factorization homology

> Currently this entry consists mainly of notes taken live in a talk by [[John Francis]] at [ESI Program on K-Theory and Quantum Fields](http://maths-old.anu.edu.au/esi/2012/) (2012), without as yet, any double-checking or polishing. So handle with care for the moment.

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###Context###
#### Higher algebra
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#### AQFT
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[[!include AQFT and operator algebra contents]]
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#Contents#
* table of contents
{:toc}

## Idea

_Factorization homology_  is a notion of [[homology theory]] for [[framed manifold|framed]] $n$-[[dimension|dimensional]] [[manifolds]] with coefficients in [[En-algebras]], due to ([Francis](#Francis)). It is similar in spirit to _[[factorization algebras]]_, _[[blob homology]]_ and _[[topological chiral homology]]_. In fact the definition of factorization homology turns out to be equivalent to that of topological chiral topology ([Francis b](#Francisb)).

## Definition

Write $Mfd_n^{\coprod}$ for the category of [[manifolds]] with [[embeddings]] as morphisms. This is naturally a [[topological category]], hence regard it as an [[(infinity,1)-category]].
Regard it furthermore as a [[symmetric monoidal (∞,1)-category]] with [[tensor product]] given by [[disjoint union]].

For $k$ a [[field]], write $Mod_k$ for the [[symmetric monoidal (∞,1)-category]] of $k$-[[chain complexes]].

Let $H(Mfd_n^{\coprod}, Mod_k)$ be the [[sub-(∞,1)-category]] of those  [[monoidal (∞,1)-functors]] $F : Mfd_n^{op} \to Mod_k$ which are "[[cosheaves]]" in that for any decomposition of a manifold $X$ into submanifolds $X'$ and $X''$ with overlap $O$, we have an [[equivalence in an (∞,1)-category|equivalence]]

$$
  F(X) \simeq F(X') \otimes_{F(O)}F(X'')
  \,.
$$

Next, let $Disk_n \subset Mfd_n$ be the full [[sub-(∞,1)-category]] on those [[manifolds]] which are finite [[disjoint unions]] of the [[Cartesian space]] $\mathbb{R}^n$.

Restriction along this inclusion gives an [[(∞,1)-functor]]

$$
  H(Mfd_n, Mod_k)
  \to
  Disk_n-Alg (Mod_k)
$$

This turns out to be an [[equivalence of (∞,1)-categories]].  The inverse is defined to be _factorization homology_

$$
  FactorizationHomology
   :
  Disk_n-Alg (Mod_k)
  \to
  H(Mfd_n, Mod_k)
  \,.
$$

which sends an $n$-disk algebra $A : Disk_n \to Mod_k$ to the functor that sends a manifold $X$ to the [[derived functor|derived]] [[coend]]

$$
  \int^X A = \mathbb{E}_X \otimes_{Disk_n} A
$$

of $A$ with

$$
  \mathbb{E}_X
    :
  Disk_n \stackrel{Emb(-,X)}{\to} Top \stackrel{C_\bullet(-)}{\to} Mod_k
  \,.
$$

This is equivalent to [[topological chiral homology]], to be thought of as a topological version of [[chiral algebras]]. A version with values in [[homotopy types]] instead of chain complexes was given by Salvatore and [[Graeme Segal]].

## Properties

### Relation to cobordism hypothsis

From a functor $F \in H(Mfd_n, Mod_k)$ we get an [[extended TQFT]] with values in $k$-linear $(\infty,n)$-categories

$Z_F : Bord_n \to Cat_n(k)$ which sends a $k$-manifold $X$ to $F(X \times \mathbb{R}^{n-k})$, regarded as a [[bimodule]] between the analogous boundary restriction, and hence as a [[n-morphism|k-morphism]] in $Cat_n(k)$.

From a  $Disk_n$-algebra $A$ we obtain the corresponding [[delooping]] $\mathbf{B}A \in (Cat_n(k)_{dualizable})^{O(n)}$ which is a $k$-linear [[(infinity,n)-category]] that is a [[fully dualizable object]]. The [[cobordism hypothesis]] identifies this with cobordism representations, and the claim is that this identification is compatible factorization homology.

## Examples

### Dimension 1

A $Disk_1$-algebra $A$ in $Mod_k$ is equivalently a [[differential graded algebra]].

The value of the corresponding $F_A \in H(Mfd_1, Mod_k)$ on the circle is the [[Hochschild homology]] of $A$

$$
  \int_{S^1} A \simeq \int_{\mathbb{R}^1} A \otimes_{\int_{S^0 \times \mathbb{R}}A} \int_{\mathbb{R}^1} A \simeq HH_\bullet(A)
  \,.
$$

### From $n$-fold loop spaces

Given a [[topological space]] $Z$ we get a $Disk_n$-algebra

$$
  Disk_n^\coprod
  \stackrel{Maps_{compact}(-,Z)}{\to}
  Top
  \stackrel{C_\ast(-)}{\to} Mod_k
$$

Where $Maps_{compact}(\mathbb{R}^n, Z) \simeq \Omega^n Z$ is the [[n-fold loop space]] of $Z$.

**Theorem** (Salvatore and [[Jacob Lurie|Lurie]])

If $Z$ is $(n-1)$-[[n-connected object of an (infinity,1)-category]]

$$
  \int_X C_\ast(\Omega^n Z) \simeq C_\ast Maps_{compact}(X,Z)
  \,.
$$

## Related entries

* [[topological chiral homology]], [[factorization algebra]], [[blob homology]]

* [[local net of observables]]

* [[representation stability]]

[[!include Isbell duality - table]]

## References
 {#References}

### General

The definition appears in section 3 of

* [[John Francis]], _The tangent complex and Hochschild cohomology of $\mathcal{E}_n$-rings_ ([arXiv:1104.0181](http://arxiv.org/abs/1104.0181))
 {#Francis}

A detailed account is in

* [[David Ayala]], [[John Francis]], _Factorization homology of topological manifolds_ ([arXiv:1206.5522](http://de.arxiv.org/abs/1206.5522))
 {#Francisb}

A survey that also covers [[factorization algebras]] is

* [[Grégory Ginot]], _Notes on factorization algebras, factorization homology and applications_, [arXiv:1307.5213](http://arxiv.org/abs/1307.5213).

See also

* [[Jacob Lurie]], _[[Higher Algebra]]_, section 5.3.

* [[Hiro Lee Tanaka]], _Manifold calculus is dual to factorization homology_, at [Talbot 2012:_ Calculus of functors](http://math.mit.edu/conferences/talbot/2012/_2012.php) ([pdf](http://math.mit.edu/conferences/talbot/2012/notes/14_Tanaka_FactorizationHomology%28hiro%29.pdf))

Generalization to [[orbifolds]]:

* [[Tim Weelinck]], _Equivariant factorization homology of global quotient orbifolds_ ([arXiv:1810.12021](https://arxiv.org/abs/1810.12021), [talk pdf](https://www.maths.ed.ac.uk/~tweelinck/efhqsp.pdf))

Some applications are

* [[Ben Knudsen]], _Betti numbers and stability for configuration spaces via factorization  homology_, [arXiv:1405.6696](http://arxiv.org/abs/1405.6696).

* [[Quoc Ho]], _Densities and stability via factorization homology_, [arXiv:1802.07948](https://arxiv.org/abs/1802.07948).

Application to higher [[Hochschild cohomology]] is discussed in

* [[Grégory Ginot]], Thomas Tradler, [[Mahmoud Zeinalian]], _Higher Hochschild cohomology, Brane topology and centralizers of $E_n$-algebra maps_, ([arXiv:1205.7056](http://arxiv.org/abs/1205.7056))

* [[Geoffroy Horel]], *Higher Hochschild homology of the Lubin-Tate ring spectrum*, Algebr. Geom. Topol. **15** (2015) 3215-3252 [[arXiv:1311.2805](https://arxiv.org/abs/1311.2805), [doi:10.2140/agt.2015.15.3215](https://doi.org/10.2140/agt.2015.15.3215)]

Application to stratified spaces with tangential structures is discussed in

* [[David Ayala]], [[John Francis]], [[Hiro Lee Tanaka]], _Factorization homology of stratified spaces_, ([arXiv:1409.0848](http://arxiv.org/abs/1409.0848))

A [[duality]] theorem for factorization homology, generalizing [[Poincare duality]] for [[manifolds]] and [[Koszul duality]] for [[E-n algebras]].

* [[David Ayala]], [[John Francis]], _Poincaré/Koszul duality_, [arXiv:1409.2478](http://arxiv.org/abs/1409.2478).

Discussion in the context of [[extended TQFT]] appears in

* [[Claudia Scheimbauer]], _Factorization homology as a fully extended topological field theory_ ([pdf](https://people.maths.ox.ac.uk/scheimbauer/ScheimbauerThesis.pdf))

* [Videos from  from the BIRS Workshop 15w5125](http://www.birs.ca/events/2015/5-day-workshops/15w5125/videos) on _Factorizable Structures in Topology and Algebraic Geometry_.

For [[surfaces]] equipped with [[flat connections]] for a [[finite group]]:

* {#KellerMüller23} [[Corina Keller]], [[Lukas Müller]], *Finite symmetries of quantum character stacks*, Theory and Applications of Categories **39** 3 (2023) 51-97  [[arXiv:2107.12348](https://arxiv.org/abs/2107.12348), [tac:39-03](http://www.tac.mta.ca/tac/volumes/39/3/39-03abs.html)]

Survey:

* [[Corina Keller]], *Twisted Character Varieties and Quantization via Factorization Homology*, talk at PIRSA (2022) [video: [doi:10.48660/21100028](https://doi.org/10.48660/21100028)]

* [[Lukas Müller]], *Deformation quantization and categorical factorization homology*, talk at [[CQTS]] (Mar 2023) [[web](Center+for+Quantum+and+Topological+Systems#MuellerMar2023), video:[YT](https://www.youtube.com/watch?v=kC488wISPX4)]

### Relation to cohomology of configuration spaces

Expressing the [[rational cohomology]] of [[ordered configuration spaces of points]] via [[factorization homology]] and [[Ran spaces]], and relation to [[representation stability]]:

* [[Quoc P. Ho]], _Higher representation stability for ordered configuration spaces and twisted commutative factorization algebras_ [[arXiv:2004.00252](https://arxiv.org/abs/2004.00252)]


factorization lemma

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

## Idea ##

The _factorization lemma_ ([Brown 73](#Brown73), prop. \ref{TheFactorizationLemma} below) is a fundamental tool in the theory of [[categories of fibrant objects]] ([[formal dual|dually]]: [[cofibration category|of cofibrant objects]]). It mimics one half of the _factorization axioms_ in a [[model category]] in that it asserts that every morphism may be factored as, in particular, a weak equivalence followed by a fibration.

A key corollary of the factorization lemma is the statement, widely known as _Ken Brown's lemma_ (prop. \ref{KenBrownLemma} below) which says that for a functor from a category of fibrant objects  to be a [[homotopical functor]], it is sufficient already that it sends acyclic fibrations to weak equivalences.

For more background, see also at _[[Introduction to Stable homotopy theory -- P|Introduction to classical homotopy theory]]_ [this lemma](Introduction+to+Stable+homotopy+theory+--+P#FactorizationLemma).

## Factorization lemma ##

Let $\mathcal{C}$ be a [[category of fibrant objects]].

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

Let

$$
X \overset{p_{X}}{\leftarrow} X \times Y \overset{p_{Y}}{\rightarrow} Y
$$

be a product in $\mathcal{C}$. Then $p_{X}$ and $p_{Y}$ are fibrations.

=--

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

By one of the axioms for a category of fibrant objects, $\mathcal{C}$ has a final object $1$. We have the following.

