wave polarization

> for other concepts of a similar name see at _[[polarization]]_

***

#Contents#
* table of contents
{:toc}

## Idea

A _[[plane wave]]_ with more than one component, i.e. being a [[section]] of a [[trivial vector bundle]], is characterized not just by its [[wave vector]] $k$, but also by that component vector $e$. Broadly speaking this $e$ is the _polarization_ of the wave.

Specifically a [[plane wave]] with [[coefficients]] in the [[cotangent bundle]] of a [[Minkowski spacetime]] $\Sigma$, such as an [[electromagnetic field|electromagnetic]] [[field history]] "[[vector potential]]" $A \in \Gamma_\Sigma(T^\ast \Sigma)$ is given by

$$
  A_\mu(x) = e_\mu e^{i k_\mu x^\mu}
  \,.
$$

In the case of [[free field theory|free]] [[electromagnetic waves]] the [[wave vector]] $k$ is [[light-like]], $k_\mu k^\mu = 0$, and in [[Gaussian-averaged Lorenz gauge]] the polarization vector $e$ has to satisfy $e^\mu k_\mu = 0$ and polarization vectors proportional to the wave vector $e_\mu \propto k_\mu$ are [[gauge equivalence|gauge equivalent]] to [[zero]]. Therefore in this case the space of physically distinguishable polarizations for given [[wave vector]] $k$ is the [[quotient space]]

$$
  \frac{
    \left\{ e \,\vert\, e^\mu k_\mu = 0 \right\}
  }{
    \left\{ e \,\vert\, e_\mu \propto k_\mu \right\}
  }
  \,.
$$

This is also called the space of _transversal polarizations_.

If one chooses [[coordinates]] such that $k = (\kappa, 0, \cdots, 0, \kappa)$ then this may be identified with the space of vectors of the form $e = (0, *, \cdots, *, 0)$.

## References

* Wikipedia, _Polarization (waves)_

* {#Dermisek09} [[Radovan Dermisek]], _Quantum Electrodynamics (QED)_ ([pdf](http://www.physics.indiana.edu/~dermisek/QFT_09/qft-II-5-4p.pdf), [[DermisekQED.pdf:file]])

[[!redirects wave polarizations]]

[[!redirects transversal polarization]]
[[!redirects transversal polarizations]]


wave vector

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

#Contents#
* table of contents
{:toc}

## Idea

A _wave vector_ is a [[vector]] that encodes [[wavelength]] and [[direction of a vector|direction]] of a _[[plane wave]]_.

## Definition

Let $n \in \mathbb{N}$ and write $\mathbb{R}^n$ the [[Cartesian space]] of [[dimension]] $n$. Thinking of $\mathbb{R}^n$ as a [[vector space]], then each point in it is a [[vector]] $\vec x \in \mathbb{R}^n$ and hence a [[smooth function]] $f \colon \mathbb{R}^n \to \mathbb{C}$ may be thought of as a function of these "position vectors".

If $f$ is a [[function with rapidly decreasing partial derivatives]], then its [[Fourier transform]] $\hat f \;\colon\; \mathbb{R}^n \to \mathbb{C}$ exists. By the [[Fourier inversion theorem]], this function is such that it expresses $f$ as a [[superposition]] of "[[plane wave]]" functions $\vec x \mapsto e^{2\pi i \vec x \cdot \vec k}$ as

$$
  f(\vec x)
  \;=\;
  \underset{\vec k \in \mathbb{R}^n}{\int}
   \hat f(k) \, e^{2 \pi i \vec k \cdot \vec x}
  \, d \vec k
  \,.
$$

Here the [[vector]] $\vec k \in \mathbb{R}^n$ determines

1. the [[wavelength]] $\lambda \coloneqq 1/{\vert \vec k\vert}$ (the inverse of the [[norm]] of $\vec k$);

1. the direction $\frac{\vec k}{{\vert \vec k\vert }} \in S(\mathbb{R}^n)$ (the corresponding unit vector in the [[unit sphere]])

of the "[[plane wave]]" $\vec x \mapsto e^{2 \pi i \vec x \cdot \vec k}$.

The product $2 \pi {\vert k \vert}$ is also called the _wave number_ and $2 \pi k$ then the _wave number vector_. Beware that elsewhere the wave number vector is denoted "$k$", which makes the "wave vector" become $k / 2 \pi$. (See e.g. [Wikipedia](#Wikipedia), "Physics definition" as opposed to "Crystallography definition".)

If here $\mathbb{R}^n \simeq \mathbb{R}^{p,1}$ is identified with [[Minkowski spacetime]] with canonical coordinates denoted $(x^0, x^1, \cdots, x^p)$, then the 0-component of the wave vector

$$
  \nu \coloneqq k_0
$$

is called the _[[frequency]]_ of the corresponding [[plane wave]] (in the chosen [[coordinate system]]); this $\omega = 2 \pi \vu$ is the _[[angular frequency]]_.

[[!include plane waves -- table]]

## Related concepts

* [[plane wave]]

* [[Fourier transform]]

* [[wave]]

* [[frequency]], [[amplitude]]

* [[wave polarization]]

## References

See also

* {#Wikipedia} Wikipedia, _[Wave vector](https://en.wikipedia.org/wiki/Wave_vector)_

[[!redirects wave vectors]]

[[!redirects wavevector]]
[[!redirects wavevectors]]


waveform

\tableofcontents

\section{Idea}

In the context of [[signal processing]], a _waveform_ is a [[periodic wave]] [[signal]] viewed as a [[function]] from [[time]] to [[amplitude]], both viewed as [[real number|real-valued]] variables. One speaks of this as viewing the signal 'in the time domain'.

This is contrast to describing the wave as a function from [[frequency]] to amplitude, where we typically have only a [[finite set]] of frequencies. One speaks in this case of viewing the signal 'in the frequency domain'.

A [[discrete Fourier transform]] can be used to pass from the time domain to the frequency domain; an inverse discrete Fourier transform can be used to pass from the frequency domain to the time domain.

\section{See also}

* [[frequency]]

* [[fundamental frequency]]

[[!redirects waveforms]]


waveform modulation

\tableofcontents

\section{Idea}

In [[signal processing]], _modulation_ refers to modification of a standard [[waveform]] (for example a [[sine wave]]) according to a fixed algorithm in order to create a [[signal]] conveying the input to that algorithm. To create a [[digital signal]], for example, the input to such an algorithm would typically be a sequence $S$ of [[bit|bits]], and the output of the algorithm would be a [[waveform]] from whose properties $S$ can be recovered.

\section{Digital modulation}

The following are forms of digital modulation.

* [[On-off keying]] (OOK)


wavefront set

+-- {: .rightHandSide}
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### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[microlocal analysis]], the _wave front set_ ([Hörmander 70](#Hoermander70)) of a [[generalized function]] such as a [[distribution]] or a [[hyperfunction]] is a characterization of the singularity structure of the generalized function, hence of how it deviates from being an ordinary smooth function.

The wave front set is the sub-bundle of the [[cotangent bundle]] that consists of all those [[direction of a vector|directions]] (non-zero [[covectors]]) such that the local [[Fourier transform]] of the distribution is not rapidly decaying in this [[direction of a vector|direction]] ([Hörmander 90, section 8.1](#Hoermander90)).  Such covectors are stable under multiplication by positive scalars, so the wave front set can also be considered as a [[sub-bundle]] of the [[unit sphere bundle]] of the [[cotangent bundle]].

The [[projection]] of the wave front set down to the base space is the [[singular support of a distribution|singular support]] of the distribution. The additional information in the "wave front" [[covectors]] over this singular support may be understood as providing the directions of _propagation of these singularities_. This is made precise by the _[[propagation of singularities theorem]]_

A notorious issue with [[distributions]] is that, when thought of as generalized functions, generally neither their [[composition of distributions]] nor their pointwise [[product of distributions]] is defined. However, closer inspection shows that the [[obstruction]] to these operations being defined for any given pair of distributions is exactly characterized by the wave front set:

For instance the [[product of distributions]] is well defined precisely if the sum of their wave front sets does not intersect the zero-section ([[Hörmander's criterion]], [Hörmander 90, theorem 8.2.10](#Hoermander90)).

## Definition

### Motivation

The definition of wavefront sets is motivated by a version of a [[Paley-Wiener theorem]] that characterizes smooth compactly supported functions ($\mathbb{R}^n \to \mathbb{R}$) by a growth condition on their [[Fourier transform]] $\mathcal{F}$:

+-- {: .num_theorem}
###### Theorem
**([[Paley-Wiener-Schwartz theorem]])**

The vector space $C_0^{\infty}(\mathbb{R}^n)$ of [[smooth function|smooth]] [[compact support|compactly supported]] functions ([[bump functions]]) is (algebraically and topologically) [[isomorphism|isomorphic]], via the [[Fourier transform]], to the space of [[entire functions]] $F$ which satisfy the following estimate: there is a positive constant $B$ such that for every [[integer]] $m \gt 0$ there is a constant $C_m$ such that:
$$
      F(z) \le C_m (1 + |z|)^{-m} \exp{ (B \; |\operatorname{Im}(z)|)}
$$

=--

### Smoothness

We call a smooth compactly supported function that is identically $1$ in a neighbourhood of a point $x_0$ a **cutoff** function at $x_0$.
Let $U \subset \mathbb{R}^n$ be open, we identify the [[cotangent bundle]] of $U$ with $U \times \mathbb{R}^n$. A subset of $U \times \mathbb{R}^n$ is said to be **conic** if it is stable under the transformation
$$
    (x, \zeta) \mapsto (x, \rho \zeta) \quad \text{with} \; \rho \gt 0
$$
Note that a [[conical set|conic]] subset is uniquely determined by its intersection with the [[unit sphere]] bundle $U\times S^{n-1}$.

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

Let $f$ be a distribution and $(x_0, \zeta_0)$ with $\zeta_0 \neq 0$ be a point of the cotangent bundle of $U$. $f$ is **smooth** in $(x_0, \zeta_0)$ if there is a cutoff function $\chi$ in $x_0$ and an open cone $\Gamma_0$ in $\mathbb{R}^n$ containing $\zeta_0$ such that for every $m \gt 0$ there is a nonnegative constant $C_m$ such that for all $\zeta \in \Gamma_0$:
$$
\| \mathcal{F}(\chi f) (\zeta)   \| \le C_m (1 + \| \zeta \|)^{-m}
$$
where $\mathcal{F}(\chi f)$ is the [[Fourier transform]] (of the variable $\zeta$) of the function $\chi f$ (of the variable $x$).

=--

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

A distribution $f$ is smooth in a [[conical set|conic]] subset $\Gamma$ of the cotangent bundle of $U$ if $f$ is smooth in a neighbourhood of every point in $\Gamma$.

=--

### Wavefront set

Let $U \subseteq \mathbb{R}^n$ be an open subset, $T^* U$ its cotangent bundle and $f$ be a distribution on $U$. The complement of the union of all [[conical set|conic]] subsets of $T^* U$ where $f$ is smooth is the **wavefront set $WF(f)$**.  Since the wavefront set is therefore itself [[conical set|conic]], it is equivalently determined by a subset of the unit sphere bundle of $T^* U$.

([Hörmander 70 (2.4.1)](#Hoermander70), [Hörmander 90, section 8.1](#Hoermander90))

This definition turns out to make invariant sense ([Hörmander 90, p. 256](#Hoermander90)).

## Examples

+-- {: .num_example #WaveFrontOfDeltaDistribution}
###### Example
**(wave front set of [[delta distribution]])**

For $n \in \mathbb{N}$, consider the [[delta distribution]]

$$
  \delta(0) \in \mathcal{D}'(\mathbb{R}^n)
$$

on $n$-dimensional [[Cartesian space]], given by [[evaluation]] at the origin. Its wave front set is

$$
  WF(\delta(0))
  =
  \left\{
    (0,k)
    \;\vert\;
    k \in \mathbb{R}^n \setminus \{0\}
  \right\}
  \subset
  \mathbb{R}^n \times \mathbb{R}^n
  \simeq
  T^\ast \mathbb{R}^n
  \,.
$$

=--

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

First of all the [[singular support of a distribution|singular support]] of $\delta(0)$ is clearly $singsupp(\delta(0)) = \{0\}$, hence the wave front set vanishes over $\mathbb{R}^n \setminus \{0\}$.

