast to the usual case, it does not require the assumption of left or right [[proper model category|properness]].

## Definition

A __weak model category__ is a [[premodel category]] that satisfies the following two [[axioms]]:

1. __Cylinder axiom__: Every [[cofibration]] $A\to X$ from a [[cofibrant object]] to a [[fibrant object]] admits a __relative strong cylinder object__
$$X\sqcup_A X\to I_A X\to X,$$
where the left map is a [[cofibration]] and its first component $X\to I_A X$
is an [[acyclic cofibration]].

1. __Path object axiom__: Every [[fibration]] $A\to X$ from a [[cofibrant object]] to a [[fibrant object]] admits a __relative strong path object__
$$A\to P_X A\to A\times_X A,$$
where the right map is a [[fibration]] and its first component $P_X A\to A$ is an [[acyclic fibration]].

## Properties

### Relation to premodel categories

A [[premodel category]] can be upgraded to a weak model category
as follows.

\begin{theorem}
If a [[premodel category]] admits a weak Quillen cylinder,
then it is a weak model category.
\end{theorem}

\begin{definition}
A __weak Quillen cylinder__ on a [[premodel category]] $C$
is a pair of [[left adjoint functors]] $I,D\colon C\to C$
together with the following commutative square of
[[natural transformations]] of [[functors]] $C\to C$:
$$\begin{matrix}
id_C\sqcup \id_C&\mathop{\longrightarrow}\limits^i&I\cr
\downarrow\nabla&&\downarrow e\cr
id_C&\mathop{\longrightarrow}\limits_j&D,\cr
\end{matrix}$$
where $\nabla$ is the [[codiagonal]],
$i$ is a [[cofibration]],
$j$ is a [[trivial cofibration]],
and the first component of $i$ is a [[trivial cofibration]].
\end{definition}

Here a [[natural transformation]] $\lambda\colon F\to G$
of [[functors]] $C\to C$ is a cofibration
if for any (trivial) cofibration $X\to Y$
the map $F(Y)\sqcup_{F(X)}G(X)\to G(Y)$ is a (trivial) cofibration.
Likewise, $\lambda$ is a trivial cofibration
if for any cofibration $X\to Y$ the above map is a trivial cofibration.

Reference: [Henry 20, Section 6](#Henry20).

This is essentially a reformulation of [[Cisinski-Olschok theory]].

### Relation to model categories

[[model category|Model categories]] can be singled out from weak model categories
by adding the following properties:

1.  Every [[fibrant object]]
admits a strong path object and every [[cofibrant object]] admits a strong [[cylinder object]].

1.  All [[acyclic cofibrations]] are [[trivial cofibrations]]
and all [[acyclic fibrations]] are [[trivial fibrations]].
(Trivial maps are given as data for a [[premodel category]],
whereas acyclic (co)fibrations are defined as (co)fibrations
that satisfy a right (left) lifting property with respect to the class of cofibrations with cofibrant source (respectively fibrations with fibrant target.)

1.  The two classes of [[weak equivalence]] corresponding
to the left and right induced [[semimodel structures]] coincide.

### Relation to combinatorial model categories

Every [[combinatorial]] weak model category can
be connected to a [[combinatorial model category]]
by a zigzag of [[Quillen equivalences]].

## Examples

\begin{example}\label{WeakModelStructureOnSemiSimplicialSets}
**([[weak model structure on semi-simplicial sets]])** \linebreak
There is a weak model structure on [[semi-simplicial sets]] which is [[Quillen equivalence|Quillen equivalen]] to [[classical model structure on simplicial sets|that on simplicial sets]]. ([Henry 18, Thm. 5.5.6](#Henry18)).
\end{example}

## Related concepts

* [[model category]]

* [[premodel category]]

## References

* {#Henry18} [[Simon Henry]], _Weak model categories in classical and constructive mathematics_, Theory and Applications of Categories, Vol. 35, 2020, No. 24, pp 875-958.  ([arXiv:1807.02650](https://arxiv.org/abs/1807.02650), [tac:35-24](http://www.tac.mta.ca/tac/volumes/35/24/35-24abs.html))

* {#Henry20} [[Simon Henry]], _Combinatorial and accessible weak model categories_ ([arXiv:2005.02360](https://arxiv.org/abs/2005.02360))

[[!redirects weak model categories]]
[[!redirects weak model structure]]
[[!redirects weak model structures]]


weak multilimit

# Weak multilimits
* table of contents
{: toc}

## Idea

A *weak multilimit* is a common generalization of [[multilimits]] and [[weak limits]].

## Definition

If $F\colon D\to C$ is a diagram in a [[category]] $C$, then a **weak multilimit** of $F$ is a (small) set $L$ of [[cones]] over $F$ such that any other cone over $F$ factors (not necessarily uniquely) through some (not necessarily unique) element of $L$.  If the factorization, and the cone factored through, are unique, then $L$ is a multilimit, whereas if $L$ is a singleton, then it is a *weak limit*.

The existence of weak multilimits is a "pure size condition" on $C$, in the sense that if $C$ is a [[small category]], then every small diagram in $C$ (that is, every functor $F\colon D\to C$ where $D$ is also small) has a weak multilimit, namely the set of *all* cones over $F$.

Of course, weak multilimits in $C^{op}$ are called **weak multicolimits** in $C$.

## Examples

* A weak multilimit of the empty diagram is a *weak multi-terminal-object*, also called a **weakly terminal set**: a small set $T$ of objects such that every object admits a morphism to some object in $T$.  The dual concept is a **weakly initial set**.  These notions play a role in some statements of the [[adjoint functor theorem]].

[[!redirects weak multilimit]]
[[!redirects weak multicolimit]]
[[!redirects weakly initial set]]
[[!redirects weakly terminal set]]


weak omega-category

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* table of contents
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## Idea

In [[higher category theory]] the term  _weak $\omega$-categories_ is essentially synonymous with _[[infinity-category]]_ in the fully general sense of _[[(infinity,infinity)-category]]_.  The terms "$\omega$-category" and "$\infty$-category" originate in different schools and their choice of use is mostly a matter of the preference of individual authors.

One slight difference is that "$\infty$-category" usually implies a "weak" (fully general) notion, while in addition to weak $\omega$-categories there are also [[strict omega-category|strict ones]]. Another difference is that definitions of weak $\omega$-categories tend to be [[algebraic definitions of higher categories|algebraic]] instead of [[geometric definition of higher categories|geometric]] (accordingly typically the central open question is whether a definition really satisfies the [[homotopy hypothesis]]), though some definitions of weak $\omega$-categories are geometry (for instance some flavors of definition of [[opetopic omega-category]]).

## Examples

The following are examples for proposals of definitions of weak $\omega$-categories.

* [[Trimble n-category|Trimble ∞-category]]

* [[simplicial weak omega-category|simplicial weak ∞-category]]

* [[Batanin ∞-category]]

* [[opetopic omega-category]]

## References

> Fore more see general references at *[[higher category theory]]*, such as:

* [[Andre Joyal]], [[Tim Porter]], [[Peter May]], _Weak categories_ ([pdf](https://web.archive.org/web/20150326110254/http://www.ima.umn.edu/talks/workshops/SP6.7-18.04/may/PorterMay.pdf))

Discussion of weak $\omega$-categories via [[computads]] construed as [[inductive types]]:

* [[Christopher J. Dean]], [[Eric Finster]], [[Ioannis Markakis]], [[David Reutter]], [[Jamie Vicary]], *Computads for weak $\omega$-categories as an inductive type* [[arXiv:2208.08719](https://arxiv.org/abs/2208.08719)]

[[!redirects weak omega-categories]]
[[!redirects weak ∞-category]]
[[!redirects weak ∞-categories]]


weak order

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#### Relations
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## Disambiguation

A **weak order** could refer to either

* a [[total preorder]]

* a [[strict weak order]]

## References

* Wikipedia, [Weak order](https://en.wikipedia.org/wiki/Weak_order)

category: disambiguation

[[!redirects weak order]]
[[!redirects weak orders]]
[[!redirects weak ordering]]
[[!redirects weak orderings]]

[[!redirects weakly ordered]]
[[!redirects weakly ordered set]]
[[!redirects weakly ordered sets]]


weak quotient > history

< [[weak quotient]]
[[!redirects weak quotient -- history]]


weak representation of a functor

# Weak representation of a functor

* table of contents
{: toc}

## Idea

A _weak representation of a functor_ $P : C^o \to Set$ ([[presheaf]]) is like a [[representable functor]] but only satisfying the representability condition up to [[retract]] rather than [[isomorphism]]. Unlike a representable, this structure is not unique up to isomorphism, and so we speak of _representations_ as [[structure]] rather than _representability_ as a [[property]].

In [[type theory|type theoretic]] terms, this is an object that only satisfies the $\beta$ equation of a universal property and not the $\eta$ equation.

## Definition

A weak representation of a presheaf $P : C^o \to Set$ consists of

1. An object $X$ in $C$
2. [[natural transformation|natural transformations]] $s : P(\Gamma) \to C(\Gamma, X)$ and $r : C(\Gamma, X) \to P(\Gamma)$ such that $r$ is a [[retract]] of $s$.