1) The following diagram in $\mathcal{C}$ is a cartesian square.

$$
   \array{
      X \times Y               &  \overset{p_{Y}}{\to} & Y \\
      p_{X} \downarrow &                                      & \downarrow  \\
      X                            & \to                                 & 1 \\
   }
$$

2) By one of the axioms for a category of fibrant objects, the arrows $Y \to 1$ and $X \to 1$ are fibrations.

By one of the axioms for a category of fibrant objects, it follows from 1) and 2) that $p_{X}$ and $p_{Y}$ are fibrations.

=--

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

Let $X$ be an object of $\mathcal{C}$. Let

$$
X \overset{p_{0}}{\leftarrow} X \times X \overset{p_{1}}{\rightarrow} X
$$

be a product in $\mathcal{C}$. By one of the axioms for a category of fibrant objects, there is a commutative diagram

$$
   \array{
      X &  \overset{c}{\to}                      & X^I \\
          &    \underset{\Delta}{\searrow} & \downarrow e \\
          &                                                   & X \times X
   }
$$

in $\mathcal{C}$ in which $c$ is a weak equivalence, and in which $e$ is a fibration.

The arrow $e_0 : X^I \to X$ given by $p_0 \circ e$ is a trivial fibration. The arrow $e_1 : X^I \to X$ given by $p_1 \circ e$ is a trivial fibration.

=--

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

We have the following.

1) The following diagram in $\mathcal{C}$ commutes.

$$
   \array{
      X &  \overset{c}{\to}                                         & X^I \\
          &    \underset{id_X}{\searrow}                    & \downarrow e_{0} \\
          &                                                                    & X
   }
$$

2 By one of the axioms for a category of fibrant objects, $id_X$ is a weak equivalence.

By one of the axioms for a category of fibrant objects, we deduce from 1), 2), and the fact that $c$ is a weak equivalence, that $e_{0}$ is a weak equivalence.

An entirely analogous argument demonstrates that $e_{1}$ is a weak equivalence.

=--

+-- {: .num_prop #TheFactorizationLemma}
###### Proposition
**(Factorization lemma)**

Let $f : X \to Y$ be an arrow of $\mathcal{C}$. There is a commutative diagram

$$
   \array{
      X &  \overset{j}{\to}             & Z \\
          & \underset{f}{\searrow} & \downarrow g \\
          &                                        & Y
   }
$$

in $\mathcal{C}$ such that the following hold.

1) The arrow $g : Z \to Y$ is a fibration.

2) There is a trivial fibration $r : Z \to X$ such that the following diagram in $\mathcal{C}$ commutes.

$$
   \array{
      X &  \overset{j}{\to}                & Z \\
          &  \underset{id}{\searrow} & \downarrow r \\
          &                                           & X
   }
$$

=--

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

We can construct the following diagram:

\begin{tikzcd}
X \arrow[rdd, "{(id,f)}", bend right] \arrow[rrrd, "f", bend left] \arrow[rd, "\exists ! j", dashed] &                                                                                &                                            &                                         \\
                                                                                & Z \arrow[d, "\pi_1^Z"] \arrow[r, "\pi_0^Z"] \arrow[dr, phantom, "\lrcorner", very near start]                                                            & Y^I \arrow[d, "e"]                         & Y \arrow[l, "c"'] \arrow[ld, "\Delta"'] \\
                                                                                & X \times Y \arrow[r, "f \times id"] \arrow[d, "\pi_0^{X \times Y}"] \arrow[ld, "\pi_1^{X \times Y}"']  \arrow[dr, phantom, "\lrcorner", very near start] & Y \times Y \arrow[d, "\pi_0^{Y \times Y}"] &                                         \\
Y                                                                                & X \arrow[r, "f"]                                                                                & Y                                          &
\end{tikzcd}

The triangle on the right exists with $c$ a weak equivalence and $e$ a fibration, as categories of fibrant objects have path objects. As $e$ is a fibration, we can pullback along $f \times id$ to get the upper-left pullback square. By the universal property of this pullback square, we can produce a unique $X \overset j \to Z$ which makes the top half of the diagram commute.

We let $g$ be the composite $Z \overset{\pi_1^Z} \to X \times Y \overset{\pi_1^{X \times Y}} \to Y$ so that $g \circ j = f$; these are fibrations due to being a pullback of the fibration $e$ and a product projection (Fact 1), so their composite $g$ is a fibration too.

We let $r$ be the composite $Z \overset{\pi_1^Z} \to X \times Y \overset{\pi_0^{X \times Y}} \to X$ so that $r \circ j = id_X$. As the upper square and lower square are both pullback squares, $r$ is the pullback of the morphism $Y^I \overset{e} \to Y \overset{\pi_0^{Y\times Y}} \to Y$ (which is an acyclic fibration by Fact 2), hence $r$ is also an acyclic fibraiton.

=--

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

That $r$ is a fibration can be demonstrated in exactly the same way as that $g$ is a fibration. It is to prove the stronger assertion that $r$ is a trivial fibration that the argument with which we concluded the proof is needed.

=--

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

By the commutativity of the diagram

$$
   \array{
      X &  \overset{j}{\to}                & Z \\
          &  \underset{id}{\searrow} & \downarrow r \\
          &                                           & X
   }
$$

and the fact that $r$ is a weak equivalence, we have, by one of the axioms for a category of fibrant objects, that $j$ is a weak equivalence.

=--

## Ken Brown's lemma ##

+-- {: .num_prop #KenBrownLemma}
###### Proposition
**(Ken Brown's lemma)**

Let $\mathcal{C}$ be a [[category of fibrant objects]]. Let $\mathcal{D}$ be a [[category with weak equivalences]]. Let $F : C \to D$ be a functor with the property that, for every arrow $f$ of $\mathcal{C}$ which is a trivial fibration, we have that $F(f)$ is a weak equivalence.

Let $w : X \to Y$ be an arrow of $\mathcal{C}$ which is a weak equivalence. Then $F(w)$ is a weak equivalence.
=--

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

By Proposition 3, there is a commutative diagram

$$
   \array{
      X &  \overset{j}{\to}             & Z \\
          & \underset{w}{\searrow} & \downarrow g \\
          &                                        & Y
   }
$$

in $\mathcal{C}$ such that the following hold.

1) The arrow $g : Z \to Y$ is a fibration.

2) There is a trivial fibration $r : Z \to X$ such that the following diagram in $\mathcal{C}$ commutes.

$$
   \array{
      X &  \overset{j}{\to}                & Z \\
          &  \underset{id}{\searrow} & \downarrow r \\
          &                                           & X
   }
$$

By the commutativity of the diagram

$$
   \array{
      X &  \overset{j}{\to}             & Z \\
          & \underset{w}{\searrow} & \downarrow g \\
          &                                        & Y
   }
$$

and the fact that both $j$ and $w$ are weak equivalences, we have that $g$ is a weak equivalence, by one of the axioms for a category of fibrant objects.

By assumption, we thus have that $F(g) : F(Z) \to F(Y)$ is a weak equivalence.

The following hold.

1) By the commutativity of the diagram

$$
   \array{
      X &  \overset{j}{\to}                & Z \\
          &  \underset{id}{\searrow} & \downarrow r \\
          &                                           & X
   }
$$

in $\mathcal{C}$, we have that the following diagram in $\mathcal{D}$ commutes.

$$
   \array{
      F(X) &  \overset{F(j)}{\to}           & F(Z) \\
             &  \underset{id}{\searrow} & \downarrow F(r) \\
             &                                           & F(X)
   }
$$

2) Since $r$ is a trivial fibration, we have by assumption that $F(r)$ is a trivial fibration. In particular, $F(r)$ is a weak equivalence.

3) By one of the axioms for a category with weak equivalences, we have that $id : F(X) \to F(X)$ is a weak equivalence.

By 1), 2), 3) and one of the axioms for a category with weak equivalences, we have that $F(j)$ is a weak equivalence.

The following diagram in $\mathcal{C}$ commutes.

$$
   \array{
      F(X) &  \overset{F(j)}{\to}             & F(Z) \\
             & \underset{F(w)}{\searrow} & \downarrow F(g) \\
             &                                             & F(Y)
   }
$$

Since $F(j)$ and $F(g)$ are weak equivalences, we conclude, by one of the axioms for a category with weak equivalences, that $F(w)$ is a weak equivalence.

=--

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

In other words, $F$ is a [[homotopical functor]].

=--

+-- {: .num_remark}
###### Remark
If $C$ is the full subcategory of fibrant objects in a [[model category]], then this corollary asserts that a [[Quillen adjunction|right Quillen functor]] $G$, which by its axioms is required only to preserve fibrations and trivial fibrations, preserves also weak equivalences between fibrant objects.
=--

+-- {: .num_remark}
###### Remark
By the dual nature of [[model categories]], we then get that a [[Quillen adjunction|left Quillen functor]] preserves weak equivalences between cofibrant objects.
=--

## Computing a homotopy pullback by means of an ordinary pullback

+-- {: .num_cor}
###### Corollary

Let $A \to C \leftarrow B$ be a diagram between fibrant objects in a [[model category]]. Then the ordinary [[pullback]] $A \times_C^h B$

$$
  \array{
     A \times_C^h B &\to& C^I
     \\
     \downarrow && \downarrow
     \\
     A \times B &\to& C \times C
  }
$$

presents the [[homotopy pullback]] of the original diagram.

=--

See the section _[Concrete constructions](http://ncatlab.org/nlab/show/homotopy+pullback#ConcreteConstructions)_ at _[[homotopy pullback]]_ for more details on this.

## Examples ##

* For $G$ an [[∞-group]] object in $C$ with [[delooping]] $\mathbf{B}G$, applying the factorization lemma to the point inclusion $* \to \mathbf{B}G$ yields a morphism $* \stackrel{\simeq}{\to} \mathbf{E}G \stackrel{p}{\to} \mathbf{B}G$. This exhibits a [[universal principal ∞-bundle]] for $G$.

## References ##

The [factorization lemma](#TheFactorizationLemma) is due to

* {#Brown73} [[Kenneth Brown]], p. 421 (4 of 41) in: _[[BrownAHT|Abstract Homotopy Theory and Generalized Sheaf Cohomology]]_, 1973, p. 4

The corollary commonly known as "[Ken Brown's lemma](#KenBrownLemma)" does not appear explicitly in [Brown 73](#Brown73); it does appear under this name in

* [[Mark Hovey]], Lemma 1.1.12 in: _Model Categories_, Mathematical Surveys and Monographs, Volume 63, AMS (1999) ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/hovey-model-cats.pdf), [Google books](http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover), [ISBN:978-0-8218-4361-1](https://bookstore.ams.org/surv-63-s))

Lecture notes:

* [here](Introduction+to+Homotopy+Theory#FactorizationLemma) and [here](Introduction+to+Homotopy+Theory#KenBrownLemma) in: *[[Introduction to Homotopy Theory]]*

A version in the setup of $\infty$-cosmoi is Lemma 2.1.6 in

* {#RiehlV17} [[Emily Riehl]], [[Dominic Verity]], _Fibrations and Yoneda’s lemma in an $\infty$-cosmos_, Journal of Pure and Applied Algebra, 221:3, 2017, pp. 499--564 ([arXiv:1506.05500](http://arxiv.org/abs/1506.05500))

[[!redirects ken brown's lemma]]

[[!redirects Ken Brown's lemma]]


factorization structure

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#### Category theory
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[[!include category theory - contents]]
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# Factorization structures for sinks and cosinks
* table of contents
{: toc}

## Idea

A *factorization structure* is a strengthening of an [[orthogonal factorization system]] which allows us to factor, not only single [[morphisms]], but arbitrary [[sinks]] (or cosinks).