At the origin, any bump function $b$ supported around the origin with $b(0) = 1$ satisfies $b \cdot \delta(0) = \delta(0)$ and hence the wave front set over the origin is the set of covectors along which the [[Fourier transform of distributions|Fourier transform]] $\hat \delta(0)$ does not suitably decay. But this Fourier transform is in fact a [[constant function]] and hence does not decay in any direction.

=--

+-- {: .num_example #WaveFrontSetOfHeavisideDistribution}
###### Example
**(wave front set of [[Heaviside distribution]])**

Let $H \in \mathcal{D}'(\mathbb{R}^1)$ be the [[Heaviside distribution]] given by

$$
  \langle H, b\rangle \coloneqq \int_0^\infty b(x)\, d x
  \,.
$$

Its wave front set is

$$
  WF(H) = \{(0,k) \vert k \neq 0\}
  \,.
$$

=--

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

For $(X,e)$ a [[globally hyperbolic spacetime]] and $P$ a [[hyperbolic differential operator]] such as the [[wave operator]]/[[Klein-Gordon operator]], then the [[propagation of singularities theorem]] says that the wave front set of any solution $f$ to $P f = 0$ is a union of [[lightlike]] [[geodesics]] and their [[cotangent vectors]].

Specifically for the [[Klein-Gordon operator]] such ditributional solutions include the [[causal propagator]] and the [[Feynman propagator]].

=--

+-- {: .num_example #WaveFrontOfTensorProductDistribution}
###### Example
**(wave front set of [[tensor product distribution]])**

Let $u \in \mathcal{D}'(X)$ and $v \in \mathcal{D}'(Y)$ be two distributions. then the wave front set of their [[tensor product distribution]] $u \otimes v \in \mathcal{D}'(X \times Y)$ satisfies

$$
  WF(u \otimes v)
  \;\subset\;
  \left(
     WF(u) \times WF(v)
  \right)
    \cup
  \left(
    \left( supp(u) \times \{0\} \right)
    \times WF(v)
  \right)
    \cup
   \left(
     WF(u)
     \times
     \left(
       supp(v) \times \{0\}
     \right)
   \right)
  \,,
$$

where $supp(-)$ denotes the [[support of a distribution]].

=--

([Hörmander 90, theorem 8.2.9](#Hoermander90))

## Properties

+-- {: .num_prop #EmptyWaveFrontSetCorrespondsToOrdinaryFunction}
###### Proposition
**(empty wave front set corresponds to ordinary functions)**

The wave front set of a [[compactly supported distribution]] is empty precisely if the distribution comes from an ordinary [[smooth function]] (hence a [[bump function]]).

=--

e.g. ([Hörmander 90, below (8.1.1)](#Hoermander90))

+-- {: .num_prop #DerivativeOfDistributionRetainsOrShrinksWaveFrontSet}
###### Proposition
**([[derivative of distributions]] retains or shrinks wave front set)**

Taking [[derivatives of distributions]] retains or shrinks the wave front set:

For $u \in \mathcal{D}'(\mathbb{R}^n)$ a distribution and $\alpha \in \mathbb{N}^n$ a multi-index with $D^\alpha$ denoting the corresponding [[partial derivative|partial]] [[derivative of distributions]], then

$$
  WF(D^\alpha u) \subset WF(u)
  \,.
$$

Hence if $P$ is any [[differential operator]] with [[smooth function]] [[coefficients]], then

$$
  WF(P u) \subset WF(u)
  \,.
$$

=--

([Hörmander 90, (8.1.10) (8.1.11), p. 256](#Hoermander90))

+-- {: .num_prop #WaveFrontSetOfCompactlySupportedDistributions}
###### Proposition
**([[wave front set]] of [[convolution of distributions|convolution of]] [[compactly supported distributions]])**

Let $u,v \in \mathcal{E}'(\mathbb{R}^n)$ be two  [[compactly supported distributions]]. Then the [[wave front set]] of their [[convolution of distributions]] is

$$
  WF(u \star v)
  \;=\;
  \left\{
    (x + y, k)
    \;\vert\;
    (x,k) \in WF(u) \,\text{and}\, (y,k) \in WF(u)
  \right\}
  \,.
$$

=--

([Bengel 77, prop. 3.1](convolution+of+distributions#Bengel77))

## Related concepts

* [[ultraviolet divergence]]

## References

### General

The concept of wave front set is due to

* {#Hoermander70} [[Lars Hörmander]], _Linear differential operators_, Actes Congr. Int. Math. Nice 1970, 1, 121-133 ([pdf](http://www.mathunion.org/ICM/ICM1970.1/Main/icm1970.1.0121.0134.ocr.pdf))

A textbook account for distributions on open subsets of [[Euclidean space]] is in

* {#Hoermander90} [[Lars Hörmander]], section 8.1 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990

and for distributions more generally on smooth manifolds is in

* {#Hoermander94} [[Lars Hörmander]], _The analysis of linear partial differential operators_, vol. III, Springer 1994

A history of the concept of wave front sets with extensive pointers to the
literature is given in [Hörmander 90, p. 322-324](#Hoermander90).

See also

* Wikipedia: [wavefront set] (http://en.wikipedia.org/wiki/Wavefront_set)

### In quantum field theory

The application of [[microlocal analysis]] via wave front sets to the discussion of [[n-point functions]] in [[quantum field theory]] and especially [[quantum field theory on curved spacetimes]] originates with the results of

* {#DuistermaatHoermander72} [[Johann Duistermaat]], [[Lars Hörmander]], sections 6.5, 6.6 of _Fourier integral operators II_, Acta Mathematica 128, 183-269, 1972 ([Euclid](https://projecteuclid.org/euclid.acta/1485889724))

which were first picked up in

* C. Moreno, _Spaces of positive and negative  frequency  solutions of  field equations in curved  space- times. I.  The Klein-Gordon  equation in stationary  space-times, II. The massive  vector field equations in static  space-times_, J. Math.  Phys. 18, 2153-61 (1977), J. Math.  Phys. 19, 92-99 (1978)

*  [[Jonathan Dimock]], _Scalar  quantum field  in  an external  gravitational  background_,  J. Math.  Phys. 20, 2549-2555 (1979)

and brought into context with the [[Hadamard distributions]] needed for the [[construction]] of [[Wick algebras]] in

* {#Radzikowski96} [[Marek Radzikowski]], *Micro-local approach to the Hadamard condition in quantum field theory on curved space-time*, Commun. Math. Phys. **179** (1996) 529-553 [[doi:10.1007/BF02100096](https://doi.org/10.1007/BF02100096), [euclid:cmp/1104287114](http://projecteuclid.org/euclid.cmp/1104287114)]

* {#BrunettiFredenhagen00} [[Romeo Brunetti]], [[Klaus Fredenhagen]], *Microlocal Analysis and Interacting Quantum Field Theories: Renormalization on Physical Backgrounds*, Commun. Math. Phys. **208** (2000) 623-661 [[math-ph/9903028](https://arxiv.org/abs/math-ph/9903028), [doi:10.1007/s002200050004](https://doi.org/10.1007/s002200050004)]

A textbook account amplifying this usage (on [[Minkowski spacetime]]) in the mathematically rigorous construction of [[perturbative quantum field theory]] via [[causal perturbation theory]] is in

* {#Scharf95} [[Günter Scharf]], _[[Finite Quantum Electrodynamics -- The Causal Approach]]_, Berlin: Springer-Verlag, 1995, 2nd edition

* {#Scharf01} [[Günter Scharf]], _[[Quantum Gauge Theories -- A True Ghost Story]]_, Wiley 2001

For more see the references at _[[locally covariant perturbative quantum field theory]]_.

### In differential cohomology

Wave-front sets of [[currents (distribution theory)|currents]] play a role in the construction of "geometric cycles" for [[differential cobordism cohomology]] by actual [[cobordism]]-classes equipped with [[differential geometry|differential geometric]] data:

* [[Ulrich Bunke]], [[Thomas Schick]], Ingo Schroeder, Moritz Wiethaup, §4.2.6 in: *Landweber exact formal group laws and smooth cohomology theories*, Algebr. Geom. Topol. **9** (2009) 1751-1790 [[arXiv:0711.1134](https://arxiv.org/abs/0711.1134), [doi:10.2140/agt.2009.9.1751](https://doi.org/10.2140/agt.2009.9.1751)]

and in the [[Hodge-filtered differential cohomology|Hodge-filtered]] version:

* {#Haus22} [[Knut Bjarte Haus]], §2.6 in: *Geometric Hodge filtered complex cobordism*, PhD thesis (2022) [[ntnuopen:3017489](https://ntnuopen.ntnu.no/ntnu-xmlui/handle/11250/3017489)]

* {#HausQuick22} [[Knut Bjarte Haus]], [[Gereon Quick]], §2.10 of: *Geometric Hodge filtered complex cobordism* [[arXiv:2210.13259](https://arxiv.org/abs/2210.13259)]

[[!redirects wavefront sets]]

[[!redirects wave front set]]
[[!redirects wave front sets]]

[[!redirects wave-front set]]
[[!redirects wave-front sets]]

[[!redirects wavefront]]
[[!redirects wavefronts]]


wavelength

+-- {: .rightHandSide}
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### Context
#### Harmonic analysis
+-- {: .hide}
[[!include harmonic analysis - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[distance]] between two [[crests]] of a [[wave]].

## Definition

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

$$
  \vec x \mapsto e^{2 \pi i \vec k \cdot \vec x}
$$

a [[plane wave]] with [[wave vector]] $\vec k \in \mathbb{R}^n$, then its _wavelength_ is the inverse of the [[norm]] of $k$:

$$
  \lambda \coloneqq 1/{\vert k\vert}
  \,.
$$

The quotient $k/{\vert k \vert} \in S(\mathbb{R}^n)$ is the _[[direction of a vector|direction]]_ of the [[plane wave]].

The product $2\pi/\lambda$ is also called the _wave number_.

[[!include plane waves -- table]]

## Examples

* [[Compton wavelength]]

## Related concepts

* [[wave vector]]

* [[frequency]]

* [[Fourier analysis]]

* [[momentum]], [[energy]]

[[!redirects wavelengths]]

[[!redirects wave-length]]
[[!redirects wave length]]
[[!redirects wave-lengths]]
[[!redirects wave lengths]]

[[!redirects wave number]]
[[!redirects wave numbers]]

[[!redirects wavenumber]]
[[!redirects wavenumbers]]


weak (star)-autonomous category

## Idea

Some [[symmetric monoidal categories]] have a notion of [[duality]] without being a [[*-autonomous category]]. However, they can be a weak $*$-autonomous category.

## Definition

\begin{definition}
A _weak $*$-autonomous category_  is a [[symmetric monoidal
category|symmetric]] [[closed monoidal category]] $\langle
C,\otimes, I,\multimap\rangle$ with an object
$\bot$ such that the canonical morphism
$$ d_A: A \to (A \multimap \bot) \multimap \bot ,$$
which is the transpose of the [[evaluation map]]
$$ ev_{A,\bot}: (A \multimap \bot) \otimes A \to \bot ,$$
is a [[monomorphism]] for all $A$.  (Here, $\multimap$ denotes the [[internal hom]].)
\end{definition}

A $*$-autonomous category is a weak $*$-autonomous category such that for every object $A$, $d_{A}: A \to (A \multimap \bot) \multimap \bot$ is an isomorphism.

A strictly weak $*$-autonomous category is a weak $*$-autonomous category such that for at least one object $A$, $d_{A}$ is not an isomorphism. Equivalently, a strictly weak $*$-autonomous category is a weak $*$-autonomous category which is not a $*$-autonomous category.

## Example

* For every [[field]] $\mathbb{K}$, the category $Vec_{\mathbb{K}}$ is a strict weak $*$-autonomous category.
* For every [[commutative ring]] $R$, the category $Mod_{R}$ is a strict weak $*$-autonomous category.