Equivalently, by the [[Yoneda lemma]], a weak representation is

1. An object $X$
2. A "weak universal morphism" $\epsilon : P(X)$
3. A natural transformation $I : P(\Gamma) \to C(\Gamma, X)$ that is a [[section]] of $P(-)(\epsilon) : C(\Gamma, X) \to P(\Gamma)$

## Relation to Representable Functors

A weak representation $(X, \epsilon, I)$ of $P$ induces an [[idempotent]] $e$ on the representable sets $C(-, X)$ such that $P$ is the [[idempotent splitting]] of the representable $C(-,X)$ in the presheaf category. By the Yoneda lemma, this induces an idempotent $e$ on $X$ in $C$, and an object $Y$ represents $P$ if and only if $Y$ is the idempotent splitting of $e$. As a consequence, a representable $Y$ has a canonical section to any weak representation $X$.

In type theoretic terms, the idempotent performs [[eta expansion|$\eta$ expansion]], and so this says that a representable can be constructed from a weak representable as the [[fixed points]] of $\eta$ expansion, or dually as a quotient equating the $\eta$ expansion with the identity.

## Related Concepts

* [[weak adjoint]]
* [[weak limit]]
* [[semifunctor]]
* [[lax idempotent 2-adjunction]]
* [[weak 2-representation of a functor]]


weak topology

> For the strong topology in functional analysis, see at [[strong operator topology]].

***

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# Induced topologies
* table of contents
{:toc}

## Definitions

See also at *[[Top]]* the section *[Universal constructions](Top#UniversalConstructions)*.

### Weak/coarse/initial topology
 {#WeakTopologyDefinition}

Suppose

1. $S$ is a [[set]],

1. $\{ (X_i, T_i) \}_{i \in I}$ is a [[family]] of [[topological spaces]]

1. $\big\{ f_i \colon S \to X_i \big\}_{i \in I}$ an [[indexed set]] of [[functions]] from $S$ to the family $\{ X_i \}_{i \in I}$.

Let $\Gamma$ denote the set of all topologies $\tau$ on $S$ such that $f_i$ is a [[continuous map]] for every $i \in I$. Then the [[intersection]] $\bigcap_{\tau \in \Gamma} \tau$ is again a topology and also belongs to $\Gamma$. Clearly, it is the coarsest/weakest topology $\tau_0$ on $X$ such that each function $f_i\colon S \to X_i$ is a [[continuous map]].

We call $\tau_0$ the **weak/coarse/initial topology induced** on $S$ by the family of mappings $\{ f_i \}_{i \in I}$.  Note that all terms 'weak topology', 'initial topology', and 'induced topology' are used.  The [[subspace topology]] is a special case, where $I$ is a [[singleton]] and the unique function $f_i$ is an [[injection]].

### Strong/fine/final topology
 {#StrongTopologyDefinition}

[[formal duality|Dually]], suppose

1. $S$ is a [[set]],

1. $\{ (X_i, T_i) \}_{i \in I}$ a [[family]] of [[topological spaces]]

1. $\big\{ f_i \colon X_i \to S \big\}_{i \in I}$ a family of [[functions]] to $S$ from the family $\{ X_i \}_{i \in I}$.

Let $\Gamma$ denote the set of all topologies $\tau$ on $S$ such that $f_i$ is a [[continuous map]] for every $i \in I$. Then the [[union]] $\bigcup_{\tau \in \Gamma} \tau$ is again a topology and also belongs to $\Gamma$. Clearly, it is the finest/strongest topology $\tau_0$ on $S$ such that each function $f_i\colon X_i \to S$ is a [[continuous map]].

We call $\tau_0$ the **strong/fine/final topology induced** on $S$ by the family of mappings $\{ f_i \}_{i \in I}$.  Note that all terms 'strong topology', 'final topology', and 'induced topology' are used.  The [[quotient topology]] is a special case, where $I$ is a [[singleton]] and the unique function $f_i$ is a [[surjection]].

## Generalisations

We can perform the first construction in any [[topological concrete category]], where it is a special case of an [[initial structure]] for a [[sink|source or cosink]].

We can also perform the second construction in any [[topological concrete category]], where it is a special case of an [[final structure]] for a [[sink]].

## In functional analysis

In [[functional analysis]], the term 'weak topology' is used in a special way.  If $V$ is a [[topological vector space]] over the [[ground field]] $K$, then we may consider the [[continuous linear functional]]s on $V$, that is the [[continuous map|continuous]] [[linear maps]] from $V$ to $K$.  Taking $V$ to be the set $X$ in the general definition above, taking each $T_i$ to be $K$, and taking the continuous linear functionals on $V$ to comprise the family of functions, then we get the __weak topology__ on $V$.

The _weak-star topology_ on the dual space $V^*$ of continuous linear functionals on $V$ is precisely the weak topology induced by the dual (evaluation) functionals on $V^*$

$$
\left\{V^* \overset{\operatorname{ev}_v}{\to} K, \text{ by } f \mapsto f(v)\right\}_{v \in V}.
$$

For the strong topology in functional analysis, see the [[strong operator topology]].

## Related entries

* [Category of topological spaces -- Universal constructions](Top#UniversalConstructions)

## References

The original version of this article was posted by [[Vishal Lama]] at [[nlabmeta:induced topology]].

See also

* Wikipedia, _[Initial topology](https://en.wikipedia.org/wiki/Initial_topology)_, _[Final topology](https://en.wikipedia.org/wiki/Final_topology)_,

[[!redirects weak topology]]
[[!redirects weak topologies]]

[[!redirects coarse topology]]
[[!redirects coarse topologies]]

[[!redirects coarsest topology]]
[[!redirects coarsest topologies]]

[[!redirects initial topology]]
[[!redirects initial topologies]]

[[!redirects strong topology]]
[[!redirects strong topologies]]
[[!redirects fine topology]]
[[!redirects fine topologies]]
[[!redirects final topology]]
[[!redirects final topologies]]
[[!redirects weak-star topology]]
[[!redirects induced topology]]
[[!redirects induced topologies]]


weak type theory

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#### Type theory
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#### Deduction and Induction
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[[!include deduction and induction - contents]]
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#### Constructivism, Realizability, Computability
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[[!include constructivism - contents]]
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#### Foundations
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[[!include foundations - contents]]
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=--

# Contents
* table of contents
{: toc}

## Idea

**Weak type theory** or **propositional type theory** is a [[dependent type theory]] without judgmental conversion; all the [[computation rules]]/[[beta-conversion|$\beta$-conversion rules]] and [[uniqueness rules]]/[[eta-conversion|$\eta$-conversion rules]] for types use [[identity types]] instead of [[judgmental equality]]. As a result, the results in weak type theory are more general than in models which use judgmental equality for computational and uniqueness rules, since judgmental equality implies typal equality, while the converse isn't necessarily the case.

Formally, weak type theories come into two general versions:

* Dependent type theories which do not have judgmental equality and its [[structural rules]]. Instead, [[definitions]] of [[types]] and [[terms]] are made using [[identifications]] or [[equivalences of types]]. See [[objective type theory]] for more details on this.

* Dependent type theories which do have judgmental equality and its [[structural rules]], and thus definitions of [[types]] and [[terms]] are still made using [[judgmental equality]]. However, in addition to the [[congruence rules]] for [[judgmental equality]] for the [[formation rules]], [[introduction rules]], and [[elimination rules]] of each type former, there are additional [[congruence rules]] for the computation and uniqueness rules, since the conversion rules are represented by the structure of an identification rather than judgmental equality, and thus this structure has to be preserved across judgmental equality.

The latter includes weak versions of [[Martin-Löf type theory]], [[cubical type theory]], and [[observational type theory]], as well as extensions thereof such as [[type theory with shapes]] and [[simplicial type theory]]. Hypothetically, the latter would also be seen in [[proof assistants]], where the base [[programming language]] used to implement the weak type theory, such as [[Coq]] or [[Agda]], already has a judgmental equality.

A hybrid of weak and non-weak type theories can occur in [[two-level type theory]] and variants thereof like [[Homotopy Type System]], where one level has weak types and the other one has strict types.

## Open problems

There are plenty of questions which are currently unresolved in weak type theory. Van der Berg and Besten listed the following problems in the context of [[objective type theory]], but equally this applies to any weak type theory:

* Does [[univalence]] imply [[function extensionality]] for types in the universe in [[weak type theory]]?

* Is (the [[homotopy type theory]] based upon) [[Martin-Löf type theory]] conservative over (the [[homotopy type theory]] based upon) [[weak type theory|weak]] Martin-Löf type theory?

* How much of the [[HoTT book]] could be done in [[weak type theory]]?

* Does [[weak type theory]] have [[homotopy canonicity]] and [[normalization]]?

Other problems include

* What is the [[categorical semantics]] of the [[homotopy type theory]] based upon [[weak type theory]], with all [[higher inductive types]] and [[weakly Tarski]] [[univalent universes]]?

* Is [[weak function extensionality]] equivalent to [[function extensionality]] in [[weak type theory]]?

* Does [[product extensionality]] hold in [[weak type theory]]? Namely, given types $A$ and $B$ and elements $a:A \times B$ and $b:A \times B$, is the canonical function $\mathrm{idtoprojectionids}:(a =_{A \times B} b) \to (\pi_1(a) \simeq \pi_1(b)) \times (\pi_2(a) \simeq \pi_2(b))$ an [[equivalence of types]]?

* Is [[function extensionality]] still provable in weak [[cubical type theory]]?