## Definition

Let $\{e_i \colon X_i \to Y\}_{i\in I}$ be a [[sink]] and $m\colon A \to B$ a [[morphism]] in some [[category]].  We say that $\{e_i\}$ is *orthogonal* to $m$ if for any sink $\{f_i \colon X_i \to A\}_{i\in I}$ and morphism $g\colon Y\to B$ such that $m f_i = g e_i$ for all $i\in I$:
$$\array{X_i & \overset{f_i}{\to} & A\\
  ^{e_i}\downarrow && \downarrow^m\\
  Y & \underset{g}{\to} & B}$$
there exists a unique arrow $h\colon Y\to A$ such that $h e_i = f_i$ for all $i$ and $m h = g$.  Clearly, if ${|I|}=1$ this reduces to the usual notion of orthogonality for morphisms.

Let $(E,M)$ be an [[orthogonal factorization system|(orthogonal) factorization system]]; we say it extends to a **factorization structure for sinks** if every sink $\{f_i \colon X_i \to Y\}_{i\in I}$ can be factored as $f_i = m e_i$, where $m\in M$ and the sink $\{e_i \colon X_i \to Z\}_{i\in I}$ is orthogonal to $M$.  Note there is no restriction on the sinks involved to be [[small category|small]].

Some authors write $\mathbf{E}$ for the collection of sinks orthogonal to $M$, and say that $(\mathbf{E},M)$ is a factorization structure for sinks, or that the category $C$ is an "$(\mathbf{E},M)$-category".  However, since $\mathbf{E}$ is uniquely determined by $(E,M)$, and $E$ is precisely the 1-ary sinks in $\mathbf{E}$, there is little harm in saying that $(E,M)$ is a factorization structure for sinks.

The dual notion is a **factorization structure for cosinks** ("sources").

### On size and foundations

Observe that if $C$ is [[large category|large]], then the collection $\mathbf{E}$ contains [[proper classes]] as elements.  Therefore, in some [[foundations]] such as [[ZF]], it is not definable as a single thing.  In [[NBG]] one may treat it as a "hyperclass" defined by a first-order formula, in the same way that one treats classes in ZF.  If we use a [[Grothendieck universe]] to define smallness, of course, there is no problem.

## Constructions

Many well-known factorization systems, and ways to construct factorization systems, extend to factorization structures for sinks and/or cosinks.

* [This theorem](/nlab/show/M-complete+category#ConstructingOFS) implies that if a category is [[M-complete category|M-complete]] for a class $M$ of morphisms which consists of monomorphisms and is closed under composition and pullback, then $M$ is the right class in a factorization structure for sinks.

* If $p\colon A\to B$ is a [[Grothendieck fibration]], then factorization structures for sinks can be lifted from $B$ to $A$.  Dually, if $p$ is an opfibration, we can lift factorization structures for cosinks.  For the "mismatched" types of lifting, we require more: $p$ must be a [[topological concrete category|topological functor]].

## Examples

* In [[Set]], the factorization system (epi, mono) extends to both a factorization structure for sinks and one for cosinks.  The epi-sinks are those that are jointly epimorphic, and the mono-sinks are those that are jointly monomorphic.

* By lifting (epi,mono) to [[Top]] (or any other topological category over $Set$), we obtain two factorization structures for sinks: (jointly surjective, subspace inclusions) and (final topologies, injections).  Dually, we have two factorization structures for cosinks: (surjections, initial topologies) and (quotient maps, jointly injective).

## Properties

### $M$ consists of monics

Recall that a factorization system $(E,M)$ is called *proper* if $E\subseteq Epi$ and $M\subseteq Mono$.  In the case of a factorization structure for sinks, the second of these is automatic.

+-- {: .un_theorem}
###### Theorem
If $(E,M)$ is a factorization structure for sinks, then $M$ consists of monomorphisms.
=--
+-- {: .proof}
###### Proof
Let $m\colon A\to B$ be in $M$, and suppose that $m r = m s$ for some $r,s\colon X\to A$, but $r\neq s$.  Consider the sink $\{m r: X \to B \}_{f\in Mor(C)}$ consisting of one copy of $m r$ ($= m s$) for each arrow of the ambient category $C$, and factor this sink as an $E$-sink $\{e_f\colon X \to Y\}_{f\in Mor(C)}$ followed by an $M$-morphism $n\colon Y\to B$.

Now there are at least $2^{|Mor(C)|}$ different sinks $\{g_f\colon X \to A\}_{f\in Mor(C)}$ such that $m g_f = n e_f$, since we may take each $g_f$ to be either $r$ or $s$.  Therefore, by orthogonality, there are at least $2^{|Mor(C)|}$ different morphisms $Y\to A$, a contradiction to [[Cantor's theorem]], since there can be at most $Mor(C)$.
=--

This proof is of course quite reminiscent of Freyd's theorem that any [[complete small category]] is a [[preorder]].  In fact, Freyd's theorem is a consequence of this one (or at least of its proof).  For given a complete small category, there is a factorization structure acting at least on *small* sinks, where $M$ is the class of all morphisms and $E$ the class of families of injections into coproducts.  (Any complete small category is also cocomplete, by the [[adjoint functor theorem]].)  Therefore, all morphisms in a complete small category are monic, including the unique maps to the [[terminal object]]; hence the category is a preorder.

Since Freyd's theorem can fail in [[constructive mathematics]], we should expect the use of [[excluded middle]] to also be essential in proving the above property of factorization structures.

[[!redirects factorization structures]]
[[!redirects factorization structure for sinks]]
[[!redirects factorization structure for sources]]
[[!redirects factorization structure for cosinks]]
[[!redirects factorization structures for sinks]]
[[!redirects factorization structures for sources]]
[[!redirects factorization structures for cosinks]]


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

Roughly speaking, a **factorization system** on a [[category]] consists of two classes of [[morphism|maps]], $L$ and $R$, such that every map factors into an $L$-map followed by an $R$-map, and the $L$-maps and $R$-maps satisfy some [[lifting property|lifting]] or [[diagonal fill-in]] property.  The various ways of filling in the details give rise to many kinds of factorization systems:

* In most of the literature, "factorization system" unqualified refers specifically to [[orthogonal factorization system]]s (OFS).

* [[weak factorization system]]s (WFS) are "more general" than orthogonal ones, in the sense that every OFS is also a WFS.  But since the most important examples of WFS (those that occur in [[model category|model categories]]) are not OFS, intuitively they are more or less independent concepts.

* [[algebraic weak factorization system]]s (AWFS) are a strengthened "algebraic" version of WFS in which the factorizations are functorial and the two classes of maps are algebraic.

* In an [[enriched category]] it is natural to consider [[enriched factorization system|enriched (orthogonal) factorization systems]].  The enriched version of WFS falls under [[enriched model category|enriched model categories]].

* In a [[bicategory]] (i.e. a possibly non-strict [[2-category]]) one wants instead the notion of [[factorization system in a 2-category]], which is like a Cat-enriched OFS "up to isomorphism."  The situation in an [[n-category]] is analogous; see for instance [[orthogonal factorization system in an (∞,1)-category]] and [[orthogonal factorization system in a derivator]].

* In a [[model category]], a [[homotopy factorization system]] leverages the ambient weak factorization systems to give a presentation of an orthogonal factorization system in the underlying $(\infty,1)$-category.

* In a [[strict 2-category]] there is also the notion of an [[enhanced factorization system|enhanced (orthogonal) factorization system]], of which the main example is [[bo-ff factorization system|(bo,ff)]] in [[Cat]].

* A [[strict factorization system]] is one in which the factorizations are specified uniquely on the nose, rather than merely up to isomorphism.

* A [[factorization system over a subcategory]] is a common generalization of orthogonal and strict factorization systems, which requires uniqueness of factorizations only up to specified zigzags.

Particular examples of factorization systems of various sorts can be found on the individual pages referred to above.

## Higher-ary factorization systems

The above notion of "binary" factorization system can be generalized to factor a morphism into more than two factors.

* The orthogonal 3-ary version is a [[ternary factorization system]].

* This has a generalization to a [[k-ary factorization system]].

* The corresponding 3-ary version for *weak* factorization systems is closely related to the notion of [[model category]] (one of the main applications of weak factorization systems).

## Cylinder factorization systems

See [[cylinder factorisation system]].

## Related concepts

* [[anodyne morphism]], [[small object argument]]

* [[Joyal-Tierney calculus]]

## References

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

Introductory texts:

* [[Introduction to Homotopy Theory]]
* {#Riehl2008} [[Emily Riehl]], [_Factorization Systems_](https://math.jhu.edu/~eriehl/factorization.pdf), 2008
* The following dissertation section is entirely written after learning from Riehl's note 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

The factorization systems were probably first introduced in

* [[S. MacLane]], _Duality for groups_,  Bull. Amer. Math. Soc. __56__,  (1950). 485--516, [MR0049192](http://www.ams.org/mathscinet-getitem?mr=0049192), [doi](http://dx.doi.org/10.1090/S0002-9904-1950-09427-0)
* J. R. Isbell, _Some remarks concerning categories and subspaces_, Canad. J. Math. __9__ (1957), 563--577; [MR0094405](http://www.ams.org/mathscinet-getitem?mr=0094405)

* [[Ross Street]], _Notes on factorization systems_, ([pdf](http://www.maths.mq.edu.au/~street/factoriz.pdf))
[[!redirects factorization systems]]

On the relationship between [[cones]] and factorisation systems:

* [[Rudolf-E. Hoffmann]], _Factorization of cones_, Mathematische Nachrichten **87** 1 (1979) 221-238. [[doi:10.1002/mana.19790870120](https://doi.org/10.1002/mana.19790870120)]

* [[Rudolf-E. Hoffmann]], _Factorization of cones II, with applications to weak Hausdorff spaces_, in: *Categorical Aspects of Topology and Analysis: Proceedings of an International Conference Held at Carleton University, Ottawa, August 11–15, 1981*, Springer (2006) [[doi:10.1007/BFb0092878](https://doi.org/10.1007/BFb0092878)]

A list of elementary examples of factorization systems
(associated with the notions of: compact, discrete, connected, and totally disconnected spaces, dense image, induced topology, and separation axioms; finite groups being nilpotent, solvable, torsion-free, p-groups, and prime-to-p groups; injective and projective modules; injective, surjective, and split homomorphisms)

* M. Gavrilovich, [_The unreasonable power of the lifting property in elementary mathematics_](http://mishap.sdf.org/by:gavrilovich/expressive-power-of-lifting-property.pdf)

* M. Gavrilovich, [_A naive diagram-chasing approach to formalisation of tame topology_](http://mishap.sdf.org/by:gavrilovich/mintsGE.pdf)

[[!redirects factorization systems]]

[[!redirects factorisation system]]
[[!redirects factorisation systems]]


factorization system on a 2-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

In a (weak) [[2-category]], the appropriate notion of an [[orthogonal factorization system]] is suitably weakened up to isomorphism.  Specifically, a factorization system on a 2-category $K$ consists of two classes $(E,M)$ of 1-morphisms in $K$ such that:

1. Every 1-morphism $f:x\to y$ in $K$ is *isomorphic* to a composite $m\circ e$ where $e\in E$ and $m\in M$, and

1. For any $e:a\to b$ in $E$ and $m:x\to y$ in $M$, the following square
   $$\array{ K(b,x) & \to & K(b,y) \\
   \downarrow & \cong & \downarrow \\
   K(a,x) & \to & K(a,y)}$$
   (which commutes up to isomorphism) is a [[2-pullback]] in $Cat$.

This second property is a "2-categorical orthogonality."  In particular, it implies that any square
$$\array{a & \to & x \\
^e\downarrow & \cong & \downarrow^m \\
b & \to & y}$$
which commutes up to specified isomorphism, where $e\in E$ and $m\in M$, has a diagonal filler $b\to x$ making both triangles commute up to isomorphisms that are coherent with the given one.  It also implies an additional factorization property for 2-cells.