## Related concepts

[[*-autonomous category]]
[[closed monoidal category]]


weak adjoint

# Weak adjoints

* table of contents
{: toc}

## Idea

A **weak adjoint** is like an [[adjoint functor]] but without the uniqueness of factorizations.

Importantly, this means that unlike adjoint functors, weak adjoints are not unique, and so a weak adjoint of a given functor is a [[structure]] whereas a functor having an adjoint is a [[property]].

## Definition

Let $G:D\to C$ be a [[functor]].  We say $G$ has a **left weak adjoint** if for each $x\in C$ there exists a morphism $\eta_x:x\to G z$ such that for any morphism $f:x\to G y$ there exists a (not necessarily unique) morphism $g:y\to z$ such that $f = G g \circ\eta_x$.

## Examples

* A [[weak limit]] is a weak right adjoint to a constant-diagram functor.

## Remarks

Of course, if the factorizations $g$ are always unique, this is precisely an ordinary [[left adjoint]].  A weak adjoint can also be regarded as a stronger form of the [[solution set condition]] in which the solution sets are required to be singletons.  The "dual" property, in which the solution sets need not be singletons but the factorizations are unique, is called a [[multi-adjoint]].

## Related Concepts

* [[weak representation of a functor]]

* [[weakly initial object]]

## References
 {#References}

Weak adjoint functors along with [[weak colimits]] were defined in:

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

[[!redirects weak adjoints]]
[[!redirects weak adjoint functor]]
[[!redirects weak adjoint functors]]
[[!redirects weak adjunction]]
[[!redirects weak left adjoint]]
[[!redirects weak left adjoint]]
[[!redirects weak right adjoint]]
[[!redirects weak right adjoint]]
[[!redirects left weak adjoint]]
[[!redirects left weak adjoints]]
[[!redirects right weak adjoint]]
[[!redirects right weak adjoints]]


weak bialgebra

#Contents#
* table of contents
{:toc}

## Idea

The notion of _weak bialgebra_ is a generalization of that of  [[bialgebra]] in which the comultiplication $\Delta$ is weak in the sense that $\Delta(1)\neq 1\otimes 1$ in general;
similarly the compatibility of counit with the multiplication
map is weakened (counit might fail to be a morphism of algebras). (Still a special case of [[sesquialgebra]].)

Correspondingly [[weak Hopf algebras]] generalize [[Hopf algebras]] accordingly. Every weak Hopf algebra defines a
[[Hopf algebroid]].

### Physical motivation

This kind of structures naturally comes in [[CFT]] models relation to quantum groups a root of unity: the full symmetry algebra is not quite a quantum group at root of unity, because if it were one would have to include the nonphysical quantum dimension zero finite-dimensional quantum group representations into the (pre)Hilbert space; those are the zero norm states which do not contribute to physics (like ghosts). If one quotients by these states then the true unit of a quantum group becomes an idempotent (projector), hence one deals with weak Hopf algebras instead as a price of dealing with true, physical, Hilbert space.

## Definitions

A __weak bialgebra__ is a tuple $(A,\mu,\eta,\Delta,\epsilon)$
such that $(A,\mu,\eta)$ is an associative unital algebra,
$(A,\Delta,\epsilon)$ is a coassociative counital coalgebra
and the following compatibilities, (i),(ii) and (iii), hold:

(i) the coproduct $\Delta$ is multiplicative $\Delta(x)\Delta(y)= \Delta(x y)$. If only (i) is satisfied, following Böhm, Caenapeel and Janssen 2011, we may speak of a __prebialgebra__.

(ii) the counit $\epsilon$ satisfies weak multiplicativity
$$
\epsilon(x y z) = \epsilon(x y_{(1)})\epsilon(y_{(2)} z),
$$
$$
\epsilon(x y z) = \epsilon(x y_{(2)})\epsilon(y_{(1)} z).
$$
A prebialgebra satisfying the first (the second) of the above properties is said to be left (right) monoidal.

(iii) Weak comultiplicativity of the unit:

$$
\Delta^{(2)} (1) = (\Delta(1) \otimes 1)(1\otimes \Delta(1))
$$
$$
\Delta^{(2)} (1) = (1 \otimes\Delta(1))(\Delta(1) \otimes 1)
$$
A prebialgebra satisfying the first (the second) of the above properties is said to be left (right) comonoidal.

As usually in the context of coassociative coalgebras, we denoted $\Delta^{(2)} := (id\otimes\Delta)\Delta =
(\Delta\otimes id)\Delta$.

A weak $k$-bialgebra $A$ is a __weak Hopf algebra__ if it has a $k$-linear map $S:A\to A$ (which is then called an antipode)
such that for all $x\in A$
$$
x_{(1)} S(x_{(2)}) = \epsilon(1_{(1)} x)1_{(2)},
$$
$$
S(x_{(1)})x_{(2)} = 1_{(1)} \epsilon(x 1_{(2)}),
$$
$$
S(x_{(1)})x_{(2)} S(x_{(3)}) = S(x)
$$
It follows that the antipode is antimultiplicative, $S(x y)=S(y)S(x)$, and
anticomultiplicative, $\Delta(S(x)) = S(x)_{(1)}\otimes S(x)_{(2)} =
S(x_{(2)})\otimes S(x_{(1)})$.

## Properties

### Idempotents ("projections")

For every weak bialgebra there are $k$-linear maps
$\Pi^L,\Pi^R:A\to A$ defined by
$$
\Pi^L(x) := \epsilon(1_{(1)} x) 1_{(2)},\,\,\,\,
\Pi^R(x) := 1_{(1)}\epsilon(x 1_{(2)}).
$$
Expressions for $\Pi^L(x),\Pi^R(x)$ are already met above as the right hand sides in two of the axioms for the antipode. Maps $\Pi^L,\Pi^R$
are idempotents, $\Pi^R\Pi^R = \Pi^R$ and $\Pi^L\Pi^L = \Pi^L$:
$$\array{
\Pi^L(\Pi^L(x)) &=& \epsilon\left(1_{(1')}\epsilon(1_{(1)}x) 1_{(2)}\right)1_{(2')}
= \epsilon(1_{(1)}x)\epsilon(1_{(1')}1_{(2)}) 1_{(2')}
\\ &=&\epsilon(1_{(1)}x)\epsilon(1_{(2)}) 1_{(3)}
= \epsilon(1_{(1)}x)1_{(2)} = \Pi^L(x).
}$$

Notice $\epsilon(x z) = \epsilon(x 1 z)  = \epsilon(x 1_{(2)})\epsilon(1_{(1)}z)) = \epsilon(x \epsilon(1_{(1)}z))1_{(2)} = \epsilon(x\Pi^L(z)) = \epsilon(\Pi^R(x)z)$.
The images of the idempotents $A^R = \Pi^R(A)$ and $A^L = \Pi^L(R)$ are dual as $k$-linear spaces: there
is a canonical nondegenerate pairing $A^L\otimes A^R\to k$
given by $(x,y) \mapsto \epsilon(y x)$.

Also $\Pi^L(x\Pi^L(y)) = \Pi^L(x y)$
 and $\Pi^R(\Pi^R(x)y) = \Pi^R(x y)$,
 dually $\Delta(A^L)\subset A\otimes A^L$
        and $\Delta(A^R)\subset A^R\otimes A$,
 and in particular $\Delta(1)\in A^R\otimes A^L$.

Sometimes it is also useful to consider the idempotents $\bar\Pi^L,\bar\Pi^R:A\to A$ defined by
$$
\bar\Pi^L(x) := \epsilon(1_{(2)} x) 1_{(1)},\,\,\,\,
\bar\Pi^R(x) := 1_{(2)}\epsilon(x 1_{(1)}).
$$
$$\array{
\bar\Pi^L(\bar\Pi^L(x))&=&\epsilon(1_{(2')}\epsilon(1_{(2)}x)1_{(1)})1_{(1')} = \epsilon(1_{(2)}x)\epsilon(1_{(2')}1_{(1)})1_{(1')} \\
&=& \epsilon(1_{(3)}x)\epsilon(1_{(2)})1_{(1)}=
\epsilon(1_{(2)}x)1_{(1)} = \bar\Pi^L(x).
}$$

### Relation to fusion categories

Under [[Tannaka duality]] (semisimple) weak Hopf algebras correspond to (multi-)[[fusion categories]] ([Ostrik](#Ostrik)).

### Relation to Frobenius algebras

As explained in [[Hopf algebra]], any finite-dimensional Hopf algebra can be given the structure of a [[Frobenius algebra]]. There is a similar result for weak Hopf algebras.

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

Any finite-dimensional weak Hopf algebra can be given the structure of a [[quasi-Frobenius algebra]].

=--

This is due to [Bohm, Nill, and Szlachanyi (1999)](#BNS99). While [Vecsernyés (2003)](Vec03) seems to show that finite-dimensional weak Hopf algebras can be turned into Frobenius algebras, it is observed in [Iovanov & Kadison (2008)](IK08) that the proof only implies they are *quasi-*Frobenius algebras.

## Literature

Weak comultiplications were introduced in

* G. Mack, [[Volker Schomerus]], _Quasi Hopf quantum symmetry in quantum theory_, Nucl. Phys. B370(1992) 185.

where also weak quasi-bialgebras are considered and physical motivation is discussed in detail. Further work in this vain is in

* [[G. Böhm]], [[K. Szlachányi]], _A coassociative $C^\ast$-quantum group with non-integral dimensions_, Lett. Math. Phys. __35__ (1996) 437--456, [arXiv:q-alg/9509008](http://arxiv. rg/abs/q-alg/9509008); _Weak $C*$-Hopf algebras: the coassociative symmetry of non-integral dimensions_, in: Quantum groups and quantum spaces (Warsaw, 1995), 9-19, Banach Center Publ. __40__, Polish Acad. Sci., Warszawa 1997.
* Florian Nill, _Axioms for weak bialgebras_, [math.QA/9805104](http://arxiv.org/abs/math/9805104)
* [[G. Böhm]], F. Nill, [[K. Szlachányi]], _Weak Hopf algebras. I. Integral theory and $C^\ast$-structure, J. Algebra __221__ (1999), no. 2, 385-438, [math.QA/9805116](https://arxiv.org/abs/math/9805116) #{BohmNillSzlachanyi}

A book exposition is in [chapter](https://link.springer.com/chapter/10.1007/978-3-319-98137-6_6) weak (Hopf) bialgebras in

* [[Gabriella Böhm]], _Hopf algebras and their generalizations from a category theoretical point of view_, Lecture Notes in Math. __2226__, Springer 2018, [doi](https://link.springer.com/book/10.1007/978-3-319-98137-6)

Now these works are understood categorically from the point of view of weak monad theory:

* [[Gabriella Böhm]], Stefaan Caenepeel, Kris Janssen, _Weak bialgebras and monoidal categories_, Comm. Algebra __39__ (2011), no. 12 (special volume dedicated to Mia Cohen), 4584-4607. [arXiv:1103.226](http://arxiv.org/abs/1103.2261)
* [[Gabriella Böhm]], [[Stephen Lack]], [[Ross Street]], _Weak bimonads and weak Hopf monads_, J. Algebra 328 (2011), 1-30, [arXiv:1002.4493](http://arxiv.org/abs/1002.4493)
* [[Gabriella Böhm]], José Gómez-Torrecillas, _On the double crossed product of weak Hopf algebras_, [arXiv:1205.2163](arXiv/1205.2163)

The relation to [[fusion categories]] is discussed in

* Takahiro Hayashi, _A canonical Tannaka duality for finite semisimple tensor categories_ ([arXiv:math/9904073](http://arxiv.org/abs/math/9904073))

* [[Victor Ostrik]], _Module categories, weak Hopf algebras and modular invariants_ ([arXiv:math/0111139](http://arxiv.org/abs/math/0111139))
 {#Ostrik}

On the relation to [[Frobenius algebra|Frobenius algebras]]

* {#BNS99} [[Gabriella Bohm]], [[Florian Nill]], [[Kornel Szlachanyi]]. *Weak Hopf Algebras I: Integral Theory and* $C^*$*-structure*. (1999). ([arXiv:math/9805116](https://arxiv.org/abs/math/9805116))

* {#Vec03} [[Peter Vecsernyés]]. *Larson–Sweedler theorem and the role of grouplike elements in weak Hopf algebras*. Journal of Algebra. Volume 270, Issue 2, 15 December 2003, Pages 471-520. ([doi](https://doi.org/10.1016/j.jalgebra.2003.02.001))

* {#IK08} [[Miodrag Iovanov]], [[Lars Kadison]]. *When weak Hopf algebras are Frobenius*. (2008). ([arXiv:0810.4777](https://arxiv.org/abs/0810.4777))

category: algebra

[[!redirects weak bialgebras]]

[[!redirects weak Hopf algebra]]
[[!redirects weak Hopf algebras]]


weak bimonad

## Idea

[[weak bialgebra|Weak bialgebra]] is like bialgebra, except that the requirement that the counit map is an algebra map and the unit is a coalgebra map are appropriately relaxed. Like bialgebras, this notion can also be properly generalized to a monoidal/bicategorical context leading to weak bimonads.

