free monoid

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### Context
#### Algebra
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[[!include higher algebra - contents]]
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#### Induction
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[[!include induction - contents]]
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#### Monoid theory
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[[!include monoid theory - contents]]
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# Finite lists and free monoids
* table of contents
{: toc}

## Idea

Given a [[set]] $S$, the _free monoid_ on $S$ is the set $S^*$ of all _[[lists]]_ (finite [[sequences]]) of [[elements]] of $S$, made into a [[monoid]] via [[concatenation]].

The [[free functor]] from [[Set]] to [[Mon]] takes $S$ to $S^*$. Combined with its [[right adjoint]] [[forgetful functor]] this is the *[[list monad]]* whose [[modules over a monad|modules]] are [[monoids]].

## Definitions

We will give three definitions, which can all be shown equivalent.

### As functions

An element of $S^*$ consists of a [[natural number]] $n$ (possibly $n = 0$) and function from $[n]$ to $S$, where $[n]$ is the [[subset]] $\{i\colon \mathbf{N} \;|\; i \lt n\}$ of $\mathbf{N} = \{0, 1, 2, \ldots\}$.  Such an element is called a __list__ or (to specify $n$) __$n$-tuple__ of elements of $S$.  The number $n$ is called the __length__ of the list.

The __[[empty list]]__ is the unique list of length $0$.  It may be written $()$, $*$, or $\epsilon$, perhaps with a subscript $S$ if desired.

If $n \gt 0$, then the list which assigns $0, \ldots, n - 1$ to $a_0, a_1, \ldots, a_{n-1}$ may be written $(a_0, a_1, \ldots, a_{n-1})$.  For example, if $a,b,c$ are elements of $S$, then $(a,b,c)$ is an element of $S^*$.

Given two lists $x$ and $y$, the former of length $m$ and the latter of length $n$, their __[[concatenation]]__ $x * y$ is a list of length $m + n$, given as follows:
$$ i \mapsto \left\{ \array { x_i     & if\; i \lt m \\
                              y_{i-m} & if\; i \geq m } \right. $$

One can now show that [[concatenation]] is [[associativity|associative]] with the empty list as identity; hence $S^*$ is a monoid.

### Recursively

The (underlying) set $S^*$ may be defined as an [[inductive type]] as follows.  There are two basic constructors, one with no arguments, and one with two arguments, of which one is an element of $S$ and the other is an element of $S^*$.  By the yoga of inductive types, that is a complete definition, but we spell it out in more detail while also giving terminology and notation.

So, a __list__ is either the __[[empty list]]__ or the __cons__ (short for 'constructor' and deriving from Lisp) of an element $a$ of $S$ and a (previously constructed) list $x$.  The empty list may may be written $()$, $*$, or $nil$, perhaps with a subscript $S$ if desired; the cons of $a$ and $x$ may be written $a : x$, $(a) * x$, or $cons(a,x)$.  We interpret the definition [[recursion|recursively]], so we can list the elements of $S^*$ in the order in which they appear:

*  $()$,
*  $a : ()$,
*  $a : b : ()$,
*  $a : b : c : ()$,
*  etc.

Here, $a, b, c, \ldots$ are elements of $S$.  We may continue the 'etc' as far as we like, but no farther; while there are lists of arbitrarily long finite length, there are no lists of infinite length.  (We would get such lists, however, if we interpreted the definition [[corecursion|corecursively]], known in computer science as a [[stream]].)  We normally abbreviate the lists above as follows:

*  $()$,
*  $(a)$,
*  $(a,b)$,
*  $(a,b,c)$,
*  etc.

We still must define the monoidal structure on $S^*$; we define the __concatenation__ $x * y$ of $x$ and $y$ recursively in $x$.  To be explicit:

*  $() * y = y$;
*  $(a : x) * y = a : (x * y)$ (with parentheses for grouping, but the parentheses can be dropped now that have this definition).

One can now show that concatenation is associative with the empty list as identity; hence $S^*$ is a monoid.

### By general abstract nonsense

To prove that the category [[Mon]] of monoids is a [[complete category]], one normally shows that the [[forgetful functor]] $U$ (from $Mon$ to the category [[Set]] of sets) preserves all [[limits]].  Then, the [[adjoint functor theorem]] defines a [[left adjoint]] to $U$ if a size condition is met; this adjoint is the functor $*$ that takes a set to its free monoid $S^*$.

To be sure, meeting the solution set condition basically requires starting the constructions in one of the other definitions above; but the proofs may all be thrown onto the adjoint functor theorem.

Another abstract approach is given in the following general theorem, which applies to more general [[monoid in a monoidal category|monoids in a monoidal category]]:

+-- {: .num_theorem #monoid_in_monoidal}
###### Theorem
Suppose $C$ is a monoidal category with countable coproducts for which the tensor product distributes over countable coproducts (for example, a [[cocomplete category|cocomplete]] [[monoidal biclosed category]]). Then a left adjoint to the forgetful functor $Mon(C) \to C$ exists, taking an object $c$ to
$$\sum_{n \geq 0} c^{\otimes n},$$
which thereby becomes the free monoid on $c$.
=--

This applies immediately to $C = Set$, as this is a cocomplete [[cartesian closed category]].