## Examples

* The following are all factorization systems on the 2-category $Cat$.  Many of them have analogues in more general 2-categories.

  * $E=$ [[essentially surjective functors]], $M=$ [[fully faithful functors]].  This is the "ur-example," and it generalizes to [[enriched category theory]], [[internal category]] theory, etc.  See [[(eso, fully faithful) factorization system]].

  * $E=$ functors $e\colon a\to b$ such that every object of $b$ is a [[retract]] of an object in the image of $a$, and $M=$ fully faithful functors whose image is closed under retracts.

  * $E=$ [[essentially surjective functor|essentially surjective]] and [[full functors]], $M=$ [[faithful functors]].  See [[(eso+full, faithful) factorization system]].

  * $E=$ (possibly [[transfinite compositions|transfinite]]) composites of [[localizations]], $M=$ [[conservative functors]].

* The 2-category [[Topos]] admits several interesting factorization systems.

  * The [[(geometric surjection, embedding) factorization system]].

  * $E=$ [[hyperconnected geometric morphisms]], $M=$ [[localic geometric morphisms]].

* The [[monadic decomposition]] is a factorization system on a suitable 2-category.

## Cat-enriched factorization systems

If instead $K$ is a [[strict 2-category]] and we require that

1. Every 1-morphism in $K$ is *equal* to a composite of a morphism in $E$ and a morphism in $M$, and

1. The above square (which commutes strictly when $K$ is a strict 2-category) is a strict 2-pullback (i.e. a $Cat$-enriched pullback).

then we obtain the notion of a $Cat$-enriched, or strict 2-categorical, factorization system.

It is important to note that in general, the strict and weak notions of 2-categorical factorization system are incomparable; neither is a special case of the other.  For example, on $Cat$ there is a weak 2-categorical factorization system where $E=$ [[essentially surjective functors]] and $M=$ [[fully faithful functors]], and a strict 2-categorical factorization system where $E=$ [[bijective on objects functors]]
and $M=$ [[fully faithful functors]].

## Related concepts

* [[factorization system]]

  * [[weak factorization system]]

  * [[orthogonal factorization system]]

* **factorization system on a 2-category**

* [[enhanced factorization system]]

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

  * [[orthogonal factorization system in an (∞,1)-category]]

* Factorization systems in a 2-category play an important role in the construction of a [[proarrow equipment]] out of [[codiscrete cofibrations]].

* Combining the (eso,ff) and (eso+full, faithful) factorization systems into a [[ternary factorization system]] has connections with the theory of [[stuff, structure, property]].

## References

For instance

* [[Mathieu Dupont]], [[Enrico Vitale]], _Proper factorization systems in 2-categories_ ([pdf](http://breckes.org/dokumenty/propfactsys.pdf))

[[!redirects factorization system in a 2-category]]
[[!redirects factorization systems in a 2-category]]
[[!redirects factorization systems on a 2-category]]
[[!redirects factorization systems in 2-categories]]
[[!redirects factorization systems on 2-categories]]
[[!redirects 2-categorical factorization system]]
[[!redirects 2-categorical factorization systems]]

[[!redirects factorisation system on a 2-category]]
[[!redirects factorisation system in a 2-category]]
[[!redirects factorisation systems in a 2-category]]
[[!redirects factorisation systems on a 2-category]]
[[!redirects factorisation systems in 2-categories]]
[[!redirects factorisation systems on 2-categories]]
[[!redirects 2-categorical factorisation system]]
[[!redirects 2-categorical factorisation systems]]


factorization system over a subcategory

+-- {: .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 *factorization system over a subcategory* is a common generalization of an [[orthogonal factorization system]] and a [[strict factorization system]], in which factorizations are only unique up to zigzags belonging to some specified subcategory.

## Definition

Let $C$ be a [[category]], and let $J$, $E$, and $M$ be [[wide subcategories]] of $C$ with $J\subseteq E$ and $J\subseteq M$.  Given a [[morphism]] $f\colon x\to y$ in $C$, let $Fact^{E,M}_J(f)$ denote the non-full [[subcategory]] of the over-under-category ([[double comma category]]) $(x/C/y)$:

* whose objects are pairs $x\to z \to y$ such that $x\to z$ is in $E$, $z\to y$ is in $M$, and the composite $x\to y$ is $f$;
* whose morphisms from $x\to z \to y$ to $x\to z' \to y$ are morphisms $z\to z'$ which are in $J$ and make the two evident triangles commute.

We say that $(E,M)$ is a **factorization system over $J$** if $Fact^{E,M}_J(f)$ is [[connected category|connected]] (and thus, in particular, [[inhabited set|inhabited]]).

## Examples

* If $J$ consists of only the identities in $C$, then a factorization system over $J$ is a [[strict factorization system]].

* If $J$ is the [[core]] of $C$, then a factorization system over $J$ is an [[orthogonal factorization system]]

* If $J$ is the canonical inclusion of (a skeleton of) $FinSet^{op}$ into a [[Lawvere theory]] $C$, then a factorization system over $J$ is a decomposition of $C$ into a [[distributive law]] of two other Lawvere theories.

## Relation to distributive laws

Suppose given a category $J$.  Then to give a category $C$ equipped with an identity-on-objects functor $J\to C$ and a factorization system over $J$ is the same as to give a [[distributive law]] between two [[monads]] on $J$ in the bicategory [[Prof]].  The two monads are the categories $E$ and $M$, and their composite is $C$.

## References

* [[Eugenia Cheng]], *Distributive laws for Lawvere theories*, [arXiv:1112.3076](http://arxiv.org/abs/1112.3076)

[[!redirects factorization systems over a subcategory]]


factorization systems - contents

**[[factorization system]]**

* [[orthogonal factorization system|orthogonal]], [[strict factorization system|strict]]

* [[weak factorization system|weak]], [[algebraic weak factorization system|algebraic weak]]

* [[enriched factorization system|enriched]]

* [[reflective factorization system|reflective]], [[stable factorization system|stable]]

[[factorization system over a subcategory]]

[[k-ary factorization system]], [[ternary factorization system]]

* [[factorization system in a 2-category]]

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

* [[orthogonal factorization system in an (∞,1)-category|orthogonal]]

**Examples**

* [[(epi, mono) factorization system|(epi, mono)]],

* [[(n-epi, n-mono) factorization system|(n-epi, n-mono)]]/[[n-image]]

  [[Postnikov system]]

* [[(eso, fully faithful) factorization system|(eso, fully faithful)]],
[[(eso+full, faithful) factorization system|(eso+full, faithful)]], [[(bo, ff) factorization system|(bo, ff)]]

* [[(hyperconnected, localic) factorization system|(hyperconnected, localic)]]


facts about chord diagrams and weight systems -- contents

**Facts about [[chord diagrams]] and their [[weight systems]]**:

* [[chord diagrams modulo 4T are Jacobi diagrams modulo STU]]

* [[weight systems are the associated graded objects of Vassiliev invariants]]

* [[weight systems are cohomology of loop space of configuration space]]

* [[weight systems are cohomology of knot graph complex]]

* [[stringy weight systems span classical Lie algebra weight systems]]

* [[all horizontal weight systems are partitioned Lie algebra weight systems]]


Facundo Mémoli

* [personal page](https://facundo-memoli.org/)

* [institute page](https://math.osu.edu/people/memoli.2)

## Selected writings

On an [[algorithm]] for [[topological data analysis]] by a variant of [[Reeb graphs]]:

* [[Gurjeet Singh]], [[Facundo Mémoli]], [[Gunnar Carlsson]], *Topological Methods for the Analysis of High Dimensional Data Sets and 3D Object Recognition*, in: M. Botsch, R. Pajarola (eds.), *Eurographics Symposium on Point-Based Graphics* (2007) $[$[doi:10.2312/SPBG/SPBG07/091-100](http://dx.doi.org/10.2312/SPBG/SPBG07/091-100), [pdf](http://diglib.eg.org/bitstream/handle/10.2312/SPBG.SPBG07.091-100/091-100.pdf?sequence=1&isAllowed=y)$]$

On [[persistent homotopy]]:

* [[Facundo Mémoli]], [[Ling Zhou]], *Persistent Homotopy Groups of Metric Spaces* $[$[arXiv:1912.12399](https://arxiv.org/abs/1912.12399), [talk](Center+for+Quantum+and+Topological+Systems#ZhouAtHomotopicalPerspectivesOnTDAJune2022)$]$

category: people

[[!redirects Facundo Memoli]]


Fadell's configuration space

> This entry is superceded by _[[configuration space of points]]_. See there for more.

#Contents#
* table of contents
{:toc}

## Idea

Given a [[manifold]] $M$, the __Fadell's configuration space__ (in [[topology]] called simply _[[configuration space]]_) is the manifold of $N$-tuples of pairwise distinct points in $M$.

It is important in the study of topological [[fibrations]], in the study of [[arrangements of hyperplanes]], of [[Knizhnik-Zamolodchikov connection]] and in study of geometry of [[renormalization]].

See at _[[configuration space of points]]_ for more.

## Examples

### Classifying space of the symmetric group

Let $X= \mathbb{R}^\infty$. Then

* the _unordered_ configuration space of $n$ points in $\mathbb{R}^\infty$ is a model for the [[classifying space]] $B \Sigma(n)$ of the [[symmetric group]] $\Sigma(n)$;

* the _ordered_ configuration space of $n$ points, equipped with the canonical $\Sigma(n)$-[[action]], is a model for the $\Sigma(n)$-[[universal principal bundle]].

## References

* [[Edward Fadell]], [[Lee Neuwirth]], _Configuration spaces_
Math. Scand. __10__ (1962) 111-118, [MR141126](http://www.ams.org/mathscinet-getitem?mr=141126), [pdf](http://www.mscand.dk/article.php?id=1623)

* [[Craig Westerland]], _Configuration spaces in geometry and topology_, 2011, [pdf](http://www.ms.unimelb.edu.au/~craigw/config.pdf)

* [[Graeme Segal]], _Configuration-spaces and iterated loop-spaces_, Invent. Math. __21__ (1973), 213–221. MR 0331377 ([pdf](http://dodo.pdmi.ras.ru/~topology/books/segal.pdf))

* [[Edward Fadell]], [[Sufian Husseini]], _Geometry and topology of configuration spaces_, Springer Monographs in Mathematics (2001), [MR2002k:55038](http://www.ams.org/mathscinet-getitem?mr=2002k:55038), xvi+313 pp.

* [[Fred Cohen]], [[Sam Gitler|S. Gitler]], _On loop spaces of configuration spaces_, Trans. Amer. Math. Soc. __354__ (2002), no. 5, 1705–1748, [MR2002m:55020](http://www.ams.org/mathscinet-getitem?mr=1881013)


faithful > history

< [[faithful]]
[[!redirects faithful -- history]]


faithful functor

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

#Contents#
* table of contents
{:toc}

## Definition

### In 1-category theory
 {#In1CategoryTheory}

A [[functor]] $F \colon \mathcal{C} \to \mathcal{D}$ from a [[category]] $\mathcal{C}$ to a category $\mathcal{D}$ is called _faithful_, if for each [[pair]] of [[objects]] $x, y \in \mathcal{C}$, its [[function]] $F_{x,y}$ on [[hom-sets]] is [[injection|injective]]:

$$
  \array{
    \mathcal{C}(x,y)
    &\xhookrightarrow{\;\; F_{x,y} \;\;}&
    \mathcal{D}(F(x), F(y))
    \\
    (x \overset{\phi}{\to} y)
    &\mapsto&
    \big(
      F(x)\overset{F(\phi)}{\to} F(y)
    \big)
    \,.
  }
$$

More abstractly, we may say that a functor is faithful if it is $2$-[[k-surjective functor|surjective]] – or loosely speaking, 'surjective on equations between given morphisms'.