Distributive laws among monads are monads in appropriate bicategory/2-category of monads. Similarly, one can understand [[weak distributive law]]s.

## Literature

* [[Gabriella Böhm]], [[Stephen Lack]], [[Ross Street]], _Weak bimonads and weak Hopf monads_, J. Algebra 328 (2011), 1-30, [arXiv:1002.4493](http://arxiv.org/abs/1002.4493)
* [[Gabriella Böhm]], [[Stephen Lack]], [[Ross Street]], _On the 2-categories of weak distributive laws_, Comm. Alg. _39_:12 (2011) 4567--4583 [doi](https://doi.org/10.1080/00927872.2011.616436)
* Yuanyuan Chen, [[Gabriella Böhm]], _Weak bimonoids in duoidal categories_, Journal of Pure and Applied Algebra __218__:12 (2014) 2240--2273 [doi](https://doi.org/10.1016/j.jpaa.2014.04.001)


weak Cayley table

For $G$ a group, its weak Cayley table is the map which sends elements in $G\times G$ to their [[higher group character|biconjugacy class]].

#References#

* Kenneth W. Johnson, Sandro Mattarei and Surinder K. Sehgal, _Weak Cayley tables_, [Journal of the London Mathematical Society 2000 61(2):395-411](http://jlms.oxfordjournals.org/cgi/content/abstract/61/2/395) ([pdf](http://journals.cambridge.org/download.php?file=%2FJLM%2FJLM61_02%2FS0024610799008571a.pdf&code=e119cd0ef145cd79d298f403f4d38dc7))


weak colimit

+-- {: .rightHandSide}
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### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

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

For example, a **[[weakly initial object]]** in a category, $C$, is such that there is at least one arrow from it to any object in $C$.

## Related concepts

* [[weak limit]]

* [[weak multilimit]]

* [[WISC]]

* [[weakly initial set]]

* [[weakly initial object]]

## References

[[weak adjoint|Weak adjoint functors]] along with [[weak colimits]] were defined in:

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

[[!redirects weak colimits]]
[[!redirects weak multicolimit]]
[[!redirects weakly initial object]]


weak complicial set

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###Context###
#### Higher category theory
+--{: .hide}
[[!include higher category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Weak complicial sets are [[simplicial sets]] with [[stuff, structure, property|extra structure]] that are closely related to the [[∞-nerve]]s of weak [[∞-categories]].

The goal of characterizing such nerves, without an *a priori* definition of "weak $\omega$-category" to start from, is called  [[simplicial weak ∞-category]] theory.  It is expected that the (nerves of) weak $\omega$-categories will be weak complicial sets satisfying an extra "saturation" condition ensuring that "every [[equivalence]] is [[thin element|thin]]."  General weak complicial sets can be regarded as "presentations" of weak $\omega$-categories.

Weak complicial sets are a joint generalization of

* [[strict omega-category|strict ∞-categories]];

* [[Kan complexes]];

* [[quasi-category|quasi-categories]].

## Definition

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

Let

* $\Delta^k[n]$ be the [[stratified simplicial set]] whose underlying simplicial set is the $n$-[[simplex]] $\Delta[n]$, and whose marked cells are precisely those simplices $[r] \to [n]$ that contain $\{k-1, k, k+1\} \cap [n]$;

* $\Lambda^k[n]$ be the stratified simplicial set whose underlying simplicial set is the $k$-[[horn]] of $\Delta[n]$, with marked cells those that are marked in $\Delta^k[n]$;

* $\Delta^k[n]'$ be obtained from $\Delta^k[n]$ by making the $(k-1)$st $(n-1)$-face and the $(k+1)$st $(n-1)$ face thin;

* $\Delta^k[n]''$ be obtained from $\Delta^k[n]$ by making all $(n-1)$-faces thin.

An **elementary anodyne extension** in $Strat$, the category [[stratified simplicial sets]] is

* a **complicial horn extension** $\Lambda^k[n] \stackrel{\subset_r}{\hookrightarrow} \Delta^k[n]$

or

* a **complicial thinness extension** $\Delta^k[n]' \stackrel{\subset_e}{\hookrightarrow} \Delta^k[n]''$

for $n = 1,2, \cdots$ and $k \in [n]$.

=--

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

A [[stratified simplicial set]] is a **weak complicial set** if it has the [[right lifting property]] with respect to all

$\Lambda^k[n] \stackrel{\subset_r}{\hookrightarrow} \Delta^k[n]$  and
$\Delta^k[n]' \stackrel{\subset_e}{\hookrightarrow} \Delta^k[n]''$

A [[complicial set]] is a weak complicial set in which such liftings are unique.

=--

## Model structure

There is a [[model category]] structure that presents the [[(infinity,1)-category]] of weak complicial sets, hence that of weak $\omega$-categories. See

* [[model structure for weak complicial sets]].

## Examples

* For $C$ a [[strict ∞-category]] and $N(C)$ its [[oriental|∞-nerve]], the _Roberts stratification_ which regards each identity morphism as a thin cell makes $N(C)$ a strict [[complicial set]], hence a weak complicial set.  This example is not "saturated."

* There is also the [[stratified simplicial set|stratification]] of $N(C)$ which regards each $\omega$-equivalence morphism as a thin cell.  $N(C)$ with this stratification is a weak complicial set (example 17 of [Ver06](http://arxiv.org/abs/math/0604414)).  This should be the "saturation" of the previous example, and exhibits the inclusion of strict $\omega$-categories into weak ones.

* A simplicial set is a weak complicial set when equipped with its maximal [[stratified simplicial set|stratification]] (every simplex of dimension $\gt 0$ is thin) if and only if it is a [[Kan complex]].  This example is, of course, saturated, and is viewed as embedding $\omega$-groupoids into  $\omega$-categories.

* A simplicial set is a [[quasi-category]] if and only if it is a  weak complicial set when equipped with the stratification in which every simplex of dimension $\gt 1$ is thin, and only degenerate 1-simplices are thin.  This example is not saturated; in its saturation the thin 1-simplices are the internal equivalences in a quasi-category (equivalently, those that become isomorphisms in its [[homotopy category]]).  It presents the embedding of $(\infty,1)$-[[(infinity,1)-category|categories]] into weak $\omega$-categories.

  Note that 1-simplex equivalences in a quasi-category are automatically preserved by simplicial maps between quasi-categories; this is why $QCat$ can "correctly" be regarded as a full subcategory of $sSet$.  This is not true at higher levels; for instance not every simplicial map between nerves of strict $\omega$-categories necessarily preserves $\omega$-equivalence morphisms.

## References

The definition of weak complicial sets is definition 14, page 9 of

* [[Dominic Verity]], _Weak complicial sets Part I: Basic homotopy theory_ ([arXiv](http://arxiv.org/abs/math/0604414))

Further developments are in

* Dominic Verity, _Weak complicial sets Part II: Nerves of complicial Gray-categories_ ([arXiv](http://arxiv.org/abs/math/0604416))

A [[model category]] structure on [[stratified simplicial sets]] modelling [[(infinity,n)-categories|$(\infty,n)$-categories]] in the guise of [[n-complicial sets|$n$-complicial sets]]:

* [[Viktoriya Ozornova]], [[Martina Rovelli]], *Model structures for (∞,n)-categories on (pre)stratified simplicial sets and prestratified simplicial spaces*, Algebr. Geom. Topol. **20** (2020) 1543-1600 [[arxiv:1809.10621](https://arxiv.org/abs/1809.10621), [doi:10.2140/agt.2020.20.1543](https://doi.org/10.2140/agt.2020.20.1543)]

A [[Quillen adjunction]] relating [[n-complicial sets|$n$-complicial sets]] to [[n-fold complete Segal spaces|$n$-fold complete Segal spaces]]:

* [[Viktoriya Ozornova]], [[Martina Rovelli]], *A Quillen adjunction between globular and complicial approaches to $(\infty,n)$-categories*, Advances in Mathematics
**421** (2023) 108980 [[doi:10.1016/j.aim.2023.108980](https://doi.org/10.1016/j.aim.2023.108980), [arXiv:2206.02689](https://arxiv.org/abs/2206.02689)]

Review:

* {#Rovelli2023} [[Martina Rovelli]], *$n$-Complicial sets as a model for $(\infty,n)$-categories*, [talk at](Center+for+Quantum+and+Topological+Systems#RovelliMay2023) *[[CQTS]]* (2023) [[web](Center+for+Quantum+and+Topological+Systems#RovelliMay2023), video: [YT](https://www.youtube.com/watch?v=T9Bg1AdaKv8)]

[[!redirects weak complicial sets]]
[[!redirects weak compicial set]]

[[!redirects n-complicial set]]
[[!redirects n-complicial sets]]


weak counterexample

+-- {: .rightHandSide}
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### Context
#### Mathematics
+-- {: .hide}
[[!include mathematicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Recall that a _[[counterexample]]_ to a [[conjecture]] is a way to satisfy the hypotheses of the conjecture such that the conclusion is a contradiction.

In [[constructive mathematics]] and the [[foundations of mathematics]], a __weak counterexample__ to a theorem from (typically) [[classical mathematics]] is a way to satisfy the hypotheses of the theorem such that the conclusion is unacceptable in some way, without being an outright contradiction.

Weak counterexamples were first (at least under that name) used informally by [[Jan Brouwer]] in [[intuitionistic mathematics]].  They can be made into bona fide counterexamples by specifying a formal system and checking one of the following:

*  [[proof theory|Proof-theoretically]], if the unacceptable conclusion is known to be unprovable in the system, then we have a counterexample to the conjecture that every instance of the classical theorem can be proved in the system.

*  [[model theory|Model-theoretically]], if a model is known in which the unacceptable conclusion fails, then we have a counterexample to the conjecture that the classical theorem is true in every model of the system.

Either way, we now know that the classical theorem cannot be proved in the formal system.  One typically uses a weak counterexample when the classical theorem cannot be outright refuted.

## Related concepts

* [[theory]]

* [[axiom]]

* [[definition]]

* [[lemma]]

* [[proposition]]

* [[theorem]]

* [[proof]]

* [[example]]

* [[counterexample]]

* [[conjecture]]

* [[folklore]]

* [[analogy]]

* [[paradox]]

## See also

* [[taboo]]

[[!redirects weak counterexamples]]


weak distributive law

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###Context###
#### Higher algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

* table of contents
{:toc}

## Idea

The notion of a **weak distributive law** between two [[monads]] is a generalisation of that of a [[distributive law]], in which forming the [[composition|composite]] monad requires [[split idempotent|splitting]] an [[idempotent]] on the [[underlying]] [[composition|composite]] [[endofunctor]].

## Related concepts

* [[weak wreath product]]

## References

See also [[weak bimonad]].