See also [[open problems in homotopy type theory]]

## See also

* [[dependent type theory]]

## References

The original paper on weak type theory, in the context of [[objective type theory]]:

* {#BB21} [[Benno van den Berg]], [[Martijn den Besten]], *Quadratic type checking for objective type theory* ([arXiv:2102.00905](https://arxiv.org/abs/2102.00905))

Talks at *Strength of Weak Type Theory*, hosted by *[[DutchCATS]]*:

* [[Daniël Otten]], *Models for Propositional Type Theory* (11 May 2023) [[slides pdf](https://dutchcats.github.io/Weak_type_theories/slides_otten.pdf)]

* [[Théo Winterhalter]], *A conservative and constructive translation from extensional type theory to weak type theory*, 11 May 2023. ([slides](https://dutchcats.github.io/Weak_type_theories/slides_winterhalter.pdf))

* [[Sam Speight]], *Higher-Dimensional Realizability for Intensional Type Theory*, 11 May 2023. ([slides](https://dutchcats.github.io/Weak_type_theories/slides_speight.pdf))

* [[Matteo Spadetto]], *Weak type theories: a conservativity result for homotopy elementary types* (12 May 2023) [[slides pdf](https://dutchcats.github.io/Weak_type_theories/slides_spadetto.pdf)]

* [[Benno van den Berg]], *Towards homotopy canonicity for propositional type theory*, 12 May 2023. ([slides](https://dutchcats.github.io/Weak_type_theories/slides_van_den_berg.pdf))

* [[Rafaël Bocquet]], *Towards coherence theorems for equational extensions of type theories*, 12 May 2023. ([slides](https://dutchcats.github.io/Weak_type_theories/slides_bocquet.pdf))

Project to convert [[extensional type theory]] to [[weak type theory]]:

* Github, [ett-to-wtt](https://github.com/TheoWinterhalter/ett-to-wtt)

[[!redirects propositional type theory]]
[[!redirects propositional type theories]]
[[!redirects weak type theory]]
[[!redirects weak type theories]]


weak wreath

[[!redirects weak wreath product]]
## Idea

A **weak wreath** is to a [[weak distributive law]] what a [[wreath]] is to a [[distributive law]].

## Related pages

* [[weak distributive law]]

## References

* Stefaan Caenepeel and Erwin De Groot. _Modules over weak entwining structures_, Contemporary Mathematics 267 (2000): 31-54.

* [[Ross Street]], *Weak distributive laws*, Theory and Appl. of Categ. **22** (2009) 313--320 [[tac:22-12](http://www.tac.mta.ca/tac/volumes/22/12/22-12abs.html)]

* [[Gabriella Böhm]], _On the iteration of weak wreath products_, Theory and Appl. of Categories __26__:2 (2012) 30--59 [arXiv:1110.0652](https://arxiv.org/abs/1110.0652)

* [[Gabriella Böhm]], [[José Gómez-Torrecillas]], _Bilinear factorization of algebras_, Bull. Belg. Math. Soc. Simon Stevin 20(2): 221-.244 (may 2013) [doi](https://doi.org/10.36045/bbms/1369316541)

* [[Gabriella Böhm]], [[Steve Lack]], [[Ross Street]], _Idempotent splittings, colimit completion, and weak aspects of the theory of monads_, Journal of Pure and Applied Algebra __216__:2 (2012) 385--403 [doi](https://doi.org/10.1016/j.jpaa.2011.07.003)

[[!redirects weak wreath]]
[[!redirects weak wreaths]]
[[!redirects weak wreath products]]


weakening rule

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# The weakening rule
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## Idea

In [[formal logic]] and [[type theory]], by "weakening rules" one means [[inference rules]] for [[context extension]],
which state that any [[premises]] may be added to the [[hypotheses]] ([[antecedents]]) of a valid [[judgement]].

Along with the [[variable rule]], the [[contraction rule]] and the [[exchange rule]], the weakening rule is one of the most commonly adopted [[structural rules]]. But the weakening rule is not used in all [[logical frameworks]], for instance in [[linear logic]] it is discarded.

Concretely, in [[intuitionistic type theory|intuitionistic]] [[dependent type theory]] the weakening rule is the [[inference rule]]

$$
  W
  \frac{
    \Gamma
    ,\,
    \Delta
    \;
      \vdash
    \;
    j \colon J
    \;\;\;\;\;\;\;\;
    \Gamma
    \;
      \vdash
    \;
    A \colon Type
  }{
    \Gamma
    ,\;
    A
    ,\,
    \Delta
    \;
    \vdash
    \;
    j \colon J
  }
$$

As usual in [[dependent type theory]], the meaning of this rule is a little less trivial than it may superficially seem, due to the generic [[dependent type|type dependency]] involved: In [[context extension|extending the contents]] in the [[antecedent]] we are implicitly making the [[succedent]] [[dependent type|depend]] on this extended context, albeit trivially.

Accordingly, in the [[categorical semantics]] [[categorical semantics of dependent types|of dependent types]] the weakening rule corresponds to [[pullback]] of [[bundles]]/[[display maps]] to the [[fiber product]] which interprets the extended context:

## Semantics

Weakening rules correspond to having [[semicartesian monoidal category|projections for the monoidal structure]] that corresponds to the logical binary operator at hand.

## Related concepts

* [[inference rule]]

* [[structural rules]]

* [[variable rule]]

* [[substitution rule]]

## References

The notion of weakening as a [[structural inference rule]] originates (under the German name *Verdünnung*) with:

* {#Gentzen35} [[Gerhard Gentzen]], §1.2.1 in: _Untersuchungen über das logische Schließen I_ _Mathematische Zeitschrift_ **39** 1 (1935) [[doi:10.1007/BF01201353](http://dx.doi.org/10.1007/BF01201353)]

* {#Gentzen69} [[Gerhard Gentzen]], §1.21 in: *Investigations into Logical Deduction*, in M. E. Szabo (ed.), *The Collected Papers of Gerhard Gentzen*, Studies in Logic and the Foundations of Mathematics **55**, Springer (1969) 68-131 [[ISBN:978-0-444-53419-4](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/55), [pdf](https://logic-teaching.github.io/prop/texts/Gentzen%201969%20-%20Investigations%20into%20Logical%20Deduction.pdf)]

On the [[categorical semantics]]

* [[Bart Jacobs]], *Semantics of weakening and contraction*, Annals of Pure and Applied Logic **69** 1 (1994) 73-106 [doi:10.1016/0168-0072(94)90020-5]

Discussion in [[intuitionistic type theory|intuitionistic]] [[dependent type theory]]:

* {#Jacobs98} [[Bart Jacobs]], p. 122, 585 in: *Categorical Logic and Type Theory*, Studies in Logic and the Foundations of Mathematics **141**, Elsevier (1998)  [[ISBN:978-0-444-50170-7](https://www.sciencedirect.com/bookseries/studies-in-logic-and-the-foundations-of-mathematics/vol/141), [pdf](https://people.mpi-sws.org/~dreyer/courses/catlogic/jacobs.pdf)]

  > (emphasis on the [[categorical model of dependent types]])

* {#UFP13} [[Univalent Foundations Project]], §A of *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

  > (in the context of [[homotopy type theory]])

* {#Rijke18} [[Egbert Rijke]], *Dependent type theory* [[pdf](https://www.andrew.cmu.edu/user/erijke/hott/dtt.pdf)], Lecture 1 in: *[[Introduction to Homotopy Type Theory]]*, lecture notes, CMU (2018) [[pdf](http://www.andrew.cmu.edu/user/erijke/hott/hott_intro.pdf), [[Rijke-IntroductionHoTT-2018.pdf:file]], [webpage](https://www.andrew.cmu.edu/user/erijke/hott/)]

  > (in the context of [[homotopy type theory]])

[[!redirects weakening rule]]
[[!redirects weakening rules]]


weakly constant function

# Weakly constant function
* table of contents
{: toc}

## Idea

By a *weakly constant function*  one means a [[function]] which is equipped with a certain sort of "witness of [[constant function|constancy]]".

However, in [[higher category theory]] and [[homotopy theory]], it is debatable whether or not this witness really exhibits "constancy", hence the use of a different word.  (The term "steady function" was suggested by [[Andrej Bauer]] but did not catch on.)

## Definition

In [[homotopy type theory]], a [[function type|function]]

$$
  f\colon A\to B
$$

is *weakly constant* if we have a [[term]] of [[type]] of the [[dependent product]] over the [[identification types]] of all the values of $f$ on all [[pairs]] of arguments:

$$
  \prod_{(x,y:A)}
  (f x = f y)
  \,.
$$

By regarding [[homotopy type theory]] as the [[internal logic]] of an [[(∞,1)-topos]], we obtain a definition that makes sense in any [[(infinity,1)-category|$\infty$-category]] with [[binary products]]: a morphism $f:A\to B$ is *weakly constant* if the two composites $A\times A \rightrightarrows A \xrightarrow{f} B$ are [[equivalence in an (infinity,1)-category|equivalent]].

## Relationship to constancy

If $f$ is [[constant function|constant]] in the sense that it factors through the [[terminal object]] (i.e. we have $f = \lambda x. b$ for some $b:B$), then $f$ is obviously weakly constant.  The converse holds if we know that the [[domain]] $A$ is [[inhabited type|inhabited]], for if $a_0:A$, then $f a = f a_0$ for all $a:A$.  However, the [[identity function]] of the [[empty type]] is weakly constant, yet not equal to $\lambda x.b$ for any $b:\varnothing$ (since no such $b$ exists).