### In higher category theory
 {#InHigherCategoryTheory}

See also _[[faithful morphism]]_ for a generalization to an arbitrary [[2-category]].

And see [[0-truncated morphism of an (∞,1)-category|0-truncated morphism]] for generalization to [[(∞,1)-categories]] (see there).

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

This generalization is about extending to morphisms in general (∞,1)-categories the fact that in $\infty{}Grpd$, $0$-truncated morphisms give a reasonable notion of faithful functor.

In particular, the notion of a "faithful morphism in the (∞,1)-category of (∞,1)-categories" does _not_ give the right notion of a "faithful functor between (∞,1)-categories".

=--

## Properties

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

A faithful functor [[reflected limit|reflects]] [[epimorphisms]] and [[monomorphisms]].

=--

(The simple proof is spelled out for instance at _[[epimorphism]]_.)

## Related concepts

* [[embedding of categories]]

[[!include properties of functors -- contents]]

[[!redirects faithful]]
[[!redirects faithful functors]]


faithful morphism

+-- {: .rightHandSide}
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=--
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=--

#Contents#
* table of contents
{:toc}

## Definition

A [[morphism]] $f\colon A\to B$ in a [[2-category]] $K$ is said to be **(representably) faithful** if for all objects $X$, the induced [[functor]]
$$K(X,A) \to K(X,B)$$
is [[faithful functor|faithful]].  In [[Cat]], this is equivalent to $f$ being faithful in the usual sense.

## Remarks

* Faithful morphisms may also be called **2-monic** and said to make their [[source]] into a **2-subobject** of their [[target]].  See [[subcategory]] for discussion.

* Faithful morphisms often form the right class of a [[factorization system in a 2-category|factorization system]].  In [[Cat]], the left class consists of [[essentially surjective functor|essentially surjective]] and [[full functor|full]] functors.

* Of course, any [[fully faithful morphism]] is also faithful, and in particular any [[inverter]] or [[equifier]] is faithful.  Moreover, any [[inserter]] is also faithful, though not generally fully-faithful.

## Related pages

* the analogous concept in [[(∞,1)-categories]] is that of _[[0-truncated morphism of an (∞,1)-category|0-truncated morphisms]]_ (see there).

* [[subcategory]]
* [[fully faithful morphism]]
* [[conservative morphism]]
* [[pseudomonic morphism]]
* [[discrete morphism]]

[[!redirects faithful morphisms]]
[[!redirects 2-monic morphism]]
[[!redirects 2-monic morphisms]]
[[!redirects 2-subobject]]
[[!redirects 2-subobjects]]


faithful representation

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

# Faithful representations
* table of contents
{: toc}

## Idea

An _[[action]]_ or _[[representation]]_ of an algebraic object such as a [[group]] or an [[algebra]] is _faithful_ when the action of two elements being equal implies that these two elements are already equal. When the representation of a group $G$ on some $V$ is thought of as a [[functor]] $\mathbf{B}G \to \mathbf{B}Aut(V)$, then it is faithful precisely if this functor is a [[faithful functor]].

A [[representation]] of an algebraic object, such as a [[group]] or an [[algebra]], is a way of studying it by making it act on some object.  The hope being that the behaviour of the object makes it easier to see the structure of the acting object.  Another way of thinking of such a representation is that it is a [[homomorphism]] of the acting object into some other object that is (presumably) better understood.  Making a group or algebra act on a [[vector space]] is the same as giving a homomorphism into the corresponding [[general linear group]] or [[endomorphism algebra]], making a group act on a set maps it into the corresponding [[permutation group]].

When studying an object via its representation then we really only "see" that part of the object that the representation sees.  Thus there is the potential for forgetting information when passing to a representation.  This can be a good thing, but it might not be.  It is important to classify the possible scenarios and the label **faithful representation** is used for when no information is lost.  This is in line with other uses of the word [[faithful]].

Thus when we have a faithful representation, we can distinguish two elements of the acting object by their actions: if they always do the same thing then they were the same element.

## Definition

### Traditional
 {#Traditional}

Let $(\mathcal{C}, \otimes)$ be a  [[closed monoidal category]], $A$ a [[monoid object]] in $\mathcal{C}$ and

$$
  \rho \;\colon\; A \otimes V \longrightarrow V
$$

an [[action]]/[[representation]] on some $V \in \mathcal{C}$. Equivalently this is given by its $((V \otimes -) \dashv [V,-] )$-[[adjunct]] $\tilde \rho$ ("[[currying]]"), which is a [[homomorphism]]

$$
 \tilde \rho \colon A \longrightarrow [V,V]
$$

from $A$ to the object of [[endomorphism]] of $V$ (the [[internal hom]]).

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

The representation $\rho$ is called _faithful_ if its adjunct $\tilde \rho$ is a [[monomorphism]].

=--

### For $\infty$-Actions
 {#ForInfinityActions}

Let $\mathbf{H}$ be an [[(∞,1)-topos]], $G \in Grp(\mathbf{H})$ be an [[∞-group]] object and $\rho$ an [[∞-action]] on some $V \in \mathbf{H}$. By the discussion at _[[∞-action]]_ this is equivalently a [[homotopy fiber sequence]] of the form

$$
  \array{
     V &\longrightarrow& V/G
     \\
     && \downarrow
     \\
     && \mathbf{B}G
  }
  \,.
$$

Suppose that $V$ is $\kappa$-[[compact object]] for some [[cardinal]] $\kappa$. By the existence of the $\kappa$-small [[object classifier]] $Obj_\kappa \in \mathbf{H}$ the above homotopy fiber sequence is itself the [[homotopy pullback]] of the universal fibration $\widehat {Obj}_\kappa \to Obj_\kappa$ along some morphism $\mathbf{B}G \longrightarrow \mathbf{B}\mathbf{Aut}(V) \hookrightarrow Obj_\kappa$ to the [[delooping]] of the [[automorphism ∞-group]] of $V$

$$
  \array{
     V &\longrightarrow& V/G &\longrightarrow& V/\mathbf{Aut}(V) &\longrightarrow& \widehat{Obj}_\kappa
     \\
     && \downarrow && \downarrow && \downarrow
     \\
     && \mathbf{B}G &\longrightarrow& \mathbf{B}\mathbf{Aut}(V) &\hookrightarrow& Obj_\kappa
  }
  \,.
$$

There is the corresponding $\infty$-group homomorphism  (see at [[looping and delooping]])

$$
  \tilde \rho \colon G \longrightarrow \mathbf{Aut}(V)
  \,.
$$

If this is an [[n-monomorphism]] for some $n$ one might call the action "$n$-faithful". If $G$ and $V$ are [[0-truncated]] then any [[∞-action]] of $G$ on $V$ is an ordinary [[action]] in the underlying 1-[[topos]] and this is faithful in the traditional sense if it is $n$-faithful for $n = 1$ in this higher sense.

## Properties
 {#Properties}

### Existence

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

If $G$ is a [[compact Lie group]], then there exists a finite-dimensional faithful representation of $G$.

=--

E.g. [Kowalski 14, proof of theorem 6.1.2](#Kowalski14)

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

For $G$ any [[algebraic group]], then the [[regular representation]] is faithful. Moreover, it has finite-dimensional faithful sub-representations.

=--

(e.g. [Milne 12, IX, theorem 9.1](#Milne12))

### Subquotients

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

If $V$ is a [[finite number|finite]] [[dimension|dimensional]] [[representation]] of an affine [[algebraic group]] $G$ over a [[field]] $k$, then every finite dimensional representation of $G$ is [[isomorphism|isomorphic]] to a [[subquotient]] of $\otimes^n(V \oplus V^\ast)$, where $V^\ast$ is the [[dual representation]].

=--

(e.g. [Milne 12, VIII, theorem 11.7](#Milne12))

## Examples
 {#Examples}

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

Let $G$ be a [[discrete group]] and $V$ a [[set]] (hence $\mathcal{C} = $ [[Set]] with its [[Cartesian product]]). The [[adjunct]] of an action map $\rho \colon G \times V \to V$ is a function

$$
  \tilde \rho \colon G \longrightarrow End(V)
$$

from $G$ to the set of [[endofunctions]] of $V$.

The action is faithful if this function $\tilde \rho$ is [[injective]].

Observe that $G$ being a group and $\rho$ being an action means that $\tilde \rho$ factors through the inclusion of the [[automorphism group]] $Aut(V) \hookrightarrow End(V)$. (If $V$ is a [[finite set]] then this is a [[symmetric group]]).

So, equivalently the action is faithful if

$$
  \tilde \rho \colon G \longrightarrow Aut(V)
$$

is injective, hence is a [[monomorphism]] of [[groups]].

More specifically, if $V$ is equipped with the structure of a [[vector space]] and the action is by [[linear functions]], hence is a [[linear representation]], then this means that $\tilde \rho$ factors through the [[general linear group]] $GL(V)$ of $V$ through a group [[homomorphism]]

$$
  \tilde \rho \colon G \longrightarrow GL(V)
  \,.
$$

Again, $\tilde \rho$ is faithful if this is an injection, hence a group monomorphism.

The same applies when $G$ is equipped with extra geometric structure, such as being a [[topological group]] or [[Lie group]].

=--

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

A [[Lie groupoid]] is called  _[[effective Lie groupoid|effective]]_ when the [[action]] of all its [[automorphism groups]] of objects on their [[germs]] is faithful. See at _[[effective Lie groupoid]]_ for more on this.

=--

## References

* {#Milne12} [[James Milne]], _Basic theory of affine group schemes_, 2012 ([pdf](http://jmilne.org/math/CourseNotes/AGS.pdf))

* {#Kowalski} Kowalski, _An introduction to the representation theory of groups_, 2014

[[!redirects faithful representation]]
[[!redirects faithful representations]]

[[!redirects faithful action]]
[[!redirects faithful actions]]

[[!redirects faithful ∞-action]]
[[!redirects faithful ∞-actions]]
[[!redirects faithful infinity-action]]
[[!redirects faithful infinity-actions]]


Falk Hassler

* [webpage](http://www.fhassler.de)

## Selected writings

On [[Poisson-Lie T-duality]]:

* {#Hassler17} [[Falk Hassler]], _Poisson-Lie T-Duality in Double Field Theory_ ([arXiv:1707.08624](https://arxiv.org/abs/1707.08624))

* {#Hassler20} [[Falk Hassler]], Thomas Rochais, _$\alpha'$-corrected Poisson-Lie T-duality_ ([arXiv:2007.07897](https://arxiv.org/abs/2007.07897))

## Related $n$Lab entries

* [[Poisson-Lie T-duality]]

category: people


Falk Lindner

* [webpage](http://wwwiexp.desy.de/sfb676/researchers/lindner/)

category: people


falling factorial

#Contents#
* table of contents
{:toc}

## Definition

For $k$ a [[natural number]] and $x$ an element of a (unital) [[ring]], the **falling factorial** or **falling power** $x^{\underline{k}}$ (also sometimes written $(x)_k$) is defined by

$$x^{\underline{k}} = x(x-1)\ldots (x-k+1).$$

If $X$ and $Y$ are two [[finite sets]] of [[cardinalities]] $|X| = k$ and $|Y| = n$, then the falling factorial $n^{\underline{k}}$ counts the number of [[injections]] from $X$ to $Y$. Dividing by the action of the permutation group of $X$ with $k!$ elements and counting the orbits, the so called **binomial coefficient** $\frac{n^{\underline{k}}}{k!} = \binom{n}{k}$ counts the number of subsets of $Y$ of cardinality $k$.