Weak distributive laws among monads:

* [[Gabriella Böhm]], _The weak theory of monads_, Adv. in Math. __225__:1 (2010) 1--32 [doi](https://doi.org/10.1016/j.aim.2010.02.015)

For the weak _mixed_ distributive law (monad and comonad) version see

* [[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)]

2-categorical context in the sense of formal theory of monads is also exposed in

* [[Gabriella Böhm]], [[Stephen Lack]], [[Ross Street]], _On the 2-categories of weak distributive laws_, Comm. Alg. _39_:12 (2011) 4567--4583 [doi](https://doi.org/10.1080/00927872.2011.616436)

* [[Daniela Petrisan]], Ralph Sarkis. _Semialgebras and weak distributive laws_, Proceedings 37th Conference on
Mathematical Foundations of Programming Semantics, EPTCS 351 (2021) 218--241. ([doi:10.4204/EPTCS.351.14](https://doi.org/10.4204/EPTCS.351.14))

An application of weak distributive laws to explain [[weak wreath product]]s (comparable to the treatment of [[wreath]]s in bicategories) and also related bilinear factorization structures

* [[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)

[[!redirects weak distributive laws]]


weak enrichment

+-- {: .rightHandSide}
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###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An analog of [[enriched category]] in [[higher category theory]]. Enrichment in a monoidal category $(V,\otimes,I)$ requires composition morphisms
$$Hom (A,B) \otimes Hom(B,C) \to Hom(A,C)$$
and identity morphisms
$$I \to Hom(A,A)$$
in $V$ satisfying the axioms of associativity and unitality. To weakly enrich, we wish to have the axioms of associativity and unitality hold up to coherent 2-morphisms. To accomplish this we can replace $V$ with a [[monoidal bicategory]] which has sufficient structure to accommodate this. This describes one approach to weakly enriching in the 2-dimensional case. For higher values of n we need to replace $V$ with a weak n-category.

## Definition

A category weakly enriched in a [[monoidal bicategory]] $W$ is called a $W$-bicategory. The data of a $W$-bicategory $C$ includes the data of an ordinary enriched category in the [[decategorification]] of $W$. In addition, $C$ must contain associator and unitor 2-morphisms in $W$ which fill the appropriate diagrams of composition and unit 1-morphisms. These 2-cells are required to satsify higher dimensional coherence axioms.

Note that this should not be confused with a [[category enriched in a bicategory]] which allows for multiple bases of enrichment. A $Cat$-bicategory is an ordinary bicategory. Many examples come from weakly enriching in the [[monoidal 2-category]] $V$-$Cat$ of categories enriched in $V$.

In the context of [[(∞,1)-category theory]] see at
_[[enriched (∞,1)-category]]_.

In the context of [[(∞,1)-operad]] theory see ([Lurie, def. 4.2.1.12](#Lurie)):

Write $\mathcal{LM}^\otimes$ the [[operad for modules over an algebra]] regarded as an [[(∞,1)-operad]], regarded as the [[(∞,1)-category of operators]]. Similarly write $\mathcal{Ass}^\otimes$ for the [[(∞,1)-category of operators]] of the [[associative operad]].

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

For $\mathcal{V}^\otimes \to \mathcal{Ass}^\otimes$ exhibiting a  [[planar (∞,1)-operad]], a **weak enrichment** of an [[(∞,1)-category]] $\mathcal{C}$ over $\mathcal{C}^\otimes$ is a [[fibration of (∞,1)-operads]]

$$
  q \colon \mathcal{O}^\otimes \to \mathcal{LM}^{\otimes}
$$

which exhibits $\mathcal{C}$ as a module over $\mathcal{V}^\otimes$ in that it is   equipped with [[equivalence in an (∞,1)-category|equivalences]]

$$
  \mathcal{V}^\otimes \simeq \mathcal{O}^\otimes_{\mathfrak{a}}
$$

and

$$
  \mathcal{C} \simeq \mathcal{O}^\otimes_{\mathfrak{n}}
  \,.
$$

=--

([Lurie, def. 4.2.1.12](#Lurie))

> maybe better: weak _[[tensoring]]_?

## Related concepts

* [[homotopical enrichment]];

* [[enriched (∞,1)-category]]

## References

$W$-bicategories are briefly explained in Section 4 of

* [[Richard Garner]], [[Nick Gurski]], The low-dimensional structures formed by tricategories, [arXiv:0711.1761](https://arxiv.org/abs/0711.1761)

Section 4.2.1 of

* [[Jacob Lurie]], _[[Higher Algebra]]_
 {#Lurie}
* [[David Gepner]], [[Rune Haugseng]], _Enriched ∞-categories via non-symmetric ∞-operads_, [arXiv:1312.3178](https://arxiv.org/abs/1312.3178)

## Discussion

This was originally at [[bicategory]]:

+-- {: .query}

_Sebastian_: Is there a formal meaning of _weak enrichment_?

[[John Baez]]: Yes there is; indeed Clark Barwick is writing a huge book on this.

_Sebastian_: If not, is there at least a method how to get the definition of a weak $n$-category if I know the definition of a (strict) $n$-category?

[[John Baez]]: that sounds harder!  That's like pushing a rock uphill.  It's easier to go down from weak to strict.

_Sebastian_: Of course, I have recognised that there are actually different definitions of what a weak n-category should be... so to give my question a bit more precision: How do I get a definition of a weak $n$-category that is as close as possible to the definition of a strict $n$-category? The weak $n$-category should be what you call "globular", I think. (Are there different definitions of globular (weak or strict) n-categories?)

[[John Baez]]: globular strict n-categories have been understood since time immemorial, or at least around 1963, and there is just one reasonable definition.   Globular weak n-categories were defined in the 1990s by Michael Batanin, and his theory relates them quite nicely to the globular strict ones.   But there is also a _different_ definition of globular weak n-categories due to Penon.  It had a mistake in it which has now been fixed.

From the preface to [[johnbaez:Towards Higher Categories|Towards Higher Categories]]:

There is a quite different and more extensively developed operadic approach to globular [[nlab:weak omega-category|weak infinity-categories]] due to Batanin (Bat1, Str2), with a variant due to Leinster (Lein3).  Penon (Penon) gave a related, very compact definition of infinity-category; this definition was later corrected and improved by Batanin (Bat2) and Cheng and Makkai (ChMakkai).

You can get the references there.

I think this discussion should be moved over to some page on n-categories, since it's not really about bicategories.

=--

[[!redirects weak tensoring]]


weak equivalence

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###Context###
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _weak equivalence_ is a [[morphism]] in a [[category]] $C$ which is supposed to be a true [[equivalence]] in a [[higher category theory|higher categorical]] refinement of $C$.

The bare minimum of axioms to be satisfied by a weak equivalence are encoded in the concepts of [[category with weak equivalences]] and [[homotopical category]].  For such categories one can consider

* the corresponding [[homotopy category]], which is the universal solution to turning all weak equivalences into [[isomorphisms]];

* the corresponding $(\infty,1)$-[[(infinity,1)-category|category]], which is, roughly, the universal solution to turning all weak equivalences into higher categorical [[equivalences]].  There are various versions of this construction depending on what model for $(\infty,1)$-categories is chosen.

  * The [[Dwyer-Kan localization]] uses [[simplicially enriched   category|simplicially enriched categories]] to model $(\infty,1)$-categories.

  * If we use [[complete Segal spaces]] or [[quasicategories]] to model $(\infty,1)$-categories, then the construction is a version of _fibrant replacement_.

Often, categories having weak equivalences also have extra structure that makes them easier to work with.  A very powerful, and commonly occurring, level of such structure is called a [[model category|model structure]].  There are also various weaker levels of structure, such as a [[category of fibrant objects]].

## Examples

* A [[weak homotopy equivalence]] is a weak equivalence in the [[classical model structure on topological spaces]].

* A [[simplicial weak equivalence]] is a weak equivalence in the [[classical model structure on simplicial sets]].

* An [[equivariant weak homotopy equivalence]] is a weak equivalence in the [[fine model structure on topological G-spaces]].

*  A weak equivalence in the [[folk model structure on Cat]] is a functor that is [[full and faithful functor|fully faithful]] and [[essentially surjective functor|essentially surjective]].

* {#EquivalencesOfCategoriesInHomotopyTypeTheory} In [[homotopy type theory]], a [[functor]] $F: A \to B$ between [[internal categories in HoTT]] $A$ and $B$ is called a "weak equivalence" if it is [[fully faithful]] and [[essentially surjective]].

  For [[univalent categories]] there is no difference between weak equivalences and [[equivalences]].

## Related concepts

* [[equality]]

* [[isomorphism]]

* [[equivalence]]

* **weak equivalence**

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

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

* [[equivalence of (∞,1)-categories]]

[[!redirects weak equivalences]]


weak equivalence of internal categories

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Internal categories
+--{: .hide}
[[!include internal infinity-categories contents]]
=--
#### Equality and Equivalence
+--{: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The naive [[2-category]] $Cat(S)$ of [[internal categories]] [[internalization|in]] an ambient [[category]] $S$ does in general not have enough [[equivalences of categories]], due to the failure of the [[axiom of choice]] in $S$. Those [[internal functors]] which *should* but may not have [[inverses]] up to internal [[natural isomorphism]], namely those which are suitably [[fully faithful functor|fully faithful]] and [[essentially surjective functor|essentially surjective]], may be regarded as [[weak equivalences]] of internal categories ([Bunge & Paré 1979](#BungePare79)). The [[bicategory of fractions|2-category theoretic localisation]] of $Cat(S)$ at this class of [[1-morphisms]] then serves as a more natural [[2-category]] of [[internal categories]].

## Definition

Let $f:X\to Y$ be a functor between categories internal to some category $S$. $f$ is **fully faithful** if the following diagram is a pullback
$$
\begin{matrix}
  X_1& \stackrel{f_1}{\to} & Y_1 \\
  \downarrow&& \downarrow \\
  X_0\times X_0 &\underset{f_0\times f_0}{\to} & Y_0\times Y_0
\end{matrix}
$$

To discuss the analogue of essential surjectivity, we need a notion of 'surjectivity', as this does not generalise cleanly from $Set$. If we are working in a topos, a natural choice is to take epimorphisms, but weaker ambient categories are sometimes needed. A natural choice is to work in a unary [[site]], where the covers are taken as the 'surjective' maps.

Given a functor $f:X\to Y$ internal to a unary site $(S,J)$, $f$ is **essentially $J$-surjective** if the map $t\circ pr_2:X_0 \times_{f_0,Y_0,s}Y_1 \to Y_0$ is a $J$-cover.

We then define an internal functor to be a **$J$-equivalence** if it is fully faithful and essentially $J$-surjective.

## Related concepts

* [[canonical model structure on Cat]]

* [[anafunctor]]

* [[equivalence of categories]], [[weak equivalence]]

[[!redirects weak equivalencs of internal categories]]

## References

* {#BungePare79} [[Marta Bunge]], [[Robert Paré]], _Stacks and equivalence of indexed categories_, [[Cahiers|Cahiers de Topologie et Géométrie Différentielle Catégoriques]], 20 no. 4 (1979), p. 373-399 ([numdam:CTGDC_1979__20_4_373_0](http://www.numdam.org/item?id=CTGDC_1979__20_4_373_0))

* [[Tomas Everaert]], [[R.W.Kieboom]] and [[Tim Van der Linden]], _Model structures for homotopy of internal categories_, Theory and Applications of Categories 15 (2005), no. 3, 66-94. ([journal](http://www.tac.mta.ca/tac/volumes/15/3/15-03abs.html))

* [[David Roberts]], _Internal categories, anafunctors and localisation_, Theory and Applications of Categories, 26 (2012) No. 29, pp 788-829. ([journal](http://www.tac.mta.ca/tac/volumes/26/29/26-29abs.html))

[[!redirects weak equivalences of internal categories]]


weak excluded middle

# Weak excluded middle

* table of contents
{: toc}

## Definition

In [[logic]], the principle of **weak excluded middle** says that for any [[proposition]] $P$ we have $\neg P \vee \neg\neg P$.

This follows from the full principle of [[excluded middle]], which says that for any $Q$ we have $Q \vee \neg Q$ (take $Q = \neg P$).  Thus, in [[classical mathematics]] weak excluded middle is just true.  But in [[constructive mathematics]] (i.e. [[intuitionistic logic]]), it is a weaker assumption than full excluded middle.