### In dependent type theory

In a general [[dependent type theory]] where we do not assume [[uniqueness of identity proofs]], the free monoid on a type $A$ is the [[0-truncation]] of the [[type of lists]] of $A$, $\mathrm{FreeMonoid}(A) \coloneqq [\mathrm{List}(A)]_0$. This is because in any dependent type theory with [[uniqueness of identity proofs]], where every type is 0-truncated, the type of lists of $A$ is the free monoid of $A$.

Alternatively, the free monoid on $A$ can be constructed via the [[James construction type]] of $A$.

## Examples

*  If $S$ is the [[empty set]], then $S^*$ consists only of the empty list; it is the trivial monoid, one manifestation of the [[point]].

*  If $S$ is the point, then $S^*$ is $\mathbf{N}$; the only information in a list of indistinguishable points is the length of the list.  The monoid operation on $\mathbf{N}$ is addition.

*  If $S$ is $\mathbf{N}$, then $\mathbf{N}^*$ is still a [[denumerable set]].  But note that $\mathbf{N} \cong \mathbf{N}^*$ only as sets (that is, ${|\mathbf{N}^*|} = {|\mathbf{N}|} = \aleph_0$ as [[cardinal numbers]]); they are quite different as monoids.

*  Generalising the above, ${|S^*|} = {|S|}$ if $S$ is an [[infinite set]], or more generally ${|S^*|} = max(\aleph_0,S)$ if $S$ is an [[inhabited set]].  (These theorems probably require the [[axiom of choice]], but I haven\'t checked thoroughly.)

In [[computer science]], the free monoid on a given [[alphabet]] is the [[categorical semantics]] for the [[data type]] "[[String (computer science)|String]]" with characters in that alphabet.

## The free monoid monad

If the free monoid functor $F\colon Set \to Mon$ is followed by the forgetful functor $U\colon Mon \to Set$, then we get a [[monad]] on $Set$.  This monad is very important as a [[monad in computer science]], where it is known as the __[[list monad]]__.

The list monad bears the same relation to [[multicategories]] as the [[identity monad]] on $Set$ bears to ordinary [[categories]].  This relation should be explained at [[generalized multicategory]].

## Foundational relevance

Every definition of free monoid makes use of some form of [[axiom of infinity]], either $\mathbf{N}$ directly or the ability to form general inductive types.  Indeed, as $\mathbf{N} = pt^*$, the axiom of infinity follows from the existence of free monoids.

In [[topos theory]] the equivalent of the above theorem \ref{monoid_in_monoidal} is due to C. J. Mikkelsen:

+-- {: .num_prop #free_monoids_in_topos}
###### Proposition
Let $\mathcal{S}$ be a topos and $\mathbf{mon}(\mathcal{S})$ its category of internal monoids. Then $\mathcal{S}$ has a NNO precisely if the forgetful functor $U:\mathbf{mon}(\mathcal{S})\to \mathcal{S}$ has a left adjoint.
=--

For a proof see Johnstone ([1977](#JT77),p.190).

Furthermore then it is a theorem due to [[Andreas Blass]] ([1989](#Blass)) that $\mathcal{S}$ has a NNO precisely if $\mathcal{S}$ has an [[classifying topos for the theory of objects|object classifier]] $\mathcal{S}[\mathbb{O}]$.

A consequence of this, discussed in sec. B4.2 of (Johnstone 2002,I p.431), is that [[classifying topos|classifying toposes]] for [[geometric theories]] over $\mathcal{S}$ exist precisely if $\mathcal{S}$ has a NNO.

From a different perspective then, in a topos the existence of free objects over various gadgets like e.g. [[algebraic theory|algebraic theories]] or [[geometric theory|geometric theories]] often hinge on the existence of free monoids, the intuition being that the free monoids permit to construct a free model _syntactically_ by providing for the (syntactic) building blocks needed for this process.

Notice that algebraic theories can nevertheless have free algebras even if the ambient topos lacks a NNO. This may happen for algebraic theories that have the property that the free algebra on a finite set of generators has a finite carrier e.g. in the topos $FinSet$ of finite sets [[graphic category|free graphic monoids]] exist, and [[free Boolean algebras]] exist.

## Stacks and queues

In [[computer science]], lists often appear as _stacks_ (not to be confused with the [[stacks]] from higher sheaf theory) and _queues_.

Fix a [[monoidal category]] that has [[coproducts]] with the [[unit object]] $I$.  Given an [[object]] $A$, an object of __stacks__ on $A$ is an object $S_A$ equipped with [[morphisms]] $push_A\colon S_A \otimes A \to S_A$ and $pop_A\colon S_A \to S_A \otimes A + I$ such that these diagrams commute:
$$ \array {
   S_A \otimes A &                     & \overset{\iota_{S_A \otimes A,I}}\to &                  & S_A \otimes A + I \\
                 & {}_{push_A}\searrow &                                      & \nearrow_{pop_A} &                                       & \searrow^{push_A + id_I} \\
                 &                     & S_A                                  &                  & \underset{\iota_{S_A \otimes A,I}}\to &                          & S_A + I
   } $$
The idea is that $push_A$ and $pop_A$ are as close to [[inverse morphism|inverses]] as reasonably possible, but $pop_A$ takes us to $S_A \otimes A + I$ rather than to $S_A \otimes A$, because of the empty stack.