## Properties

In the calculus of finite differences, where an analogy is set up between the ordinary "continuous" derivative $(D f)(x) = \lim_{h \to 0} \frac{f(x+h) - f(x)}{h}$ and the "discrete" derivative (= difference operator)

$$(\Delta f)(x) \coloneqq f(x+1) - f(x),$$

it is the falling factorial $x^{\underline{k}}$ that plays a role analogous to the ordinary power $x^k$. For example, we have $\Delta x^{\underline{k}} = k x^{\underline{k-1}}$. Compare the formula $\Delta \frac{x^{\underline{k}}}{k!} = \frac{x^{\underline{k-1}}}{(k-1)!}$ which interpolates the Pascal triangle recurrence

$$\binom{n+1}{k} - \binom{n}{k} = \binom{n-1}{k}.$$

The discrete analogue of the identity $x^k x^l = x^{k+l}$ is the identity

$$x^{\underline{k}} (x-k)^{\underline{l}} = x^{\underline{k+l}}.$$

This may be used to motivate the definition of $x^{\underline{k}}$ for all [[integers]] $k$. For $k$ a natural number, define

$$x^{\underline{-k}} \coloneqq \frac1{(x+k)^{\underline{k}}} = \frac1{(x+k)(x+k-1) \ldots (x+1)}.$$

Then the difference formula $\Delta x^{\underline{k}} = k x^{\underline{k-1}}$ and the multiplicative identity $x^{\underline{k}} (x-k)^{\underline{l}} = x^{\underline{k+l}}$ continue to hold for all integers $k, l$. These facts have numerous applications throughout discrete mathematics.

## Related entries

* [[factorial]]

* [[binomial theorem]]

## References

* [Falling Factorial](http://mathworld.wolfram.com/FallingFactorial.html) on Wolfram MathWorld.

* Ronald Graham, Donald Knuth, and Oren Patashnik, _Concrete Mathematics_, 2$^{nd}$ edition, Addison-Wesley (1994).

[[!redirects falling power]]
[[!redirects falling powers]]
[[!redirects falling factorials]]


false vacuum

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Vacua
+-- {: .hide}
[[!include vacua -- contents]]
=--
#### Algebraic Quantum Field Theory
+--{: .hide}
[[!include AQFT and operator algebra contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

A [[classical field theory]] with [[action functional]] having the standard [[kinetic action]] and non-derivative [[interactions]] has stable, time-independent solutions to its [[equations of motion]] where all [[field (physics)|fields]] are [[constant function|constant]] on a value that is a local [[minimum]] of the interaction term [[potential energy]] (if any).

In the corresponding [[quantum field theory]], however, a [[state]] that is initially concentrated this way on a local minimum which is not also a global minimum is not time-independent, but will exhibit [[quantum tunneling]] behaviour by which the fields "tunnel" through the local potential barrier into the global minimum. Therefore one speaks of a _false [[vacuum]]_ in this case.

## Examples

* [Higgs field -- Mass and Vacuum (in-)stability](Higgs+field#MassAndVacuumInstability)

## Related concepts

[[!include states and observables -- content]]

## References

The classical articles are

* [[Sidney Coleman]], _Fate of the false vacuum: Semiclassical theory_ , Phys. Rev. D 15, 2929–2936 (1977) ([doi:10.1103/PhysRevD.15.2929](https://doi.org/10.1103/PhysRevD.15.2929))

* [[Sidney Coleman]], [[Frank De Luccia]], _Gravitational effects on and of vacuum decay_ , Phys. Rev. D 21, 3305–3315 (1980) ([doi:10.1103/PhysRevD.21.3305](https://doi.org/10.1103/PhysRevD.21.3305))

More recent development (in view of [[Higgs field|Higgs metastability]]):

* Anders Andreassen, David Farhi, William Frost, Matthew D. Schwartz, _Precision decay rate calculations in quantum field theory_, Phys. Rev. D 95, 085011 (2017) ([arXiv:1604.06090](https://arxiv.org/abs/1604.06090))

[[!redirects false vacuum]]
[[!redirects false vacua]]
[[!redirects false vacuums]]

[[!redirects unstable vacuum]]
[[!redirects unstable vacua]]

[[!redirects vacuum decay]]
[[!redirects vacuum decays]]


false vacuum eternal inflation

> under construction

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

#Contents#
* table of contents
{:toc}

## Idea

False vacuum eternal inflation (FVEI) can occur whenever the [[inflaton]] [[potential energy|potential]] $V(\phi)$ has multiple [[minima]] — in the simplest case one [[false vacuum]] (a [[local minimum]] $\phi = \phi_{\mathrm{F}}$ in the potential where $V(\phi) \gt 0$) and one true [[vacuum]] (the global minima of the potential).

\begin{imagefromfile}
    "file_name": "FalseVacuumEternalInflationBubbles.jpg",
    "float": "right",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 10
    },
    "caption": "Caption goes here..."
\end{imagefromfile}

At any given point in space, [[quantum tunneling]] from the [[false vacuum]] to the true vacuum will eventually occur, as first found by [Coleman & De Luccia 1980](#ColemanDeLuccia1980). In such a tunneling event, these authors found that a true vacuum bubble will form that will expand at a rate close to the [[speed of light]]. The interior of such a bubble will be described by an open (i.e. $k = - 1$) [[FRW model|Friedmann universe]] of infinite size. For this reason, false vacuum inflation also goes by the name of "open inflation".

If the tunneling rate per unit [[time]] and [[volume]] is sufficiently small, then once bubbles nucleate they will not be able to coalesce due to the intervening rapidly inflating false vacuum between the bubbles. In this case [[inflation]] never ends globally, leading to a [[multiverse]] picture where bubble universes nucleate ad infinitum (hence the name "[[eternal inflation]]"). Our [[observable universe]] would, in this picture, only be an infinitesimal portion of a single bubble universe out of a coutless others.

\linebreak

## Details

(...)

## Related entries

* [[cosmology]]

* [[false vacuum]]

* [[Coleman-De Luccia instanton]], [[bubble of nothing]]

* [[eternal inflation]]

* [[multiverse]]

* [[landscape of string theory vacua]]

## References

The classical reference on [[vacuum decay]] as such:

* {#ColemanDeLuccia1980} [[Sidney Coleman]], [[Frank De Luccia]], Gravitational effects on and of vacuum decay, Phys. Rev. D21:3305, 1980 ([doi:10.1103/PhysRevD.21.3305](https://doi.org/10.1103/PhysRevD.21.3305))

The original article on false vacuum eternal inflation:

* {#GarrigaGuthVilenkin06} [[Jaume Garriga]], [[Alan Guth]], [[Alexander Vilenkin]], _Eternal inflation, bubble collisions, and the persistence of memory_, Phys. Rev. D76:123512, 2007 ([arXiv:hep-th/0612242](https://arxiv.org/abs/hep-th/0612242), [doi:10.1103/PhysRevD.76.123512](https://doi.org/10.1103/PhysRevD.76.123512))


falsehood

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

# Contents
* table of contents
{: toc}

## Idea

In [[logic]], the __false__ proposition, called __falsehood__ or __falsity__, is the [[proposition]] which is always false.

The faleshood is commonly denoted $false$, $F$, $\bot$, or $0$.  These may be pronounced 'false' even where it would be ungrammatical for an adjective to appear in ordinary English.

## Definitions

### In classical logic

In [[classical logic]], there are two [[truth values]]: false and [[true]].  Classical logic is perfectly symmetric between falsehood and truth; see [[de Morgan duality]].

### In constructive logic

In [[constructive logic]], $false$ is the [[bottom element]] in the [[poset]] of [[truth values]].

Constructive logic is still [[two-valued logic|two-valued]] in the sense that any truth value is false if it is not [[true]].

In [[natural deduction]] the [[inference rules]] for falsehood are given as

$$\frac{\Gamma \; \mathrm{ctx}}{\Gamma \vdash \bot \; \mathrm{prop}} \quad \frac{\Gamma \vdash P \; \mathrm{prop}}{\Gamma, \bot \; \mathrm{true} \vdash P \; \mathrm{true}}$$

### In linear logic

In [[linear logic]], there is both _additive_ falsity, denoted $0$, and _multiplicative_ falsity, denoted $\bot$.  Despite the notation, it is $0$ that is the [[bottom element]] of the lattice of linear truth values.  (In particular, $0 \vdash \bot$ but $\bot \nvdash 0$.)

### In a topos

In terms of the [[internal logic]] of a [[topos]] (or other [[category]]), $false$ is the [[bottom element]] in the [[poset of subobjects]] of any given [[object]] (where each object corresponds to a [[context]] in the internal language).

However, not every topos is [[two-valued topos|two-valued]], so there may be other truth values besides $false$ and $true$.

### In type theory

In [[type theory]] with [[propositions as types]], falsehood is represented by the [[empty type]].

### In homotopy type theory

In [[homotopy type theory]], falsehood is represented by the [[empty space]].

## Examples

### In the topos $Set$

In the archetypical [[topos]] [[Set]], the terminal object is the [[singleton]] [[set]] $*$ (the [[point]]) and the poset of subobjects of that is classically $\{\emptyset \hookrightarrow *\}$.  Then falsehood is the [[empty set]] $\emptyset$, seen as the [[empty subset]] of the point.  (See [Internal logic of Set](http://ncatlab.org/nlab/show/internal+logic#LogicOfSet) for more details).

The same is true in the archetypical [[(∞,1)-topos]] [[∞Grpd]]. From that perspective it makes good sense to think of

* a set as a 0-[[truncated]] $\infty$-groupoid: a [[0-groupoid]];

* a [[subsingleton]] set as a $(-1)$-[[truncated]] $\infty$-groupoid: a [[(-1)-groupoid]].

In this sense, the object $false$ in [[Set]] or [[∞Grpd]] may canonically be thought of as being [[generalized the|the]] unique [[empty groupoid]].

## Related concepts

* [[true]], [[unit type]]

* [[empty type]], [[ex falso quodlibet]]

* [[contradiction]], [[inconsistency]], [[consistency]]

[[!redirects false]]
[[!redirects False]]
[[!redirects false proposition]]
[[!redirects false propositions]]
[[!redirects false statement]]
[[!redirects false statements]]
[[!redirects false sentence]]
[[!redirects false sentences]]

[[!redirects falsity]]
[[!redirects falsehood]]


family

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

# Indexed families
* table of contents
{: toc}

## Idea

The term 'family' is often used as a synonym for '[[collection]]' (especially in the sense of [[subset]]).  However, we can use it more precisely, for the concept of an _indexed_ family.

## Definitions

A __family__, or __indexed family__ (or sometimes __indexed set__, etc), consists of an __index set__ $I$ (whose [[elements]] are the __indices__ of the family) and, for each index $k$, some __element__ $x_k$.  One can also speak of an __$I$-indexed family__.  An __[[ordered pair]]__ is a family indexed by $2 = \{0,1\}$; an __[[infinite sequence]]__ is an $\mathbb{N}$-indexed family.

### In concrete categories

In a [[concrete category]] $C$, given an object $A:Ob(C)$ an **$A$-indexed family of objects in $C$** is a function $B:El(A) \to Ob(C)$.

## Notation

As a whole, this family may be denoted $(x_k \;|\; k\colon I)$, $(x_k)_{k\colon I}$, $(x_k)_k$, or simply $x$.  Sometimes one sees braces used instead of parentheses, giving the same notation for a family as for a [[collection]], although this is falling out of fashion; the parentheses ultimately come from notation for [[ordered pairs]].  One can also use notation for [[functions]], such as $\lambda\, k\colon I.\; x_k$ or $(k \mapsto x_k)$.  Finally, instead of $k\colon I$, one can see the [[type]] of $k$ indicated using any other method, especially $k \in I$ (which ultimately derives from [[material set theory]]).