## Equivalent forms

### De Morgan's Law

In intuitionistic logic, **de Morgan's law** often refers to the one of [[De Morgan duality|de Morgan's four laws]] that is *not* an intuitionistic tautology, namely $\neg (P \wedge Q) \to (\neg P \vee \neg Q)$ for any $P,Q$.

+-- {: .un_theorem}
###### Theorem
De Morgan's law is equivalent to weak excluded middle.
=--
+-- {: .proof}
###### Proof
If de Morgan's law holds, then since $\neg (P \wedge \neg P)$, we have $\neg P \vee \neg\neg P$, as desired.  Conversely, if weak excluded middle holds and we have $\neg (P\wedge Q)$, then from weak excluded middle we get $\neg P \vee \neg\neg P$ and $\neg Q \vee \neg\neg Q$ which give four cases.  In three of those cases $\neg P \vee \neg Q$ holds, while in the fourth we have $\neg\neg P$ and $\neg\neg Q$, which together imply $\neg\neg(P\wedge Q)$ (see the first lemma [here](https://ncatlab.org/nlab/show/Heyting+algebra#ToBooleanAlgebras) and its proof), contradicting the assumption of $\neg (P\wedge Q)$; so the fourth case is impossible.
=--

## See also

* [[De Morgan's law]]

* [[De Morgan Heyting algebra]]

* [[De Morgan Heyting category]]

[[!redirects principle of weak excluded middle]]
[[!redirects law of weak excluded middle]]
[[!redirects weak principle of excluded middle]]
[[!redirects weak law of excluded middle]]
[[!redirects weak PEM]]
[[!redirects weak LEM]]


weak factorization system

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _weak factorization system_ on a [[category]] is a [[pair]] $(\mathcal{L},\mathcal{R})$ of [[classes]] of [[morphisms]] ("[[projective morphisms]]" and "[[injective morphisms]]") such that 1) every morphism of the category factors as the composite of one in $\mathcal{L}$ followed by one in $\mathcal{R}$, and 2) $\mathcal{L}$ and $\mathcal{R}$ are closed under having the [[lifting property]] against each other.

If the liftings here are unique, then one speaks of an _orthogonal factorization system_. A classical example of an orthogonal factorization system is the [[(epi,mono)-factorization system]] on the category [[Set]] or in fact on any [[topos]].

Non-orthogonal weak factorization systems are the key ingredient in [[model categories]], which by definition carry a weak factorization system called ($\mathcal{L} = $ [[cofibrations]],$\mathcal{R} = $ [[acyclic fibrations]]) and another one called ($\mathcal{L} =$ [[acyclic cofibrations]], $\mathcal{R} =$ [[fibrations]]). Indeed most examples of non-orthogonal weak factorization systems arise in the context of model category theory.  A key tool for constructing these, or verifying their existence, is the _[[small object argument]]_.

There are other [[properties]] which one may find or impose on a weak factorization system, for instance [[functorial factorization]]. There is also [[extra structure]] which one may find or impose, such as for [[algebraic weak factorization systems]]. For more variants see at _[[factorization system]]_.

## Definition

### Weak factorization systems

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

A **weak factorization system** (WFS) on a [[category]] $\mathcal{C}$ is a [[pair]] $(\mathcal{L},\mathcal{R})$ of [[classes]] of [[morphisms]] of $\mathcal{C}$ such that

1. Every [[morphism]] $f \colon X\to Y$ of $\mathcal{C}$ may be factored as the [[composition]] of a morphism in $\mathcal{L}$ followed by one in $\mathcal{R}$

   $$
     f\;\colon\;  X \overset{\in \mathcal{L}}{\longrightarrow} Z \overset{\in \mathcal{R}}{\longrightarrow} Y
     \,.
   $$

1. The classes are closed under having the [[lifting property]] against each other:

   1. $\mathcal{L}$ is precisely the class of morphisms having the [[left lifting property]] against every morphism in $\mathcal{R}$;

   1. $\mathcal{R}$ is precisely the class of morphisms having the [[right lifting property]] against every morphism in $\mathcal{L}$.

=--

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

For $\mathcal{C}$ a [[category]], a **[[functorial factorization]]** of the morphisms in $\mathcal{C}$ is a [[functor]]

$$
  fact \;\colon\; \mathcal{C}^{\Delta[1]} \longrightarrow \mathcal{C}^{\Delta[2]}
$$

which is a [[section]] of the [[composition]] functor $d_1 \;\colon \;\mathcal{C}^{\Delta[2]}\to \mathcal{C}^{\Delta[1]}$.

=--

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

In def. \ref{FunctorialFactorization} we are using the following notation, see at _[[simplex category]]_ and at _[[nerve of a category]]_:

Write $\Delta[1] = \{0 \to 1\}$ and $\Delta[2] = \{0 \to 1 \to 2\}$ for the [[ordinal numbers]], regarded as [[posets]] and hence as [[categories]]. The [[arrow category]] $Arr(\mathcal{C})$ is equivalently the [[functor category]] $\mathcal{C}^{\Delta[1]} \coloneqq Funct(\Delta[1], \mathcal{C})$, while $\mathcal{C}^{\Delta[2]}\coloneqq Funct(\Delta[2], \mathcal{C})$ has as objects pairs of composable morphisms in $\mathcal{C}$. There are three injective functors $\delta_i \colon [1] \rightarrow [2]$, where $\delta_i$ omits the index $i$ in its image.
By precomposition, this induces [[functors]] $d_i  \colon \mathcal{C}^{\Delta[2]} \longrightarrow \mathcal{C}^{\Delta[1]}$. Here

* $d_1$ sends a pair of composable morphisms to their [[composition]];

* $d_2$ sends a pair of composable morphisms to the first morphism;

* $d_0$ sends a pair of composable morphisms to the second morphism.

=--

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

A weak factorization system, def. \ref{WeakFactorizationSystem}, is called a **functorial weak factorization system** if the factorization of morphisms may be chosen to be a [[functorial factorization]] $fact$, def. \ref{FunctorialFactorization}, i.e. such that $d_2 \circ fact$ lands in $\mathcal{L}$ and $d_0\circ fact$ in $\mathcal{R}$.

=--

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

Not all weak factorization systems are functorial, although most (including those produced by the [[small object argument]], with due care) are. But all [[orthogonal factorization systems]], def. \ref{OrthogonalFactorizationSystem}, automatically are functorial.  An example of a weak factorization that is not functorial can be found in [Isaksen 2001](https://arxiv.org/abs/math/0106152).

=--

### Orthogonal factorization systems

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

An **[[orthogonal factorization system]]** (OFS) is a weak factorization system $(\mathcal{L},\mathcal{R})$, def. \ref{WeakFactorizationSystem} such that the lifts of elements in $\mathcal{L}$ against elements in $\mathcal{R}$ are _unique_.

=--

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

While every OFS (def. \ref{OrthogonalFactorizationSystem}) is a WFS (def. \ref{WeakFactorizationSystem}), the primary examples of each are different:

A "basic example" of an OFS is [[(epi,mono)-factorization]] in [[Set]] (meaning $L$ is the collection of [[epimorphisms]] and $R$ that of [[monomorphisms]]), while a "basic example" of a WFS is (mono, epi) in $Set$.
The superficial similarity of these two examples masks the fact that they generalize in very different ways.

The OFS (epi, mono) generalizes to any [[topos]] or [[pretopos]], and in fact to any [[regular category]] if we replace "epi" with [[regular epimorphism|regular epi]].  Likewise it generalizes to any [[quasitopos]] if we instead replace "mono" with [[regular monomorphism|regular mono]].

On the other hand, saying that (mono,epi) is a WFS in [[Set]] is equivalent to the [[axiom of choice]].  A less loaded statement is that $(L,R)$ is a WFS, where $L$ is the class of inclusions $A\hookrightarrow A\sqcup B$ into a binary [[coproduct]] and $R$ is the class of [[split epimorphism|split epis]].  In this form the statement generalizes to any [[extensive category]]; see also [[weak factorization system on Set]].

=--

### Algebraic weak factorization systems

An [[algebraic weak factorization system]] enhances the *properties* of lifting and factorization to algebraic [[stuff, structure, property|structure]].

### Accessible weak factorization systems

An [[accessible weak factorization system]] is a wfs on a [[locally presentable category]] whose factorization is given by an [[accessible functor]].

## Properties

### Closure properties
 {#ClosureProperties}

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

Let $(\mathcal{L},\mathcal{R})$ be a weak factorization system, def. \ref{WeakFactorizationSystem} on some [[category]] $\mathcal{C}$. Then

1. Both classes contain the class of [[isomorphism]] of $\mathcal{C}$.

1. Both classes are closed under [[composition]] in $\mathcal{C}$.

   $\mathcal{L}$ is also closed under [[transfinite composition]].

1. Both classes are closed under forming [[retracts]] in the [[arrow category]] $\mathcal{C}^{\Delta[1]}$ (see remark \ref{RetractsOfMorphisms}).

1. $\mathcal{L}$ is closed under forming [[pushouts]] of morphisms in $\mathcal{C}$ ("[[cobase change]]").

   $\mathcal{R}$ is closed under forming [[pullback]] of morphisms in $\mathcal{C}$ ("[[base change]]").

1. $\mathcal{L}$ is closed under forming [[coproducts]] in $\mathcal{C}^{\Delta[1]}$.

   $\mathcal{R}$ is closed under forming [[products]] in $\mathcal{C}^{\Delta[1]}$.

=--

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

We go through each item in turn.

**containing isomorphisms**

Given a [[commuting square]]

$$
  \array{
    A &\overset{f}{\rightarrow}& X
    \\
    {}_{\mathllap{\in Iso}}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{p}}
    \\
    B &\underset{g}{\longrightarrow}& Y
  }
$$

with the left morphism an isomorphism, the a [[lift]] is given by using the [[inverse]] of this isomorphism ${}^{{f \circ i^{-1}}}\nearrow$. Hence in particular there is a lift when $p \in \mathcal{R}$ and so $i \in \mathcal{L}$. The other case is [[formal dual|formally dual]].

**closure under composition**

Given a [[commuting square]] of the form

$$
  \array{
    A &\longrightarrow& X
    \\
    \downarrow && \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in \mathcal{R}}}
    \\
    {}^{\mathllap{i}}_{\mathllap{\in \mathcal{L}}}\downarrow && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in \mathcal{R}}}
    \\
    B &\longrightarrow& Y
  }
$$

consider its [[pasting]] decomposition as

$$
  \array{
    A &\longrightarrow& X
    \\
    \downarrow
    &\searrow& \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in \mathcal{R}}}
    \\
    {}^{\mathllap{i}}_{\mathllap{\in \mathcal{L}}}\downarrow && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in \mathcal{R}}}
    \\
    B &\longrightarrow& Y
  }
  \,.
$$

Now the bottom commuting square has a lift, by assumption. This yields another [[pasting]] decomposition

$$
  \array{
    A &\longrightarrow& X
    \\
    {}^{\mathllap{i}}_{\mathllap{\in \mathcal{L}}}\downarrow
    && \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in \mathcal{R}}}
    \\
    \downarrow &\nearrow& \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in \mathcal{R}}}
    \\
    B &\longrightarrow& Y
  }
$$

and now the top commuting square has a lift by assumption. This is now equivalently a lift in the total diagram, showing that $p_1\circ p_1$ has the right lifting property against $\mathcal{L}$ and is hence in $\mathcal{R}$. The case of composing two morphisms in $\mathcal{L}$  is [[formal dual|formally dual]]. From this the closure of $\mathcal{L}$ under [[transfinite composition]] follows since the latter is given by [[colimits]] of sequential composition and successive lifts against the underlying sequence as above constitutes a [[cocone]], whence the extension of the lift to the colimit follows by its [[universal property]].