More generally, if $f$ factors through the [[propositional truncation]] ${\|A\|}$, then it is weakly constant, since any two [[elements]] of ${\|A\|}$ are equal (i.e. it is an [[h-proposition]]).  In fact, this is true if $f$ factors through *any* [[h-proposition]] (in which case it in fact also factors through ${\|A\|}$, by the [[universal property]] of the latter).

The converse to this last implication does hold for some specific $f:A\to B$, such as:

* If $B$ is an [[h-set]].  For then $f$ factors through the [[0-truncation]] ${\|A\|_0}$, and the set-[[coequalizer]] of the two [[projections]] ${\|A\|_0} \times {\|A\|_0} \to {\|A\|_0}$ is the [[propositional truncation]].

* If $A$ has [[split support]].  For then we have a composite ${\|A\|} \to A \xrightarrow{f} B$, whose restriction to $A$ is equal to $f$ by weak constancy.

* If $A=P+Q$, with $P$ and $Q$ [[h-propositions]].  For then ${\|A\|}$ is the [[join]] $P*Q$ of $P$ and $Q$, i.e. the [[pushout]] of the two [[projections]] $P \leftarrow P\times Q \to Q$.  The [[universal property]] of this pushout says exactly that any weakly constant map $P+Q\to B$ factors through $P*Q = \Vert P+Q\Vert$.

* If $A=B$ (see below).

However, it can fail in general, even when $A$ is [[mere proposition|merely]] inhabited (i.e. ${\|A\|}= 1$).  For instance, let $A=P+Q+R$ for h-propositions $P$, $Q$, and $R$, and let $B$ be the triple pushout of $P$, $Q$, and $R$ under $P\times Q$, $P\times R$, and $Q\times R$.  Then there is a steady map $f:A\to B$, but there exist models in which ${\|A\|} = 1$ but $B$ has no [[global element]].  The most straightforward such model is [[presheaves]] on the [[poset]] of proper [[subsets]] of $\{a,b,c\}$, with $P=\{a,b\}$, $Q=\{b,c\}$, and $R=\{a,c\}$.  In this model, we have $B(S) = 1$ for all nonempty proper subsets $S$, while $B(\emptyset) = S^1$, and $B$ has no global sections.

See [this discussion](https://groups.google.com/d/msg/homotopytypetheory/FeBAScTgwzg/Dx6E3-ezdxIJ).

In general, being weakly constant may be regarded as an "incoherent approximation" to constancy in the sense of factoring through an [[h-proposition]].  Indeed for a set $A$, its propositional truncation is the set-[[coequalizer]] of $A\times A\rightrightarrows A$.  However, in general such a construction requires the realization of a whole [[simplicial diagram]] (the simplicial [[kernel]] of the map $A\to 1$).

## Weakly constant endomaps

While an arbitrary weakly constant function is not very [[coherence law|coherent]], a weakly constant [[endofunction]] $f:A\to A$ has some extra degree of "coherence", as witnessed by the following results of [(KECA)](#KECA).

+-- {: .un_lemma}
###### Lemma
If $f:A\to A$ is weakly constant, then the type $Fix(f) \coloneqq \sum_{x:A} (f x = x)$ is an [[h-proposition]], and equivalent to ${\|A\|}$.
=--
+-- {: .proof}
###### Proof
Suppose $H: \prod_{(x,y:A)} (f x = f y)$, and let $(a,p),(b,q):Fix(f)$; we want to show $(a,p)=(b,q)$.  Let $r:a=b$ be the concatenated path
$$ a \xrightarrow{p^{-1}} f a \xrightarrow{H_{a,a}^{-1}} f a \xrightarrow{H_{a,b}} f b \xrightarrow{q} b. $$
It will suffice to show that $p \bullet r = ap_f(r) \bullet q$, where $ap_f$ denotes the action of $f$ on paths.  However, the dependent action of $H$ on paths implies that $H_{x,y} \bullet ap_f(s) = H_{x,y'}$ for any $x:A$ and any $s:y=y'$, and in particular $ap_f(r) = H_{a,a}^{-1} \bullet H_{a,b}$.  From this $p \bullet r = ap_f(r) \bullet q$ is immediate.  Thus, $Fix(f)$ is an h-proposition.

Now we have a map $g:A\to Fix(f)$ defined by $g(x) \coloneqq (f x, H_{f x,x})$, so by the universal property of ${\|A\|}$, we have ${\|A\|} \to Fix(f)$.  On the other hand, we have the first projection $pr_1:Fix(f) \to A$, and hence $Fix(f) \to {\|A\|}$.  Thus, these two h-propositions are equal.
=--

+-- {: .un_theorem}
###### Theorem
A type $A$ has [[split support]] if and only if it admits a weakly constant endomap $f:A\to A$.
=--
+-- {: .proof}
###### Proof
Given ${\|A\|} \to A$, the composite $A\to {\|A\|} \to \A$ is weakly constant.  Conversely, if $f$ is steady, $Fix(f) = {\|A\|}$ by the lemma, so $pr_1:Fix(f) \to A$ splits the support of $A$.
=--

Note that $pr_1 \circ g = f$, so that if we start with a weakly constant endomap of $A$, deduce a splitting of the support of $A$, and then reconstruct a weakly constant endomap, we obtain the same map.  However, the proof of steadiness is generally different from the one we began with, so this "logical equivalence" is not an equivalence of types.

+-- {: .un_cor}
###### Corollary
A type $A$ is an [[h-set]] if and only if every [[identity type]] $x=_A y$ admits a weakly constant endomap.
=--
+-- {: .proof}
###### Proof
We know that $A$ is an h-set just when all $x=_A y$ have split support.
=--

## References

* {#KECA} [[Nicolai Kraus]] and [[Martin Escardo]] and [[Thierry Coquand]] and [[Thorsten Altenkirch]], "Generalizations of Hedberg's theorem", M. Hasegawa (Ed.): TLCA 2013, LNCS 7941, pp. 173-188. Springer, Heidelberg 2013. [PDF](http://www.cs.bham.ac.uk/~mhe/papers/hedberg.pdf).  In this paper, steady functions are called "constant".

* [[Nicolai Kraus]], [[Martín Escardó]], [[Thierry Coquand]], [[Thorsten Altenkirch]], *Notions of Anonymous Existence in Martin-Löf Type Theory*,  Logical Methods in Computer Science **13** 1 [doi:10.23638/LMCS-13(1:15)2017, [arXiv:1610.03346](https://arxiv.org/abs/1610.03346)]

[[!redirects weakly constant function]]
[[!redirects weakly constant functions]]

[[!redirects steady function]]
[[!redirects steady functions]]


weakly distributive category > history

see _[[linearly distributive category]]_


weakly globular n-fold category

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

#Contents#
* table of contents
{:toc}

## Idea

Classically, [[n-category|models of weak $n$-categories]] comprise sets of cells in dimension 0 up to $n$. This is also called the globularity condition. Geometrically, it corresponds to cells having a globular shape
\begin{imagefromfile}
"file_name": "Glob_Shape_N.jpg",
"width": 300
\end{imagefromfile}

On the other hand, non-globular structures exist in higher category theory: for instance, [[n-fold category|$n$-fold categories]], defined by iterated [[internalization|internalization]]
\[
Cat^{0}=Set,\qquad\qquad Cat^{n}=Cat(Cat^{n-1})\;.
\]
The category $n\text{-}Cat$ of [[strict n-category|strict $n$-categories]] is defined by iterated [[enriched category|enrichment]]:

$$
0\text{-}Cat=Set,\qquad\qquad n\text{-}Cat=((n-1)\text{-}Cat,\times)\text{-}Cat\;.
$$
There is an embedding
$$
n\text{-}Cat\hookrightarrow Cat^{n}
$$
such that a strict $n$-category is an $n$-fold category in which certain substructures are discrete (that is, sets).
This discreteness condition is precisely the globularity condition and the sets underlying these discrete substructures are the sets of cells in the strict $n$-category.

For instance, in the case $n=2$, strict 2-categories are [[double category|double categories]] in which the category of objects and vertical arrows is discrete. In pictures:
\begin{imagefromfile}
"file_name": "2-Cat.jpg",
"width": 800
\end{imagefromfile}
We see that the picture on the right becomes the one on the left when all vertical morphisms are identities.

In the weakly globular approach to higher categories, the cells in each dimension instead of forming a set, have a higher categorical structure which is homotopically discrete, that is only equivalent (in a higher categorical sense) to a set. This condition, called weak globularity condition, is a new paradigm to weaken higher categorical structures and allows to use rigid structures, namely $n$-fold categories, to model weak $n$-categories. There are strict embeddings
$$
n\text{-} Cat \hookrightarrow Cat_{wg}^{n} \hookrightarrow Cat^{n}
$$
so that the category $Cat_{wg}^{n}$ of weakly globular $n$-fold categories is intermediate between strict $n$-categories and $n$-fold categories.