## Families vs collections

Formally, a family of things should be distinguished from a [[collection]] of things; properly, it is the *[[range]]* of a family of things that is a collection, such as a [[subset]] of an appropriate ambient set of things.  On the other hand, often the difference between a family and a collection is unimportant, and the two may be used interchangeably.  (For example, one can take the [[union]] of either a family of subsets or a collection of subsets, with equivalent results; but one can take the sum of only a family of [[cardinal numbers]].)

## Formalization

We have been vague so far about what a family is a family *of* (what the elements are).  The easiest case is when the elements are from a [[set]] $S$; then an $I$-indexed __family of elements__ of $S$, also called an __indexed [[subset]]__ of $S$, is simply a [[function]] to $S$ from $I$.  If the elements are from a category $C$, then an $I$-indexed __family of objects__ of $C$ is a [[functor]] (or [[anafunctor]]) to $C$ from the [[discrete category]] on $I$, and an $I$-indexed __family of morphisms__ of $C$, also called an __indexed [[subcategory]]__, is similarly a functor to the [[arrow category]] of $C$.  In general, the elements ought to be from (at the very least) some sort of $\infty$-[[infinity-groupoid|groupoid]] $G$, in which case the family is a functor to $G$ from the discrete $\infty$-groupoid on $I$.

In [[foundations]] without [[proper classes]] or higher categories, it may be tricky to specify exactly what a [[family of sets]] is, if one cannot literally speak of a functor from a discrete category to the [[large category]] [[Set]]; see the article. On the other hand, there is no difficulty in speaking of a [[family of subsets]] of a given set; even in [[predicative mathematics]] (where one cannot speak of the [[power set]]), a __family of subsets__ of $S$ is simply a [[binary relation]] between $S$ and some index set $I$, writing $a \in x_k$ to denote that the $S$-element $a$ is related to the index $k$.

## Related concepts

* [[free coproduct completion]]

* [[dependent type]]

* [[empty family]]

* [[quantum set]]

[[!redirects family]]
[[!redirects families]]
[[!redirects indexed family]]
[[!redirects indexed families]]
[[!redirects indexed set]]
[[!redirects indexed sets]]

[[!redirects index set]]
[[!redirects index sets]]
[[!redirects index preset]]
[[!redirects index presets]]

[[!redirects family of elements]]
[[!redirects families of elements]]
[[!redirects indexed subset]]
[[!redirects indexed subsets]]

[[!redirects family of objects]]
[[!redirects families of objects]]
[[!redirects family of morphisms]]
[[!redirects families of morphisms]]
[[!redirects indexed subcategory]]
[[!redirects indexed subcategories]]

[[!redirects family of subsets]]
[[!redirects families of subsets]]

[[!redirects family of objects in a concrete category]]
[[!redirects families of objects in a concrete category]]


family of (infinity,1)-operads

[[!redirects family of (∞,1)-operads]]

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

#Contents#
* table of contents
{:toc}

## Definition

For the following we model [[(∞,1)-operads]] $\mathcal{O}$ specifically by their [[(∞,1)-categories of operators]] $\mathcal{O}^\otimes$ and these specifically as [[quasi-categories]].

Let $\mathcal{C}$ be an [[(∞,1)-category]]. A **$\mathcal{C}$-family of [[(∞,1)-operads]]**, is a [[fibration]] in the [[model structure for quasi-categories]]

$$
  p \colon \mathcal{O}^{\otimes}  \to \mathcal{C} \times FinSet_*
$$

such that

* For $C \in \mathcal{C}$ any [[object]], for $X \in \mathcal{O}^\otimes_C$ over $\langle n\rangle \in FinSet_*$, and for $\alpha \colon \langle n\rangle \to \langle k\rangle$ any [[inert morphism]], then there exists a $p$-[[coCartesian morphism]] $\overline \alpha \colon X \to Y$ in $\mathcal{O}^\otimes_C$.

([Lurie, 2.3.2.10](#Lurie))

## Properties

### Relation to generalized $(\infty,1)$-operads

For $\mathcal{C}$ an [[(∞,1)-category]], a  [[fibration]] $\mathcal{O}^\otimes \to \mathcal{C} \times FinSet_*$ in the [[model structure for quasi-categories]] is a $\mathcal{C}$-family of $(\infty,1)$-operads precisely if it is a fibration of [[generalized (∞,1)-operads]] such that the underlying map $\mathcal{O}^\otimes_{\langle 0\rangle} \to \mathcal{C}$ is an [[acyclic Kan fibration]].

## References

* [[Jacob Lurie]], _[[Higher Algebra]]_
 {#Lurie}

[[!redirects families of (∞,1)-operads]]


family of finite types

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Deduction and Induction
+-- {: .hide}
[[!include deduction and induction - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
#### Induction
+-- {: .hide}
[[!include induction - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea ##

In [[type theory]], the *family of finite types* is the analog of the [[natural numbers]]-[[indexed set]] of [[finite sets]] in [[set theory]].

## Definition ##

In [[type theory]], given a [[natural number]] $n \colon\mathbb{N}$, the __family of finite types__ $\mathrm{Fin}_\mathcal{U}(n)$ of a [[type universe]] $\mathcal{U}$

* with [[empty type]] $\varnothing_\mathcal{U} \colon \mathcal{U}$,

* unit type $\ast_\mathcal{U}:\mathcal{U}$,

* and binary [[coproducts]] $(-) \sqcup_\mathcal{U}(-) \colon \mathcal{U} \times \mathcal{U} \to \mathcal{U}$

is an [[inductive family]], [[induction|inductively]] defined by:

$$
  \begin{array}{lcl}
  \mathrm{Fin}_\mathcal{U}(0)
  &\coloneqq&
  \underset
    {A \colon \mathcal{U}}
    {\sum}
  \big(
    A
    \cong_\mathcal{U}
    \varnothing_\mathcal{U}
  \big)
  \\
  \mathrm{Fin}_\mathcal{U}(s(n))
  &\coloneqq&
  \underset
    {A:\mathcal{U}}
    {\sum}
  \bigg[
    \sum_{B:\mathrm{Fin}_\mathcal{U}(n)}
    \Big(
      A \cong_\mathcal{U}
      \big(
        B
        \sqcup_\mathcal{U}
        \ast_\mathcal{U}
      \big)
    \Big)
  \bigg]
  \end{array}
$$

(where "$\sum$" denotes [[dependent sum]] and "$\cong$" denotes [[identity types]], as usual).

## See also ##

* [[natural numbers]]

* [[inductive family]]

* [[finite type]], [[finite set]]

## References ##

* [[Univalent Foundations Project]], *[[HoTT book|Homotopy Type Theory – Univalent Foundations of Mathematics]]* (2013)

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: *[[Symmetry]]* (2021)

* [[Egbert Rijke]], *[[Introduction to Homotopy Type Theory]]*


family of sets

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Category theory
+-- {: .hide}
[[!include category theory - contents]]
=--
#### Discrete and concrete objects
+-- {: .hide}
[[!include discrete and concrete objects - contents]]
=--
=--
=--

# Contents
* table of contents
{:toc}

## In set theory

A __[[family]] of [[sets]]__ consists of an __index set__ $I$ and, for each [[element]] $k$ of $I$, a [[set]] $S_k$.

Given $I$, a [[set]] $F$ and a [[function]] $p\colon F \to I$, we get a family of sets by defining $S_k$ to be the [[preimage]] $p^*(k)$.

Conversely, given a family of sets, let $F$ be the [[disjoint union]]
$$ \biguplus_k S_k = \{ (k,x) | k \in I, x \in S_k \} $$
and let $f(k,x)$ be $k$.

(We should talk about ways to formalise this concept in various forms of [[set theory]] and when the latter construction above requires the [[axiom of collection]].)

## In category theory

A __[[family]] of [[sets]]__ consists of an [[discrete groupoid]] $I$ called the __index set__, and a [[functor]] $F:I \to Set$, where [[Set]] is the large [[category of sets]].

## In dependent type theory

A **[[family]] of [[sets]]** consists of a type $I$ called the **index type**, and a [[type family]] $S(x)$ indexed by elements $x:I$, such that every dependent type $S(x)$ is an [[h-set]] by the typal [[axiom K]] definition or the typal [[uniqueness of identity proofs]] definition.

Equivalently, it is a type family $(I, S)$ which satisfies the familial [[axiom K]].

## Inconsistency of the universal family of sets

A **universal family of sets** is a set $U$ with a set $I$ and a function $E:U \to I$ such that for all sets $A$, there is a unique element $i_A \in I$ and a [[bijection]] $\delta_A:A \cong E^*(i_A)$ from $A$ to the [[fiber]] of $E$ at $i_A$. This is inconsistent in any [[set theory]], due to the [[set-theoretic Girard's paradox]].

## See also

* [[family]]
* [[family of subsets]]

[[!redirects family of sets]]
[[!redirects families of sets]]

[[!redirects universal family of sets]]
[[!redirects universal families of sets]]


family of subsets

\tableofcontents

## Definition

Given any [[set]] $S$, a __[[family]] of [[subsets]]__ of $S$ consists of an index set $I$ and and a [[function]] from $I$ to the [[power set]] of $S$.  This is actually nothing special; this is the normal meaning of 'family' when taking a family of things that form a set: a family of foos is any function to the set of foos.

However, in [[predicative mathematics]], we do not accept that the subsets of $S$ form a set; that is, we believe that the power set $\mathcal{P}S$ is really a [[proper class]].  So on the face of it, a family of subsets would also be a large thing.  However, there is an alternative definition that is not only appropriate predicatively but also often provides a useful perspective in any case:  A __family of subsets__ of $S$ consists of an index set $I$ and a [[binary relation]] $\in$ between elements of $S$ and elements of $I$.

Note that we may have $x \in A \Longleftrightarrow x \in B$ for all $x$ without having $A = B$; this is because (as in any family) two different indices may index the same subset.  However, many of the operations performed on families of subsets, such as [[intersection]] and [[union]], depend only on the [[image]] of the function $I \to \mathcal{P}S$.  So it is common to assume that a family of subsets of $S$ is simply a [[subset]] of $\mathcal{P}S$.  (Besides this, not all authors agree with this page on even the *general* meaning of the word 'family'.)

## See also

* [[family of sets]]
* [[union]]

[[!redirects family of subsets]]
[[!redirects families of subsets]]

[[!redirects indexed subset]]
[[!redirects indexed subsets]]


family of supports

A __family of supports__ in a [[topological space]] $X$ is a family $\phi$ of closed subsets $S\subset X$ such that

1. if $S_1,S_2\in \phi$ then $S_1\cup S_2\in \phi$;
2. if $S_1\in \phi$ and $S_2\subset S_1$ is _closed_, then $S_2\in \phi$.

In other words, it is an [[ideal]] in the lattice of closed subsets.

Families of supports are used to introduce a variant of __[[sheaf cohomology]] with supports in $\phi$__ and also for developing certain [[homology]] theories using sheaves (see the book by Bredon, Sheaf theory). Especially useful are the so-called paracompactifying families of supports on non-[[paracompact space]]s.

Let $F$ be a sheaf of [[abelian groups]] over a topological space $X$. Denote by $\Gamma_\phi(X,F)$ the subset of the space of all sections $f\in \Gamma(X,F) = F(X)$ for which $supp\,f\in \phi$. This gives rise to a covariant [[left exact functor]] $F\mapsto \Gamma_\phi(X,F)$. Its right-[[derived functor]]s

$$H_\phi^k(X,F) := R^k\Gamma_\phi(X,F)$$

are called the __cohomology groups of $X$ with coefficients in the sheaf $F$ and with supports in the family $\phi$ of supports__. Or sometimes one simply says __sheaf cohomology with supports__.