**closure under retracts**

Let $j$ be the [[retract]] of an $i \in \mathcal{L}$, i.e. let there be a [[commuting diagram]] of the form.

$$
  \array{
    id_A \colon & A &\longrightarrow& C &\longrightarrow& A
    \\
    & \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{\mathrlap{\in \mathcal{L}}} && \downarrow^{\mathrlap{j}}
    \\
    id_B \colon & B &\longrightarrow& D &\longrightarrow& B
  }
  \,.
$$

Then for

$$
  \array{
     A &\longrightarrow& X
     \\
     {}^{\mathllap{j}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}}
     \\
     B &\longrightarrow& Y
  }
$$

a [[commuting square]], it is equivalent to its [[pasting]] composite with that retract diagram

$$
  \array{
    A &\longrightarrow& C &\longrightarrow& A &\longrightarrow& X
    \\
    \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{\mathrlap{\in \mathcal{L}}} && \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}}
    \\
    B &\longrightarrow& D &\longrightarrow& B &\longrightarrow & Y
  }
  \,.
$$

Now the pasting composite of the two squares on the right has a lift, by assumption,

$$
  \array{
    A &\longrightarrow& C &\longrightarrow& A &\longrightarrow& X
    \\
    \downarrow^{\mathrlap{j}}
      &&
    \downarrow^{\mathrlap{i}}_{\mathrlap{\in \mathcal{L}}}
      &&
    \nearrow
      &&
    \downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}}
    \\
    B &\longrightarrow& D &\longrightarrow& B &\longrightarrow & Y
  }
  \,.
$$

By composition, this is also a lift in the total outer rectangle, hence in the original square. Hence $j$ has the left lifting property against all $p \in \mathcal{R}$ and hence is in $\mathcal{L}$. The other case is [[formal duality|formally dual]].

**closure under pushout and pullback**

Let $p \in \mathcal{R}$ and and let

$$
  \array{
    Z \times_f X &\longrightarrow& X
    \\
    {}^{\mathllap{{f^* p}}}\downarrow && \downarrow^{\mathrlap{p}}
    \\
    Z &\stackrel{f}{\longrightarrow} & Y
  }
$$

be a [[pullback]] diagram in $\mathcal{C}$. We need to show that $f^* p$ has the [[right lifting property]] with respect to all $i \in \mathcal{L}$. So let

$$
  \array{
     A &\longrightarrow& Z \times_f X
     \\
     {}^{\mathllap{i}}_{\mathllap{\in \mathcal{L}}}\downarrow && \downarrow^{\mathrlap{\mathrlap{f^* p}}}
     \\
     B &\stackrel{g}{\longrightarrow}& Z
  }
$$

be a [[commuting square]]. We need to construct a diagonal lift of that square. To that end, first consider the [[pasting]] composite with the pullback square from above to obtain the commuting diagram

$$
  \array{
     A &\longrightarrow& Z \times_f X &\longrightarrow& X
     \\
     {}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{f^* p}}
     && \downarrow^{\mathrlap{p}}
     \\
     B &\stackrel{g}{\longrightarrow}& Z &\stackrel{f}{\longrightarrow}& Y
  }
  \,.
$$

By the right lifting property of $p$, there is a diagonal lift of the total outer diagram

$$
  \array{
    A &\longrightarrow& X
    \\
    \downarrow^{\mathrlap{i}} &{}^{\hat {(f g)}}\nearrow&  \downarrow^{\mathrlap{p}}
    \\
    B &\stackrel{f g}{\longrightarrow}& Y
  }
  \,.
$$

By the [[universal property]] of the [[pullback]] this gives rise to the lift $\hat g$ in

$$
  \array{
     && Z \times_f X &\longrightarrow& X
     \\
     &{}^{\hat g} \nearrow& \downarrow^{\mathrlap{f^* p}}
     && \downarrow^{\mathrlap{p}}
     \\
     B &\stackrel{g}{\longrightarrow}& Z &\stackrel{f}{\longrightarrow}& Y
  }
  \,.
$$

In order for $\hat g$ to qualify as the intended lift of the total diagram, it remains to show that

$$
  \array{
     A &\longrightarrow& Z \times_f X
     \\
     \downarrow^{\mathrlap{i}} & {}^{\hat g}\nearrow
     \\
     B
  }
$$

commutes. To do so we notice that we obtain two [[cones]] with tip $A$:

* one is given by the morphisms
  1. $A \to Z \times_f X \to X$
  2. $A \stackrel{i}{\to} B \stackrel{g}{\to} Z$

  with universal morphism into the pullback being

  * $A \to Z \times_f X$

* the other by
  1. $A \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X \to X$
  2. $A \stackrel{i}{\to} B \stackrel{g}{\to} Z$.

  with universal morphism into the pullback being

  * $A \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X$.

The commutativity of the diagrams that we have established so far shows that the first and second morphisms here equal each other, respectively. By the fact that the universal morphism into a pullback diagram is _unique_ this implies the required identity of morphisms.

The other case is [[formal dual|formally dual]].

**closure under (co-)products**

Let $\{(A_s \overset{i_s}{\to} B_s) \in \mathcal{L}\}_{s \in S}$ be a set of elements of $\mathcal{L}$. Since [[colimits]] in the [[presheaf category]] $\mathcal{C}^{\Delta[1]}$ are computed componentwise, their [[coproduct]] in this [[arrow category]] is the universal morphism out of the coproduct of objects $\underset{s \in S}{\coprod} A_s$ induced via its [[universal property]] by the set of morphisms $i_s$:

$$
  \underset{s \in S}{\sqcup} A_s
  \overset{(i_s)_{s\in S}}{\longrightarrow}
  \underset{s \in S}{\sqcup} B_s
  \,.
$$

Now let

$$
  \array{
    \underset{s \in S}{\sqcup} A_s &\longrightarrow& X
    \\
    {}^{\mathllap{(i_s)_{s\in S}}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}}
    \\
    \underset{s \in S}{\sqcup} B_s
    &\longrightarrow&
    Y
  }
$$

be a [[commuting square]]. This is in particular a [[cocone]] under the [[coproduct]] of objects, hence by the [[universal property]] of the coproduct, this is equivalent to a set of commuting diagrams

$$
  \left\{
  \;\;\;\;
  \array{
    A_s &\longrightarrow& X
    \\
    {}^{\mathllap{i_s}}_{\mathllap{\in \mathcal{L}}}\downarrow
      &&
    \downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}}
    \\
    B_s
    &\longrightarrow&
    Y
  }
  \;\;\;\;
  \right\}_{s\in S}
  \,.
$$

By assumption, each of these has a lift $\ell_s$. The collection of these lifts

$$
  \left\{
  \;\;\;\;
  \array{
    A_s &\longrightarrow& X
    \\
    {}^{\mathllap{i_s}}_{\mathllap{\in \mathcal{L}}}\downarrow
      &{}^{\ell_s}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}}
    \\
    B_s
    &\longrightarrow&
    Y
  }
  \;\;\;\;
  \right\}_{s\in S}
$$

is now itself a compatible [[cocone]], and so once more by the [[universal property]] of the coproduct, this is equivalent to a lift $(\ell_s)_{s\in S}$ in the original square

$$
  \array{
    \underset{s \in S}{\sqcup} A_s &\longrightarrow& X
    \\
    {}^{\mathllap{(i_s)_{s\in S}}}\downarrow
     &{}^{(\ell_s)_{s\in S}}\nearrow&
    \downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}}
    \\
    \underset{s \in S}{\sqcup} B_s
    &\longrightarrow&
    Y
  }
  \,.
$$

This shows that the coproduct of the $i_s$ has the left lifting property against all $f\in \mathcal{R}$ and is hence in $\mathcal{L}$. The other case is [[formal dual|formally dual]].

=--

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

Beware, in the situation of prop. \ref{ClosuredPropertiesOfWeakFactorizationSystem},
that $\mathcal{L}$ is not in general closed under all [[colimits]] in $\mathcal{C}^{\Delta[1]}$, and similarly $\mathcal{R}$ is not in general closed under all [[limits]] in $\mathcal{C}^{\Delta[1]}$. Also $\mathcal{L}$ is not in general closed under forming [[coequalizers]] in $\mathcal{C}$, and $\mathcal{R}$ is not in general closed under forming [[equalizers]] in $\mathcal{C}$. However, if $(\mathcal{L},\mathcal{R})$ is an [[orthogonal factorization system]], def. \ref{OrthogonalFactorizationSystem}, then $\mathcal{L}$ is closed under all colimits and $\mathcal{R}$ is closed under all limits.

=--

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

Here by a _retract_ of a [[morphism]] $X \stackrel{f}{\longrightarrow} Y$ in some [[category]] $\mathcal{C}$ is meant a [[retract]] of $f$ as an object in the [[arrow category]] $\mathcal{C}^{\Delta[1]}$, hence a morphism $A \stackrel{g}{\longrightarrow} B$ such that in $\mathcal{C}^{\Delta[1]}$ there is a factorization of the identity on $g$ through $f$

$$
  id_g \;\colon\;
  g \longrightarrow f \longrightarrow g
  \,.
$$

This means equivalently that in $\mathcal{C}$ there is a [[commuting diagram]] of the form

$$
  \array{
    id_A \colon & A &\longrightarrow& X &\longrightarrow& A
    \\
    & \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}}
    \\
    id_B \colon & B &\longrightarrow& Y &\longrightarrow& B
  }
  \,.
$$

=--

### Retract argument

+-- {: .num_lemma #RetractArgument}
###### Lemma
**([[retract argument]])**

Consider a [[composition|composite]] [[morphism]]

$$
  f \;\colon\; X\stackrel{i}{\longrightarrow} A \stackrel{p}{\longrightarrow} Y
  \,.
$$

Then:

1. If $f$ has the [[left lifting property]] against $p$, then $f$ is a [[retract]] of $i$.

1. If $f$ has the [[right lifting property]] against $i$, then $f$ is a [[retract]] of $p$.

=--

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

We discuss the first statement, the second is [[formal duality|formally dual]].

Write the factorization of $f$ as a [[commuting square]] of the form

$$
  \array{
    X &\stackrel{i}{\longrightarrow}& A
    \\
    {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{p}}
    \\
    Y &= & Y
  }
  \,.
$$

By the assumed [[lifting property]] of $f$ against $p$ there exists a diagonal filler $g$ making a [[commuting diagram]] of the form

$$
  \array{
    X &\stackrel{i}{\longrightarrow}& A
    \\
    {}^{\mathllap{f}}\downarrow &{}^{\mathllap{g}}\nearrow& \downarrow^{\mathrlap{p}}
    \\
    Y &= & Y
  }
  \,.
$$

By rearranging this diagram a little, it is equivalent to

$$
  \array{
    & X &=& X
    \\
    & {}^{\mathllap{f}}\downarrow
     &&
    {}^{\mathllap{i}}\downarrow
    \\
    id_Y \colon & Y &\underset{g}{\longrightarrow}& A &\underset{p}{\longrightarrow}&
    Y
  }
  \,.
$$

Completing this to the right, this yields a diagram exhibiting the required retract according to remark \ref{RetractsOfMorphisms}:

$$
  \array{
    id_X \colon & X &=& X &=& X
    \\
    & {}^{\mathllap{f}}\downarrow
     &&
    {}^{\mathllap{i}}\downarrow
     &&
     {}^{\mathllap{f}}\downarrow
    \\
    id_Y \colon & Y &\underset{g}{\longrightarrow}& A &\underset{p}{\longrightarrow}&
    Y
  }
  \,.
$$

=--

## Examples

* [[model category|Model categories]] provide many examples of weak factorization systems.  In fact, most applications of WFS involve model categories or model-categorical ideas.

* The existence of certain [[weak factorization system on Set|WFS on Set]] is related to the [[axiom of choice]].

* See the [[joyalscatlab:Weak factorisation systems|Catlab]] for more examples.