In the case $n=2$, a weakly globular double category $X$ is a double category satisfying two conditions:
the first is the weak globularity condition, stating that the category $X_0$ of objects and vertical arrows is equivalent to a discrete category, that is it is an equivalence relation. The set $pX_0$ of connected components of $X_0$ plays the role of 'set of objects' in the weak structure. We therefore need to define a composition of horizontal arrows whose source and target are in the same vertical connected component
\begin{imagefromfile}
"file_name": "Horiz-Comp.jpg",
"width": 700
\end{imagefromfile}

For this purpose, we impose a second condition: for each 'staircase path'

\begin{imagefromfile}
"file_name": "Staircase.jpg",
"width": 600
\end{imagefromfile}
A more concise way to express this second condition is in term of the so called induced Segal maps condition.

Weakly globular double categories have been shown by Paoli and Pronk to be suitably equivalent to [[bicategory|bicategories]].

In dimension $n\geq2$, a weakly globular $n$-fold category $X$ is an $n$-fold category which first of all needs to satisfy the weak globularity condition; namely those substructures that are discrete in the image of the embedding $n\text{-} Cat\hookrightarrow Cat^{n}$, are now instead only equivalent to sets. The precise notion we use for these substructures is the one of homotopically discrete $n$-fold categories. The latter are defined inductively, starting with equivalence relations in the case of $n=1$. When $n\geq1$, the idea of a homotopically discrete $n$-fold category is that it is an $n$-fold category suitably equivalent to a discrete one both 'globally' and in each simplicial dimension.

The additional conditions in the definition of $Cat_{wg}^{n}$ are given inductively in terms of induced Segal maps conditions. They guarantee the existence of weakly associative and weakly unital compositions but, like in the Tamsamani-Simpson model, their [[coherence law|coherences]] are not given explicitly but they are automatically encoded in the [[multisimplicial set|multi-simplicial]] combinatorics.

Weakly globular $n$-fold categories have been shown to satisfy the [[homotopy hypothesis|homotopy hypothesis]] and to be suitably equivalent to the [[Homotopy Theory of Higher Categories|Tamsamani-Simpson model]].

## Homotopically discrete $n$-fold categories

The definition of weakly globular $n$-fold category requires a preliminary notion, the one of homotopically discrete $n$-fold category.
\begin{definition}\label{def01}
Let $Cat_{hd}^{0}=Set$. Suppose, inductively, we have defined the subcategory $Cat_{hd}^{n-1}\subset Cat^{n-1}$ of homotopically discrete $(n-1)$-fold categories. We say that the $n$-fold category $X\in Cat^{n}$ is homotopically discrete if:

a) $X$ is a levelwise equivalence relation, that is for each $(k_1,\ldots,k_{n-1})\in\Delta^{{n-1}^{op}}$, $X_{k_1,\ldots,k_{n-1}}\in Cat$ is an [[equivalence relation|equivalence relation]] (that is, a category equivalent to a discrete one).

b) $p^{(n-1)}X\in Cat_{hd}^{n-1}$ where $(p^{(n-1)}X)_{k_1,\ldots,k_{n-1}}=p X_{k_1,\ldots,k_{n-1}}$ with $p: Cat\rightarrow  Set$ the [[isomorphism class|isomorphism classes of objects]] functor.

When $n=1$ we denote by $Cat_{hd}^{1}=Cat_{hd}^{}$ the subcategory of $ Cat$ consisting of equivalence relations.
\end{definition}

\begin{definition}\label{def02}
    Let $X\in Cat_{hd}^{n}$. Denote by $\gamma^{(n-1)}_X:X\rightarrow  p^{(n-1)}X$ the morphism given by
$$
(\gamma^{(n-1)}_X)_{s_1...s_{n-1}} :X_{s_1...s_{n-1}} \rightarrow p X_{s_1...s_{n-1}}\;.
$$
Denote by
$$
X^d =p p^{(1)}...p^{(n-1)}X
$$
and by $\gamma_{n}$ the composite
$$
X\xrightarrow{\gamma^{(n-1)}}p^{(n-1)}X \xrightarrow{\gamma^{(n-2)}} p^{(n-2)}p^{(n-1)}X \rightarrow  \cdots \xrightarrow{\gamma^{(0)}} X^d\;.
$$
We call $\gamma_{n}$ the discretization map.
\end{definition}

\begin{definition}\label{def03}
    Given $X\in Cat_{hd}^{n}$, for each $a,b\in X_0^d$ denote by $X(a,b)$ the fiber at $(a,b)$ of the map
$$
X_1 \xrightarrow{(d_0,d_1)} X_0\times X_0 \xrightarrow{\gamma_{n}\times\gamma_{n}} X_0^d\times X_0^d\;.
$$

$X(a,b)\in Cat_{hd}^{n-1}$ should be thought of as a hom-$(n-1)$-category.
\end{definition}

\begin{definition}\label{def04}
Define inductively $n$-equivalences in $Cat_{hd}^{n}$. For $n=1$, a 1-equivalence is an [[equivalence of categories|equivalence of categories]]. Suppose we have defined $(n-1)$-equivalences in $Cat_{hd}^{n-1}$. Then a map $f:X\rightarrow  Y$ in $Cat_{hd}^{n}$ is an $n$-equivalence if

a) For all $a,b \in X_0^d$,
$$
   f(a,b):X(a,b) \rightarrow  Y(f a,f b)
$$
is an $(n-1)$-equivalence.

b) $p^{(n-1)}f$ is an $(n-1)$-equivalence.
\end{definition}

The next proposition means that homotopically discrete $n$-fold categories are a higher categorical 'fattening' of sets.
\begin{proposition}\label{pro01}
Let $X\in Cat_{hd}^{n}$. Then the maps $\gamma^{(n-1)}:X\rightarrow  p^{(n-1)}X$ and $\gamma_{n}:X\rightarrow  X^d$ are $n$-equivalences.
\end{proposition}

## The three Segal-type models

There are three Segal-type models of weak $n$-categories
\begin{tikzcd}
	& Ta_{wg}^{n} \\
	Ta^{n} && Cat_{wg}^{n}
	\arrow[hook, from=2-1, to=1-2]
	\arrow[hook', from=2-3, to=1-2]
\end{tikzcd}
Here $Ta^{n}$ is the Tamsamani-Simpson model and $Ta_{wg}^{n}$ (called weakly globular Tamsamani $n$-categories) is a generalization of it using weak globularity; the latter contains weakly globular $n$-fold categories as special case.

### Weakly globular Tamsamani ${n}$-categories

The definition of $Ta_{wg}^{n}$ is by induction on $n$, starting with $Ta_{wg}^{1}= Cat$ and 1-equivalences being equivalences of categories. Suppose, inductively, that we defined $Ta_{wg}^{n-1}$ and $(n-1)$-equivalences. Then we define $Ta_{wg}^{n}$ through the following conditions:

a) There is an embedding of $Ta_{wg}^{n}$ into [[functor category|functor categories]]
\[
\label{eq3}
  Ta_{wg}^{n}\hookrightarrow [\Delta^{{}^{op}},{Ta_{wg}^{n-1}}]\hookrightarrow [\Delta^{{n-1}^{op}}, Cat]
\]

b) There is a truncation functor
\[
\label{eq4}
  p^{(n-1)}:Ta_{wg}^{n}\rightarrow  Ta_{wg}^{n-1}
\]

$$
  (p^{(n-1)}X)_{k_1,\ldots, k_{n-1}}=p X_{k_1,\ldots, k_{n-1}}
$$
for each $(k_1,\ldots, k_{n-1})\in\Delta^{{n-1}^{op}}$ and $X\in Ta_{wg}^{n}$, where $p: Cat\rightarrow  Set$ is the isomorphism classes of objects functor.

c) $X_0$ is a homotopically discrete $(n-1)$-fold category. This comes with a $(n-1)$-equivalence $\gamma:X_0\rightarrow  X_0^d$ where $X_0^d$ is a discrete $(n-1)$-fold category.

d) For each $k\geq 2$ the induced Segal maps
\[
\label{eq5}
  \hat{\mu}_k:X_k\rightarrow {X_1\times_{X_0^d}\overset{k}{\cdots}\times_{X_0^d}X_1}
\]
are $(n-1)$-equivalences in $Ta_{wg}^{n-1}$. The maps $\hat{\mu}_k$ arise from the commutativity of the diagram

\begin{tikzcd}[column sep=scriptsize]
	&&&& {X_k} \\
	& {X_1} && {X_1} && \cdots && {X_1} \\
	{X_0^d} && {X_0^d} && {X_0^d} & \cdots & {X_0^d} && {X_0^d}
	\arrow["{\nu_1}"', from=1-5, to=2-2]
	\arrow["{\nu_k}", from=1-5, to=2-8]
	\arrow["{\nu_2}"', from=1-5, to=2-4]
	\arrow["{\gamma d_1}"', from=2-2, to=3-1]
	\arrow["{\gamma d_0}", from=2-2, to=3-3]
	\arrow["{\gamma d_0}", from=2-8, to=3-9]
	\arrow["{\gamma d_0}", from=2-4, to=3-5]
	\arrow["{\gamma d_1}"', from=2-4, to=3-3]
	\arrow["{\gamma d_1}"', from=2-8, to=3-7]
\end{tikzcd}

e) To complete the inductive step, we define $n$-equivalences in $Ta_{wg}^{n}$. For this, given $X\in Ta_{wg}^{n}$ and $(a,b)\in X_0^d\times X_0^d$, we let $X(a,b)\subset X_1$ be the fiber at $(a,b)$ of the map
$$
  X_1\xrightarrow{(d_0,d_1)} X_0\times X_0 \xrightarrow{\gamma\times\gamma}  X^d_0\times X^d_0\;.
$$

We define a map $f:X\rightarrow  Y$ in $Ta_{wg}^{n}$ to be an $n$-equivalence if the following conditions hold:

i) For all $a,b\in X_0^d$
$$
  f(a,b): X(a,b)\rightarrow  Y(f a,f b)
$$
is a $(n-1)$-equivalence.

ii) $p^{(n-1)}f$ is a $(n-1)$-equivalence.