[[!redirects sheaf cohomology with supports]]


fan theorem

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### Context
#### Analysis
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[[!include analysis - contents]]
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# The fan theorem
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## Introduction

The fan theorem is one of the basic principles of [[intuitionism]] that make it more specific (even in [[mathematics|mathematical]] practice, independent of any [[philosophy|philosophical]] issues) than garden-variety [[constructive mathematics]].  Its main use is to justify pointwise [[analysis]]; without it, one really needs [[locale theory]] for [[point-free topology]] instead.  In [[classical mathematics]], the fan theorem is [[true]].

## Statement

Consider the [[list|finite]] and [[infinite sequence|infinite]] sequences of [[binary digits]].  Given an infinite sequence $\alpha$ and a [[natural number]] $n$, let $\bar \alpha n$ be the finite sequence consisting of the first $n$ [[elements]] of $\alpha$.

Let $B$ be a collection of [[finite set|finite]] sequences of bits (or _bitlists_), that is a [[subset]] of the [[free monoid]] on the [[boolean domain]].  Given an infinite sequence $\alpha$ and a natural number $n$, we say that $\alpha$ _$n$-bars_ $B$ if $\bar \alpha n \in B$; given only $\alpha$, we say that $\alpha$ _bars_ $B$ if $\alpha$ $n$-bars $B$ for some $n$.

We are interested in these three properties of $B$:

*  $B$ is _[[decidable subset|decidable]]_:  For every finite sequence $u$, either $u \in B$ or $u \notin B$.  (This is trivial in [[classical logic]] but will hold constructively only for some subsets $B$.)
*  $B$ is _barred_:  For every infinite sequence $\alpha$, $\alpha$ bars $B$.
*  $B$ is _uniform_:  For some natural number $M$, for every infinite sequence $\alpha$, if $\alpha$ bars $B$ at all, then $\alpha$ $n$-bars $B$ for some $n \leq M$.

A __bar__ is a barred subset $B$.

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###### Decidable Fan Theorem

Every decidable bar is uniform.
(In other words, if a collection of bitlists is decidable and barred, then it is also uniform.)
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###### (Full) Fan Theorem
Every bar is uniform. (In other words, if a collection of bitlists is barred, then it is also uniform.)
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Although the fan theorem is about bars, it is different from the [[bar theorem]], also known as "bar induction", which is related but stronger. The fan theorem discusses bars on lists of elements of $\mathbb{B}$; the bar theorem discusses bars on lists of elements of $\mathbb{N}$.

Bar induction has three forms; decidable bar induction, monotone bar induction, and full bar induction (each stronger than the previous). The decidable fan theorem follows from decidable bar induction; the full fan theorem follows from monotone bar induction. Full bar induction, the strongest of the three, is a classical theorem; however, full bar induction implies Bishop's limited [[principle of omniscience]]. [[function realizability|Kleene's second realizability model]] provides a model of monotone bar induction plus Brouwer's continuity principle.

### Obfuscation

Let $\mathbb{B}$ be the set $\{0,1\}$ of binary digits (bits) and $\mathbb{N}$ the set $\{0,1,2,\ldots\}$ of natural numbers (numbers).  Given a [[set]] $A$, let $A^*$ be the set of finite sequences of elements of $A$, let $A^{\mathbb{N}}$ be the set of infinite sequences of elements of $A$, and let $\mathcal{P}_{\Delta}A$ be the set of decidable subsets of $A$.  Then the fan theorem is about (elements of) $\mathbb{B}^*$, $\mathbb{B}^{\mathbb{N}}$, and $\mathcal{P}_{\Delta}\mathbb{B}^*$.

However, the sets $\mathbb{N}$, $\mathbb{B}^*$, and $\mathbb{N}^*$ are all isomorphic.  Similarly, the sets $\mathbb{B}^{\mathbb{N}}$, $\mathcal{P}_{\Delta}\mathbb{N}$, $\mathcal{P}_{\Delta}\mathbb{B}^*$, and $\mathcal{P}_{\Delta}\mathbb{N}^*$ are all isomorphic.  In much of the literature on bars, one tacitly uses all of these isomorphisms, taking $\mathbb{N}$ and $\mathbb{B}^{\mathbb{N}}$ as chosen representatives of their isomorphism classes.  Thus, everything in sight is either a natural number or an infinite sequence of bits.

The fan theorem is hard enough to understand when $\alpha$ is an infinite sequence of bits and $\bar \alpha n$ is a finite sequence of bits; it is even harder to understand when $\bar \alpha n$ is a natural number that bears no immediate relationship to the digits in the sequence $\alpha$.

## Use in analysis

In [[classical mathematics]], the fan theorem is simply true.

In [[constructive mathematics]], the full fan theorem is equivalent to any and all of the following statements:

*  As a [[locale]], [[Cantor space]] has enough points (is [[topological locale|topological]]).
*  As a [[topological space]], Cantor space is [[compact space|compact]].

Assuming some amount of countable/dependent choice (probably pretty weak: haven't checked myself though), the full fan theorem is equivalent to each of the following statements:

*  As a topological space, the (located Dedekind) [[unit interval]] is [[compact space|compact]] (the [[Heine-Borel theorem|Heine–Borel theorem]]).
*  As a topological space, the (located Dedekind) [[real line]] is [[locally compact space|locally compact]].
*  There exists a class of "kontinuous" [[partial functions]] from the set $\mathbb{R}$ of (located Dedekind) [[real numbers]] to itself (see Waaldijk) such that
   *  the [[restriction]] of a kontinuous function to any smaller domain is kontinuous;
   *  the [[identity function]] on $\mathbb{R}$ is kontinuous;
   *  the [[composite]] of two kontinuous functions is kontinuous;
   *  a function whose domain is the unit interval is kontinuous *if and only if* it is uniformly continuous (in the usual metric-space sense); and
   *  the function $(x \mapsto 1/x)$ defined on $\dot{\mathbb{R}}^+$ is kontinuous.

However, some of these equivalences fail in the absence of countable choice. See

*  Moerdijk, Ieke. “Heine-Borel Does Not Imply the Fan Theorem.” Journal of Symbolic Logic, vol. 49, no. 2, 1984, pp. 514–519., doi:10.2307/2274182

which shows that the compactness of $[0, 1]$ does not necessarily imply the fan theorem constructively if one does not have some amount of countable choice available.

The decidable fan theorem (assuming some amount of countable/dependent choice) is equivalent to any of the following statements:

*  Every [[uniformly continuous map|uniformly continuous]] function from Cantor space to the [[metric space]] $\dot{\mathbb{R}}^+$ of positive real numbers has a positive lower bound.
*  Every [[uniformly continuous map|uniformly continuous]] function from the unit interval to $\dot{\mathbb{R}}^+$ has a positive lower bound.

For the above equivalences with the decidable fan theorem, see

*  Julian, W., & Richman, F. (1984). A uniformly continuous function on [0,1] that is everywhere different from its infimum. Pacific Journal of Mathematics, 111(2), 333–340. https://doi.org/10.2140/pjm.1984.111.333

The above paper unfortunately uses different terminology than this article; for example, using "branch-bounded" and "bounded" in place of "barred" and "uniform". The above paper also proves that, assuming all functions $\mathbb{N} \to \mathbb{N}$ are computable, there exists a decidable bar which is not uniform (and, as the title suggests, a uniformly continuous function $[0, 1] \to (0, \infty)$ whose range has an infimum of zero).

Some form of the fan theorem follows from any of these statements (some work needs to be done to verify these and get citations):

*  The [[bar theorem]] holds (see above)
*  As a locale, [[Baire space of irrational numbers|Baire space]] has enough points.
*  Every pointwise-continuous function on Cantor space is uniformly continuous.
*  Every pointwise-continuous function on the unit interval is uniformly continuous.
*  Every [[uniformity]] (or indeed any [[metric]]) compatible with the usual topology of Cantor space is [[totally bounded space|totally bounded]].
*  The property of being "complete and totally bounded", i.e. [[Bishop-compact]], is invariant under topological homemorphism between uniform (or metric) spaces.

I need to figure out how it relates to the various versions of [[Konig's lemma|König's Lemma]], as well as these statements (which are mutually equivalent):

*  As a locale, the unit interval has enough points.
*  As a locale, the [[real line]] (the [[locale of real numbers]]) has enough points.

Some of the results above may use [[countable choice]], but probably no more than $AC_{0,0}$ (which is choice for relations between $\mathbb{N}$ and itself).

### Analysis without the fan theorem

Point-wise real analysis without the fan theorem is very difficult, as shown by the example above from Waaldijk regarding "kontinuous" functions: without the fan theorem there isn't really even a good notion of continuity!  This was [[L.E.J. Brouwer|Brouwer]]\'s motivation for introducing the fan theorem.

However, the fan theorem (and bar theorem) can be avoided by instead using [[locales]] or another [[point-free topology|point-free]] approach to analysis.

### Sheaf models
[Fourman and Hyland](#FourmanHyland) prove that the Fan theorem is true for all toposes of sheaves on a topological space, while also providing a sheaf model not satisfying the fan theorem. In other words, for Localic toposes, the underlying locale being spatial implies that the internal locale of real numbers is spatial, and that the internal cantor locale is spatial.

The reference also proves that sheaf toposes over locally countably compact topological spaces satisfy the stronger bar theorem.

## Proofs

I should write down the classical proof (which uses [[excluded middle]]), as well as Brouwer\'s argument.

Ideally, a page on [[bar induction]] would be added. This page would contain the classical proof of full bar induction, as well as Brouwer's reasoning for why bar induction should hold. We would then provide constructive proofs here that monotone bar induction implies the full fan theorem, and that decidable bar induction implies the decidable fan theorem.

## References

*  Thanks to Giovanni Curi on [constructive news](http://groups.google.com/group/constructivenews/browse_thread/thread/9d57fa99e67e8782).

*  [Frank Waaldijk](http://www.fwaaldijk.nl/mathematics.html) pointed out exactly why point-wise analysis needs the fan theorem.

I need to read the relevant parts here:

*  [[Mike Fourman]], R. Grayson, _Formal Spaces_. In: The L.E.J. Brouwer Centenary Symposium,  A.S. Troelstra and D. van Dalen, eds. North Holland (1982), pp. 107--122.

More links that I need to keep in mind:

*  <http://golem.ph.utexas.edu/category/2008/12/the_status_of_coalgebra.html>
*  <http://www.jaist.ac.jp/is/labs/ishihara-lab/wcalm2010/berger.pdf>
*  <http://www.cairn.info/revue-internationale-de-philosophie-2004-4-page-483.htm>

Also:

* [[Mike Fourman]], [[Martin Hyland]], _Sheaf models for analysis_. [PDF](https://www.dpmms.cam.ac.uk/~martin/Research/Oldpapers/analysis79.pdf)
  {#FourmanHyland}

[[!redirects fan theorem]]
[[!redirects Fan theorem]]

[[!redirects classical fan theorem]]


Fano plane

#Contents#
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## Idea

The Fano plane is the [[projective plane]] over the [[finite field]] $\mathbf{F}_2$ with two elements. It is denoted either $\mathbb{F}_2\mathbb{P}^2$, $\mathbb{P}^2_{\mathbb{F}_2}$ (in [[algebraic geometry]]), or $PG(2,2)$ (in finite geometry). It is often pictured as at right, but there are many different representations.

## Properties

### Automorphisms

The [[automorphism group]] $PGL(3,\mathbb{F}_2)$ is isomorphic to the [[projective special linear group]] $PSL(2,\mathbb{F}_7)$.

## Related concepts

* [[octonion]]

## References

* Wikipedia, _[Fano plane](https://en.wikipedia.org/wiki/Fano_plane)_

* [[John Baez]], _[The Fano plane](http://math.ucr.edu/home/baez/octonions/node4.html)_