## Related concepts

* [[retract argument]]

* [[injective and projective morphisms]]

* [[algebraic weak factorization system]]

[[!include algebraic model structures - table]]

## References

* [[joyalscatlab:HomePage|Joyal's CatLab]], _[[joyalscatlab:Weak factorisation systems]]_

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

* Jiri Rosicky, Walter Tholen , *Factorization, Fibration and Torsion*, [arxiv](https://arxiv.org/abs/0801.0063) (2007)

Introductory texts:

* [[Introduction to Homotopy Theory]]

* {#Riehl2008} [[Emily Riehl]], [_Factorization Systems_](https://math.jhu.edu/~eriehl/factorization.pdf), 2008

* [[Emily Riehl]], Chapter 11 in: *[[Categorical Homotopy Theory]]*, Cambridge University Press (2014) [[doi:10.1017/CBO9781107261457](https://doi.org/10.1017/CBO9781107261457), [pdf](http://www.math.jhu.edu/~eriehl/cathtpy.pdf)]

The following dissertation section is entirely written after learning of [Riehl (2008)](#Riehl2008) above, but has complementary examples and may dive deeper into some proofs:

* {#Nuyts2020} [[Andreas Nuyts]], _Contributions to Multimode and Presheaf Type Theory, section 2.4: Factorization Systems_, [PhD thesis](https://lirias.kuleuven.be/retrieve/581985), KU Leuven, Belgium, 2020

[[!redirects weak factorization systems]]


weak factorization system on Set

The "prototypical" example of a [[weak factorization system]] is ([[injection|mono]], [[surjection|epi]]) on [[Set]].  That this is a WFS is equivalent to the [[axiom of choice]], but various weaker things can be said in more generality.

* (complemented mono, split epi) is a WFS in any [[extensive category]], without any need for choice and even in [[intuitionistic logic]].  A factorization of $X\to Y$ is $X\to X+Y \to Y$.  In this WFS, every object is "cofibrant" (that is, $0\to X$ is in the left class) while an object is "fibrant" (i.e. $X\to 1$ is in the right class) if and only if it admits a [[global element]].

* Thus, [[classical logic]] (which says that any subset is complemented) is equivalent to saying that (mono, split epi) is a WFS on Set.

* Since the [[axiom of choice]] is equivalent to saying that any epic in Set is split, (mono, epi) is a WFS if and only if AC holds.

We might also ask for WFS on Set (or more general categories) whose left class or right class consists of monics or epics, respectively, with the other class being whatever it must be.

* On any [[topos]], there is a WFS (mono, relative injective), where a map $f\colon X\to Y$ in a category $C$ is *relatively injective* if it is an [[injective object]] in the [[slice category]] $C/Y$.  Factorizations are provided by the fact that any topos has enough injectives (i.e. any object can be embedded in an injective), and any slice category of a topos is again a topos.  See also [this answer](http://mathoverflow.net/questions/10246/model-category-structure-on-set-without-axiom-of-choice/10368#10368) by [[Denis-Charles Cisinski]] on MO.  In this model structure every object is "cofibrant," while the "fibrant" objects are the injectives; thus we might call this the *injective model structure*.

* On any extensive category, there is a WFS whose right class is the epimorphisms if and only if there exist enough [[projective objects]] (i.e. every object admits an epimorphism from a projective one).  For if such a WFS exists, then clearly the "cofibrant" objects are the projectives, so cofibrant replacement provides a projective cover of any object.  Conversely, if there are enough projectives, then we take the left class to be the complemented monics with projective complement; a factorization of $X\to Y$ is provided by $X\to X+Y' \to Y$, where $Y'\to Y$ is a projective cover.  In particular, there is a WFS on Set whose right class is the epimorphisms if and only if [[COSHEP]] holds.  As remarked above, the "cofibrant" objects here are the projective ones, while the "fibrant" objects are the "well-supported ones;" thus we might call this the *projective model structure*.

Looking at the "projective" and "injective" model structures above, by analogy with the [[model structures on chain complexes]] we might also be tempted to call (complemented mono, split epi) the *Hurewicz WFS*.

[[!redirects weak factorization systems on Set]]
[[!redirects WFS on Set]]
[[!redirects wfs on Set]]


weak function extensionality

\tableofcontents

## Definition

The principle of **weak function extensionality** states that the [[dependent product type]] of a family of contractible types is itself contractible.

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash p(x):\mathrm{isContr}(B(x))}{\Gamma \vdash \mathrm{weakfunext}(\lambda x.p(x)):\mathrm{isContr}\left(\prod_{x:A} B(x)\right)}$$

This is equivalent to the condition that the dependent product type of a family of [[h-propositions]] itself is an [[h-proposition]]:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash p(x):\mathrm{isProp}(B(x))}{\Gamma \vdash \mathrm{weakfunext}(\lambda x.p(x)):\mathrm{isProp}\left(\prod_{x:A} B(x)\right)}$$

Weak function extensionality is not equivalent to the [[principle of unique choice]]. The principle of unique choice states that the dependent product type of a family of contractible types is pointed

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma, x:A \vdash B \; \mathrm{type} \quad \Gamma, x:A \vdash p(x):\mathrm{isContr}(B(x))}{\Gamma\vdash \mathrm{uniquechoice}(\lambda x.p(x)):\prod_{x:A} B(x)}$$

## Properties

* Weak function extensionality is equivalent to [[function extensionality]].

## See also

* [[function extensionality]]
* [[principle of unique choice]]

## References

Weak function extensionality appears in section 13.1 of

* [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press ([pdf](https://raw.githubusercontent.com/martinescardo/HoTTEST-Summer-School/main/HoTT/hott-intro.pdf)) (478 pages)


weak gravity conjecture

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

#Contents#
* table of contents
{:toc}

## Idea

The _weak gravity conjecture_ is a [[conjecture]] ([Arkani-Hamed, Motl, Nicolis, Vafa 06](#ArkaniHamedMotlNicolisVafa06)) often summarized as saying that for [[gravity]] or [[quantum gravity]] to be consistent, the [[force]] of the field of gravity must be weaker than that of any of the [[gauge field]] forces, in suitable units.

## Loose statement

More precisely, what is argued in ([AH-M-N-V 06, section 2](#ArkaniHamedMotlNicolisVafa06)) is that there alwayst must be "[[elementary particles]]" for which the ratio $m/{\vert q\vert}$ of their [[mass]] over their [[gauge field]] [[charge]] (e.g. [[electric charge]], [[magnetic charge]]) is smaller than one: $m/{\vert q\vert} \lt 1$ (in natural [[units]]).

The argument for this in ([AH-M-N-V 06, section 2.2](#ArkaniHamedMotlNicolisVafa06)) is the following. The [[extremal black hole|extremal]] ([[BPS state]]) bound for [[charged black holes]] is that their mass $M$ _exceeds or equals_ their charge ${\vert Q\vert}$, hence  $M/{\vert Q\vert} \geq 1$. Now via [[black hole radiation]] any black hole is supposed to eventually decay by [[radiation|radiating]] away elementary particle quanta. If there were no elementary particle with $m/{\vert q\vert} \leq 1$, so the argument goes, then by radiating away particles with $m/{\vert q\vert} \gt 1$ the black hole would eventually lose all its mass, while still having some non-vanishing charge left. But such "black hole remnants" have been argued to lead to inconsistency ([Susskind 96](#Susskind96)).

## In string theory

The weak gravity conjecture was originally motivated from [[string theory]], which is argued to validate the "weak gravity" assumption.

In ([AH-M-N-V 06, section 4](#ArkaniHamedMotlNicolisVafa06)) it is argued that the closed [[superstring]] excitation states in [[heterotic string theory]] [[KK-compactification|KK-compactified]] on to 4d satisfy the condition $m/{\vert q\vert} \lt 1$:

<img src="https://ncatlab.org/nlab/files/WeakGravityConjectureHetStates.jpg" width="600">

## Relation to cosmic censorship

In 2015 [[Cumrun Vafa]] has argued that the weak gravity conjecture implies the [[cosmic censorship hypothesis]]: The latter turns out to be generally false, even in in 4 dimensions ([Crisford-Santos 17](CrisfordSantos17)), but there are arguments that configurations violating it also violate the assumption of "weak gravity".

> Subsequent calculations by Santos and Crisford supported Vafa's hunch; the simulations they're running now could verify that naked singularities become cloaked in black holes right at the point where gravity becomes the weakest force. ([Wolchover, June 20 2017](https://www.quantamagazine.org/where-gravity-is-weak-and-naked-singularities-are-verboten-20170620/))

For more on this see [Horowitz-Santos 19](#HorowitzSantos19)

## Related concepts

* [[cosmic censorship hypothesis]]

* [[landscape of string theory vacua]]

## References

The statement is due to

* {#ArkaniHamedMotlNicolisVafa06} [[Nima Arkani-Hamed]], Lubos Motl, Alberto Nicolis, [[Cumrun Vafa]], _The String Landscape, Black Holes and Gravity as the Weakest Force_, JHEP 0706:060,2007 ([arXiv:hep-th/0601001](https://arxiv.org/abs/hep-th/0601001))

with reference to arguments in

* {#Susskind96} [[Leonard Susskind]], _Trouble For Remnants_ ([arXiv:hep-th/9501106](https://arxiv.org/abs/hep-th/9501106))

Review is in

* [[Eran Palti]], _The Swampland: Introduction and Review_, lecture notes ([arXiv:1903.06239](https://arxiv.org/abs/1903.06239))

Further discussion includes
 {#FurtherDiscussion}

* Clifford Cheung, Grant N. Remmen, _Naturalness and the Weak Gravity Conjecture_, Phys. Rev. Lett. 113, 051601 (2014) ([arXiv:1402.2287](https://arxiv.org/abs/1402.2287))

* Yu Nakayama, Yasunori Nomura, _Weak Gravity Conjecture in AdS/CFT_, Phys. Rev. D 92, 126006 (2015) ([arXiv:1509.01647](https://arxiv.org/abs/1509.01647))

* [[Ben Heidenreich]], [[Matthew Reece]], [[Tom Rudelius]], _Sharpening the Weak Gravity Conjecture with Dimensional Reduction_, JHEP02(2016)140 ([arXiv:1509.06374](https://arxiv.org/abs/1509.06374))

* Clifford Cheung, Junyu Liu, Grant N. Remmen, _Proof of the Weak Gravity Conjecture from Black Hole Entropy_ ([arXiv:1801.08546](https://arxiv.org/abs/1801.08546))

* Miguel Montero, _A Holographic Derivation of the Weak Gravity Conjecture_,  JHEP 03 (2019) 157 ([arXiv:1812.03978](https://arxiv.org/abs/1812.03978))

* [[Ben Heidenreich]], [[Matthew Reece]], [[Tom Rudelius]], _Repulsive Forces and the Weak Gravity Conjecture_ ([arXiv:1906.02206](https://arxiv.org/abs/1906.02206))

Relation to [[cosmic censorship hypothesis]]

* {#CrisfordSantos17} Toby Crisford, [[Jorge Santos]], _Violating weak cosmic censorship in AdS$_4$_, Phys. Rev. Lett. 118, 181101 (2017) ([arXiv:1702.05490](https://arxiv.org/abs/1702.05490))

* {#HorowitzSantos19} [[Gary Horowitz]], [[Jorge Santos]], _Further evidence for the weak gravity - cosmic censorship connection_ ([arXiv:1901.11096](https://arxiv.org/abs/1901.11096))

Relation, via [[AdS/CFT]] and the [[conformal bootstrap]], to the solution of the [[sphere packing problem]] in [[dimensions]] 8 and 24:

* {#HMR19} [[Thomas Hartman]], [[Dalimil Mazáč]], [[Leonardo Rastelli]], _Sphere Packing and Quantum Gravity_ ([arXiv:1905.01319](https://arxiv.org/abs/1905.01319))

See also:

* Lars Aalsma, [[Alex Cole]], Gregory J. Loges, [[Gary Shiu]],
_A New Spin on the Weak Gravity Conjecture_ ([arXiv:2011.05337](https://arxiv.org/abs/2011.05337))

Proof of the weak gravity conjecture in [[bosonic string theory]] is claimed in:

* [[Ben Heidenreich]], Matteo Lotito, *Proving the Weak Gravity Conjecture in Perturbative String Theory, Part I: The Bosonic String* [[arXiv:2401.14449](https://arxiv.org/abs/2401.14449)]