### Tamsamani ${n}$-categories

The category $Ta^{n}$ of Tamsamani $n$-categories is the full subcategory of $Ta_{wg}^{n}$ whose objects $X$ are such that $X_0$ and $X_{k_1\ldots k_r 0}$ are discrete for all $(k_1\ldots k_r)\in    \Delta^{{r}^{op}}$, $1\leq r\leq n-2$.

The embedding \eqref{eq3} restricts to an embedding
$$
  Ta_{n}\hookrightarrow [\Delta^{{}^{op}},{Ta_{n-1}}]
$$
and the functor $p^{(n-1)}$ in \eqref{eq4} restricts to
$$
  p^{(n-1)}:Ta^{n}\rightarrow Ta^{n-1}
$$
The induced Segal maps $\hat{\mu}_k$ coincide with the [[Segal map|Segal maps]]
$$
  X_k\rightarrow {X_1\times_{X_0}\overset{k}{\cdots}\times_{X_0}X_1}
$$
and are $(n-1)$-equivalences in $Ta^{n-1}$.

### Weakly globular $n$-fold categories

Let $Cat_{wg}^{1}= Cat$. Having defined the full subcategory $Cat_{wg}^{n-1}\subset Ta_{wg}^{n-1}$, the category $Cat_{wg}^{n}$ of weakly globular $n$-fold categories is the full subcategory of $Ta_{wg}^{n}$ whose objects $X$ are such that $X\in Cat^{n}$, $X_k\in Cat_{wg}^{n-1}$ for all $k\geq 0$ and $p^{(n-1)}X\in Cat_{wg}^{n-1}$.

## Main results

\begin{theorem}\label{th1}
a) There is a functor 'rigidification'
$$
Q_n: Ta_{wg}^{n}\rightarrow  Cat_{wg}^{n}
$$
and for each $X\in Ta_{wg}^{n}$ an $n$-equivalence in $Ta_{wg}^{n}$ natural in $X$
$$
  s_n(X):Q_n X\rightarrow  X.
$$

b) There is a functor 'discretization'
$$
  Disc_{n}:Cat_{wg}^{n}\rightarrow  Ta^{n}
$$
and, for each $X\in Cat_{wg}^{n}$, a zig-zag of $n$-equivalences in $Ta_{wg}^{n}$ between $X$ and $Disc_{n} X$.
\end{theorem}

The functors discretization and rigidification are used in the comparison result between Tamsamani $n$-categories and weakly globular $n$-fold categories as follows:

\begin{theorem}\label{th2}
The functors
$$
    Q_n:Ta^{n}\leftrightarrows Cat_{wg}^{n}:Disc_{n}
$$
induce an equivalence of categories after [[localization|localization]] with respect to the $n$-equivalences
$$
    Ta^{n}/\!\!\sim^n\;\simeq \; Cat_{wg}^{n}/\!\!\sim^n\;.
$$
\end{theorem}

## The groupoidal case

There is a subcategory $GCat_{wg}^{n}\subset Cat_{wg}^{n}$ of groupoidal weakly globular $n$-fold categories which is an algebraic model of [[homotopy n-type|homotopy $n$-types]]. This means that weakly globular $n$-fold categories satisfy the homotopy hypothesis.
To define $GCat_{wg}^{n}$ we first consider the groupoidal version of the largest of the three Segal-type models:

\begin{definition}\label{def1}
The full subcategory $GTa_{wg}^{n}\subset Ta_{wg}^{n}$ of groupoidal weakly globular Tamsamani $n$-categories is defined inductively as follows.

For $n=1$, $GTa_{wg}^{1}=Gpd$ is the category of [[groupoid|groupoids]]. Note that $Cat_{hd}^{}\subset GTa_{wg}^{1}$. Suppose inductively we have defined $GTa_{wg}^{n-1}\subset Ta_{wg}^{n-1}$. We define $X\in GTa_{wg}^{n}\subset Ta_{wg}^{n}$ such that

i) $X_k\in GTa_{wg}^{n-1}$ for all $k\geq 0$.

ii)  $p^{(n-1)}X\in GTa_{wg}^{n-1}$.
\end{definition}

From this definition it is immediate that homotopically discrete $n$-fold categories are a full subcategory of $GTa_{wg}^{n}$.

\begin{definition}\label{def2}
The category $GCat_{wg}^{n}\subset Cat_{wg}^{n}$ of groupoidal weakly globular $n$-fold categories is the full subcategory of $Cat_{wg}^{n}$ whose objects $X$ are in $GTa_{wg}^{n}$.

The category $GTa^{n}\subset Ta^{n}$ of groupoidal Tamsamani $n$-categories is the full subcategory of $Ta^{n}$ whose objects $X$ are in $GTa_{wg}^{n}$.
\end{definition}

\begin{proposition}\label{pro1}
The functors
$$
  Q_n:GTa^{n}\leftrightarrows GCat_{wg}^{n}:Disc_{n}
$$
induce an equivalence of categories after localization with respect to the $n$-equivalences
$$
GTa^{n}/\!\!\sim^n \;\simeq\; GCat_{wg}^{n}/\!\!\sim^n\;.
$$
\end{proposition}

Since Tamsamani $n$-groupoids are a model of $n$-types, as a consequence of the previous proposition we obtain that weakly globular $n$-fold categories are an algebraic model of $n$-types.

There is an alternative more explicit way to obtain a fundamental functor
$$
  \mathcal{H}_{n}:n\text{-types}\rightarrow  GCat_{wg}^{n}
$$
based on the work of Blanc and Paoli. This work shows in particular how the notion of weak globularity arises naturally in topology.

The functor $\mathcal{H}_n$ is given by the composite
$$
 \mathcal{H}_n: n\text{-types}\xrightarrow{\mathcal{S}} [\Delta^{{}^{op}},Set] \xrightarrow{Or_{n}} [\Delta^{{n}^{op}},Set]\xrightarrow{\mathcal{P}_n} Gpd^{n}\;.
$$
Here $\mathcal{S}$ is the [[singular simplicial complex|singular functor]] and $Or_{n}$ is the functor induced by [[ordinal sum|ordinal sum]] $or_{n}:    \Delta^{{n}^{op}} \rightarrow  \Delta$, that is
$$
  (Or_{n} X)_{p_1\ldots p_n}=X_{n-1+p_1+\ldots+ p_n}
$$
The functor $\mathcal{P}_n$ is left adjoint to the $n$-fold [nerve](https://ncatlab.org/nlab/show/nerve#internal_nerve) functor $N_n:Gpd^{n}\rightarrow
[\Delta^{{n}^{op}},Set]$. While the computation of $\mathcal{P}_n$ is in general very difficult, this becomes easy on those multi-simplicial sets in the essential image of the functor $Or_{n}\mathcal{S}$ and one obtain
$$
  \mathcal{H}_n X=\hat{\pi}^{1} \hat{\pi}^{2} \cdots \hat{\pi}^{n} Or_{n} \mathcal{S} X\;.
$$
where $\hat{\pi}^{(i)}$ denotes the [[fundamental groupoid|fundamental groupoid]] functor in the $i^{th}$ direction. Using this expression of $\mathcal{H}_n X$ one can check that $\mathcal{H}_n X\in GCat_{wg}^{n}$. In conclusion

\begin{theorem}
The functors
$$
  B\circ Disc_{n}:GCat_{wg}^{n}\rightleftarrows n\text{-types}:\mathcal{H}_n
$$
induce an equivalence of categories
$$
  GCat_{wg}^{n}/\!\!\sim^{n}\,\simeq \,\mathcal{H}\!o(n\text{-types})\;.
$$
where $\mathcal{H}\!o(n\text{-types})$ is the [[homotopy category|homotopy category]] of $n\text{-types}$.
\end{theorem}

## References

The main reference for the theory of weakly globular $n$-fold categories is the following [research monograph](https://doi.org/10.1007/978-3-030-05674-2), which contains an account of all three Segal-type models

* [[Simona Paoli|S. Paoli]]. _Simplicial Methods for Higher Categories: Segal-type models of weak n-categories_, volume 26 of Algebra and Applications. Springer, 2019 [toc pdf](https://link.springer.com/content/pdf/bfm%3A978-3-030-05674-2%2F1.pdf).

The case $n=2$ was originally introduced in the following paper

* S. Paoli, [[Dorette Pronk|D. Pronk]], A double categorical model of weak 2-categories, _Theory and Application of categories_, 28, (2013), 933-980.

The theory or Tamsamani weak $n$-categories was originally developed in

* Z. Tamsamani. Sur des notions de $n$-categorie et $n$-groupoide non-strictes via des ensembles multisimpliciaux. _K-theory_, 16:51–99, 1999.

Tamsamani weak $n$-categories were further studied (from a model category theoretic perspective) in

* [[Carlos Simpson|C. Simpson]]. _Homotopy theory of higher categories_, volume 19 of New Math. Monographs. Cambridge University Press, 2012.

The groupoidal case of weakly globular $n$-fold structures started in

* [[David Blanc|D. Blanc]] and S. Paoli. Segal-type algebraic models of $n$-types. _Algebr. Geom. Topol._, 14:3419–3491, 2014.

An even earlier precursor in the groupoidal case, restricted to modelling path connected $n$-types and related to the [[cat-n-group|$cat_n$-groups]] model, can be found in

* S. Paoli. Weakly globular $cat_n$-groups and Tamsamani’s model. _Adv. in Math._, 222:621–727, 2009.

[[!redirects weakly globular n-fold categories]]


weakly Hausdorff topological space

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# Weakly Hausdorff spaces
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## Definition

A [[topological space]] $X$ is **weakly Hausdorff** (or weak Hausdorff) if for any [[compact Hausdorff space]] $K$ and every [[continuous map]] $f\colon K\to X$, the [[image]] $f(K)$ is [[closed subspace|closed]]. Every weakly Hausdorff space is [[T1-space|$T_1$]] (that is every point is [[closed point|closed]]), and every [[Hausdorff space]] is weakly Hausdorff. For the most common purposes for which Hausdorff spaces are used, the assumption of being weakly Hausdorff suffices.  See also [[compactly generated space]].

The notion also also makes sense as stated for [[locales]].  Where their theory overlaps (in [[sober spaces]] and [[topological locales]]), the notions of weak Hausdorffness agree, given the [[ultrafilter theorem]] (which implies that all compact Hausdorff spaces/locales are sober/topological).

## Properties

### Weak Hausdorffification

(this is a [[left adjoint]] ...) See for now ([Strickland 2009](#Strickland), Proposition 2.22), for the weak Hausdorffification of a compactly-generated topological space. Note in particular that the construction of the weak Hausdorffification of a compactly-generated space is a quotient by a closed equivalence relation given by a one-step construction (the intersection of all closed equivalence relations on the space), as opposed to the equivalence relation generated by a possibly transfinite procedure as in the Hausdorffification of an arbitrary topological space.

### Pushouts

Write CGWH for the category of [[compactly generated topological spaces|compactly generated]] weakly Hausdorff topological spaces, and $CGH$ for compactly generated [[Hausdorff topological spaces]]. Both are [[convenient categories of topological spaces]] that both admit a [[homotopy hypothesis]]-comparison to [[simplicial sets]], but CGWH has a key further property:

>The construction of [[pushouts]] is better behaved in CGWH than in CGH. Specifically, CHWH is closed under pushouts, one leg of which is the inclusion of a closed subspace. CGH does not have such nice behavior, and pushouts like that are used all over _The Geometry of Iterated Loop Spaces_, specifically in the construction of a monad from an operad and in the use of geometric realizations of simplicial spaces.

([[Peter May]], [MO answer, April 2015](http://mathoverflow.net/a/204221/))

## Related concepts

* [[model structure on compactly generated weak Hausdorff spaces]]

## References

The category of [[compactly generated topological space|compactly generated]] weakly Hausdorff topological spaces was introduced in

* {#McCord69} [[Michael C. McCord]], Section 2 of: *Classifying Spaces and Infinite Symmetric Products*, Transactions of the American Mathematical Society, Vol. 146 (Dec., 1969), pp. 273-298  ([jstor:1995173](https://www.jstor.org/stable/1995173), [pdf](https://www.ams.org/journals/tran/1969-146-00/S0002-9947-1969-0251719-4/S0002-9947-1969-0251719-4.pdf))

as a more convenient setting than Steenrod's compactly generated Hausdorff spaces, given that the latter is not closed under many colimits (for instance quotients) as computed in $Top$.

* [[André Henriques]] et al, _Why the “W” in CGWH (compactly generated weakly Hausdorff spaces)?_, [MO discussion](http://mathoverflow.net/q/47702/), 2010.

A survey is in

* {#Strickland} Neil Strickland, _The category of CGWH spaces_ (2009) (pdf)

[[!redirects weak Hausdorff space]]
[[!redirects weak Hausdorff spaces]]
[[!redirects weakly Hausdorff space]]
[[!redirects weakly Hausdorff spaces]]

[[!redirects weakly Hausdorff topological spaces]]


weakly initial object

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# Contents
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## Definition
 {#Definition}

An [[object]] in a [[category]] is **weakly initial** if there exists a [[morphism]] from it to every other object in the category.

(So a weakly initial object is an actual [[initial object]] if this morphism is unique.)

Weak initiality is an instance of a [[weak colimit]]. It is also an instance of a [[weakly initial set]] that happens to be a [[singleton]] set.

## Related concepts

* [[strict initial object]]

## References

[[weak adjoint|Weak adjoint functors]] along with [[weak colimits]], with weakly initial objects as a special case, were defined in:

* [[Paul Kainen]], *Weak adjoint functors*, Mathematische Zeitschrift **122** 1 (1971) 1-9 [[dml:171575](https://eudml.org/doc/171575)]

See also:

* [[Philip Wadler]], _[Recursive types for free!](https://homepages.inf.ed.ac.uk/wadler/papers/free-rectypes/free-rectypes.txt)_

[[!redirects weak initial object]]
[[!redirects weakly initial objects]]
[[!redirects weak initial objects]]
[[!redirects weak initial]]
[[!redirects weakly initial]]
[[!redirects weakly initial algebra]]
[[!redirects weakly initial algebras]]
[[!redirects weak initial algebra]]
[[!redirects weak initial algebras]]

[[!redirects weakly terminal object]]
[[!redirects weakly terminal objects]]


weakly interacting massive particle

[[!redirects weakly interacting massivle particle]]

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## Idea

What are called _weakly interacting massive particles_ or "WIMP"s, for short, are hypothetical [[fundamental particles]]/[[fields (physics)|field quanta]] which have [[mass]] but which, besides via [[gravity]], [[interaction|interact]] only via the [[weak nuclear force]], or via some yet weaker force, but in particular not via [[electromagnetism]].

WIMPs used to be thought of as likely candidates for [[dark matter]], a conclusion suggested by analysis of relic abundancies (see [below](#MotivationViaRelicAbundancy)), but more recently a series of direct detection [[null results]] puts that assumption increasingly into question -- unless one assumes that the WIMPs interact strictly _only_ via [[gravity]], in which case they would have to be very massive ("WIMPzillas", [CKR 98](#CKR98), [CCKR 01](#CCKR01)).

## Motivation via relic abundancy
  {#MotivationViaRelicAbundancy}

In detail, the argument for WIMP dark matter proceeds as follows (recalled e.g. in [CCKR 01](#CCKR01)):

> The case for [[dark matter|dark]], non[[baryon|baryonic]] [[matter]] in the [[observable universe|universe]] is today stronger than ever [1]. The observed [[cosmic structure formation|large-scale structure]] suggests that [[dark matter]] (DM) accounts for at least 30% of the critical mass density of the universe $\rho_C = 3 H_0^2 M_{Pl}^2 / 8 \pi = 1.88 \times 1-^{-29} g cm^{-3}$, where  $H_0 = 100 h km sec^{-1} Mpc^{-1}$ is the present [[Hubble constant]] and $M_{Pl}$ is the [[Planck mass]].

> The most familiar assumption is that dark matter is a thermal relic, i.e., it was initially in chemical equilibrium in the early universe. A particle species, $X$, tracks its equilibrium abundance as long as reactions which keep the species in chemical equilibrium can proceed on a timescale more rapid than the expansion rate of the universe, $H$. When the reaction rate becomes smaller than the expansion rate, the particle species can no longer track its equilibrium value. When this occurs the particle species is said to be "frozen out.”" The more strongly interacting the particle, the longer it stays in local thermal equilibrium and the smaller its eventual freeze-out abundance.  Conversely, the more weakly interactingthe particle, the larger its present abundance. If freeze out occurs when the particles $X$ areno nrelativistic, the freeze-out value of the particle number per comoving volume $Y$ is related to the mass of the particle and its annihilation cross section (here characterized by $\sigma_0$) by $Y \propto (1/M_X M_{Pl} \sigma_0)$ where $M_X$ is the mass of the particle $X$. Since the contribution to $\Omega_X = \rho_X/\rho_C$ is proportional to $M_X Y$, the present contribution to $\Omega_X$ from a thermal relic roughly is independentof its mass and depends only upon the annihilation cross section. The cross section that results in $\Omega_X h^2\sim 1$ is of order $10^{-37 } cm^2$, which is of the order the [[weak nuclear force|weak]] scale.

## References

* Jonathan L. Feng, *The WIMP Paradigm: Theme and Variations* [[arXiv:2212.02479](https://arxiv.org/abs/2212.02479)]

See also

* Wikipedia _[Weakly interacting massive particle](https://en.wikipedia.org/wiki/Weakly_interacting_massive_particles)_

On super-heavy WIMPs

* {#CKR98} Daniel J. H. Chung, Edward W. Kolb, Antonio Riotto, _Superheavy dark matter_, Phys. Rev. D 59, 023501 (1998) ([arXiv:hep-ph/9802238](https://arxiv.org/abs/hep-ph/9802238))

* {#CCKR01} Daniel J. H. Chung, Patrick Crotty, Edward W. Kolb, Antonio Riotto, _On the gravitational production of superheavy dark matter_,  	Phys. Rev. D64:043503, 2001 ([arXiv:hep-ph/0104100](https://arxiv.org/abs/hep-ph/0104100))

[[!redirects weakly interacting massive particles]]

[[!redirects WIMP]]
[[!redirects WIMPs]]


weakly Lindelöf spaces with countably locally finite base are second countable

## Statement

Recall that a [[topological space]] is [[weakly Lindelöf topological space|weakly Lindelöf]] if every [[open cover]] has a [[countable set|countable]] subcollection the union of which is dense.

\begin{theorem}
Every [[weakly Lindelöf topological space|weakly Lindelöf]] space with [[countably locally finite set of subsets|$\sigma$-locally finite]] [[topological base|base]] is [[second countable space|second countable]].
\end{theorem}

\begin{proof}
Let $\mathcal{V}$ be a countably locally finite base. For each $x \in X$, there is a neighborhood $N_x$ meeting countably many members of $\mathcal{V}$. If $X$ is weakly Lindelöf, there is a countable $\{N_n\}_n$ which covers a dense subset of $X$. Then $ \mathcal{U} = \{V\in \mathcal{V} \mid N_n \cap V \neq \emptyset\}$ is a countable basis for X.
\end{proof}

## Related statements and properties

[[!include topology - global countability axioms]]


weakly Lindelöf topological space

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Named after [[Ernst Leonard Lindelöf]].

#Contents#
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## Definition

A [[topological space]] is a _Lindelöf space_ if every [[open cover]] has a [[countable set|countable]] subcover.
## Related properties and theorems

[[!include topology - countability axioms]]

## References

* [Weakly Lindelöf](https://topology.jdabbs.com/properties/P000062), $\pi$-Base.

[[!redirects weakly Lindelöf topological spaces]]

[[!redirects weakly Lindelöf space]]
[[!redirects weakly Lindelöf spaces]]

[[!redirects weakly Lindeloef topological space]]
[[!redirects weakly Lindeloef topological spaces]]

[[!redirects weakly Lindeloef space]]
[[!redirects weakly Lindeloef spaces]]

[[!redirects weakly Lindelof topological space]]
[[!redirects weakly Lindelof topological spaces]]

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weakly open subtopos

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## Idea

The concept of **weakly open subtopos** is weakening of the concept of a [[dense subtopos]] in topos theory.

## Definition

Let $\mathcal{E}$ be a [[topos]] and $Sh_j(\mathcal{E})$ a subtopos of $\mathcal{E}$ with $j$ the corresponding [[Lawvere-Tierney topology]], $a_j$ the corresponding [[associated sheaf functor]] and $\neg$ and $\neg^j$ the pseudo-complementation operators on the subobject lattices in $\mathcal{E}$ and $Sh_j(\mathcal{E})$, respectively. The subtopos $Sh_j(\mathcal{E})$ is called _weakly open_ if for all subobjects $A\rightarrowtail E$ in $\mathcal{E}$, $a_j(\neg A)\cong \neg^j a_j(A)$.

## Remark

In other words, the sheafification $a_j$ commutes with pseudo-complementation.

If $i:Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ is weakly open then the topology $j$ and the inclusion $i$ are called _weakly open_ as well.

Since the associated sheaf functor $a_j$ is just the [[inverse image]] of the inclusion $i: Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ the definition can be generalized to general [[geometric morphisms]].

## Properties

As already the name suggests, the concept of a weakly open subtopos can equally be viewed as a weakening of the concept of an [[open subtopos]]. Indeed, since the associated sheaf functor of an open inclusion is [[logical functor|logical]] we have

+-- {: .num_prop}
###### Proposition
An open subtopos $i:Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ is weakly open. $\qed$
=--

Less straightforward is the following

+-- {: .num_prop}
###### Proposition
A dense subtopos $i:Sh_j(\mathcal{E})\hookrightarrow\mathcal{E}$ is weakly open.
=--

This occurs as prop.6.3 in [Caramello (2012a)](#Caramello12a). See also [Johnstone (1982)](#Johnstone82).

## Remark

The concept and terminology goes back to [Johnstone (1982)](#Johnstone).

The equivalent concept for [[frames]] is called '_nearly open_' in [Banaschewski-Pultr (1996)](#Banaschewski_Pultr96). The maps they call 'weakly open' are called '[[skeletal geometric morphism|skeletal]]' by Johnstone.

The concept of map in point-set topology that corresponds to weakly open is that of a continuous map $f$ such that $f(U)$ is dense in some open set, for $U$ open (cf. [Pták (1958)](#Ptak58)).

## Related entries

* [[dense subtopos]]

* [[open subtopos]]

* [[closed subtopos]]

* [[locally closed subtopos]]

* [[skeletal geometric morphism]]

* [[double negation]]

* [[De Morganization]]

## References

* {#Banaschewski_Pultr94} [[B. Banaschewski]], A. Pultr, *Variants of openness*, Appl. Cat. Struc. **2** (1994) 1-21 [[doi:10.1007/BF00873038](https://doi.org/10.1007/BF00873038)]

* {#Banaschewski_Pultr96} [[B. Banaschewski]], A. Pultr, *Booleanization*, Cah. Top. Géom. Diff. Cat. **XXXVII** **1** (1996)  41-60 [[numdam:CTGDC_1996__37_1_41_0](http://numdam.mathdoc.fr/numdam-bin/fitem?id=CTGDC_1996__37_1_41_0)]

* {#Caramello12a}[[Olivia Caramello]], _Universal models and definability_ , Math. Proc. Cam. Phil. Soc. (2012) pp.279-302. ([arXiv:0906.3061](http://arxiv.org/abs/0906.3061))

* {#Caramello12}[[Olivia Caramello]], _Topologies for intermediate logics_ , arXiv:1205.2547 (2012). ([abstract](http://arxiv.org/abs/1205.2547))

* {#Johnstone82}[[Peter Johnstone]], _Factorization theorems for geometric morphisms II_ , pp.216-233 in LNM **915** Springer Heidelberg 1982.

* {#Johnstone02} [[Peter Johnstone]], _[[Sketches of an Elephant]] I_, Oxford UP 2002. (pp.209f)

* {#Ptak58}V. Pták, _Completeness and the open mapping theorem_ , Bull. Soc. Math. France **86** (1958) pp.41-74. ([pdf](http://archive.numdam.org/article/BSMF_1958__86__41_0.pdf))

[[!redirects weakly open subtoposes]]
[[!redirects weakly open topology]]
[[!redirects weakly open topologies]]
[[!redirects implicationally open subtopos]]
[[!redirects implicationally open topology]]


weakly periodic cohomology theory

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## Definition

A [[multiplicative cohomology theory|multiplicative]] [[cohomology theory]] $A$ is **weakly periodic** if the natural map

$$
  A^2({*}) \otimes_{A^0({*})} A^n({*}) \stackrel{\simeq}{\to}
  A^{n+2}({*})
$$

is an [[isomorphism]] for all $n \in \mathbb{Z}$.

e.g. [Lurie, remark 1.4](#Lurie)

Compare with the notion of a _[[periodic cohomology theory]]_.

## Properties

### Relation to formal groups

One reason why weakly periodic cohomology theories are of interest is that their [[cohomology ring]] over the space $\mathbb{C}P^\infty$ defines a [[formal group]].

To get a [[formal group]] from a [[weakly periodic cohomology theory|weakly periodic]], [[even cohomology theory|even]] [[multiplicative cohomology theory|multiplicative]] [[cohomology theory]] $A^\bullet$, we look at the induced map on $A^\bullet$ from a morphism

$$
  i_0 : {*} \to \mathbb{C}P^\infty
$$

and take the kernel

$$
  J := ker(i_0^* : A^0(\mathbb{C}P^\infty) \to A^0({*}))
$$

to be the [[ideal]] that we complete along to define the [[formal scheme]] $Spf A^0(\mathbb{C}P^\infty)$ (see there for details).

Notice that the map from the point is unique only up to [[homotopy]], so accordingly there are lots of chocies here, which however all lead to the same result.

The fact that $A$ is weakly periodic allows to reconstruct the [[cohomology theory]] essentially from this [[formal scheme]].

To get a [[formal group law]] from this we proceed as follows: if the [[Lie algebra]] $Lie(Spf A^0(\mathbb{C}P^\infty))$ of the [[formal group]]

$$
  Lie(Spf A^0(\mathbb{C}P^\infty))
  \simeq
  ker(i_0^*)/ker(i_0^*)^2
$$

is a free $A^0({*})$-module, we can pick a generator $t$ and this gives an [[isomorphism]]

$$
  Spf(A^0(\mathbb{C}P^\infty)) \simeq Spf(A^0({*})[[t]])
$$

if $A^0(\mathbb{C}P^\infty) A^0({*})[ [t] ]$ then $i_0^*$ "forgets the $t$-coordinate".

## Examples

* [[complex K-theory]]

* [[elliptic cohomology]]

## Related entries

* [[periodic cohomology theory]]

* [[periodic ring spectrum]]

* [[A Survey of Elliptic Cohomology - formal groups and cohomology]]

## References

* {#Lurie} [[Jacob Lurie]], _[[A Survey of Elliptic Cohomology]]_
[[!redirects weakly periodic cohomology theories]]