Queues are a little more complicated.  An object of __queues__ on $A$ is an object $Q_A$ equipped with morphisms $ins_A\colon A \otimes Q_A \to Q_A$ ('insert') and $rem_A\colon Q_A \to Q_A \otimes A + I$ ('remove').  These operations are far from inverses; whereas popping a stack returns the last item to be pushed onto it, removing an item from a queue returns the *first* item to have been inserted into it.

+-- {: .query}
What are the diagrams for this?  I seem to recall that we need a [[distributive category]]; in particular, we need a [[cartesian monoidal category]], so that $\otimes$ is $\times$.  But perhaps a [[2-rig]] will be sufficient?
=--

## Related concepts

* [[list]]

* [[string (computer science)]], [[stream]]

* [[classifying topos for the theory of objects]]

* [[natural numbers object]]

* [[arithmetic pretopos]]

* [[tensor algebra]]

* [[cofree coalgebra]]

* [[James construction type]]

## References

* {#Ben91}[[Jean Bénabou]], _Some Remarks on Free Monoids in a Topos_ , pp.20-29 in LNM **1488** Springer Heidelberg 1991.

* {#Blass} [[Andreas Blass]], _Classifying topoi and the axiom of infinity_ , Algebra Universalis **26** (1989) pp.341-345.

* {#JT77}[[Peter Johnstone]], _Topos Theory_, Academic Press New York 1977. (Dover reprint Minneola 2014, chap. 6)

The list [[monad]] as a [[monad in computer science]]:

* [[Tarmo Uustalu]], p.25 of: *Monads and Interaction Lecture 1* [[pdf](https://cs.ioc.ee/~tarmo/mgs21/mgs1.pdf), [[Uustalu-Monads1.pdf:file]]], lecture notes for [MGS 2021](https://staffwww.dcs.shef.ac.uk/people/G.Struth/mgs21.html) (2021):

[[!redirects free monoids]]

[[!redirects finite list]]
[[!redirects finite lists]]
[[!redirects finite sequence]]
[[!redirects finite sequences]]

[[!redirects free monoid]]

[[!redirects queue]]
[[!redirects queues]]

[[!redirects list object]]
[[!redirects list objects]]


free object

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

## Definition

Let $U: C\to D$ be a [[forgetful functor]] and $x\in D$ an [[object]] of the [[category]] $D$.

A **free $C$-object** on $x$ with respect to $U$ is an [[object]] of $C$ that satisfies the [[universal property]] that $F(x)$ would have, if $F$ were a [[left adjoint]] to $U$ (the corresponding [[free functor]]) (the _[[free construction]]_ on $x$).

If $U$ actually has a [[left adjoint]], then $F(x)$ is a free $C$-object on $x$ for every $x$, and conversely if there exists a free $C$-object on every $x\in D$ then $U$ has a left adjoint.  But individual free objects can exist without the whole left adjoint functor existing.  In general, we have a "partially defined adjoint", or $J$-[[relative adjoint]] where $J$ is the inclusion of a [[full subcategory]] (on those objects admitting free objects).

More precisely: a __free $C$-object on $x$__ consists of an object $y\in C$ together with a [[morphism]] $\eta_x \colon x\to U y$ in $D$ such that for any other $z\in C$ and morphism $f\colon x\to U z$ in $D$, there exists a unique $g\colon y\to z$ in $C$ with $U(g) \circ \eta_x = f$.

In other words, it is an [[initial object]] of the [[comma category]] $(x/U)$.  A free $C$-object on $x$ is also sometimes called a **universal arrow** from $x$ to the functor $U$.  It can also be identified with a [[semi-final lift]] of an empty $U$-structured [[sink]].

Sometimes one says an object $c$ of $C$ is free (relative to a forgetful functor $U: C \to D$ which is often tacitly understood) if there is some object $x$ of $D$ and some arrow $x \to U c$ that is initial in $(x/U)$. For example, the [[Quillen-Suslin theorem]] says that [[finitely generated module|finitely generated]] [[projective modules]] over [[polynomial algebras]] over a [[field]] are free; the tacit forgetful functor is from the category of modules over a polynomial algebra to $Set$. In this way, freeness is understood as a [[stuff, structure, property|property]] of an object.

Similarly, a __cofree object__ is given by a [[cofree functor]].

## Examples

* [[free abelian group]]

* [[free module]]

* [[distributivity pullback]]

For more examples see at _[[free construction]]_.

A general way to construct free objects is with a [[transfinite construction of free algebras]] (in [[set theory|set-theoretic]] foundations), or with an [[inductive type]] or [[higher inductive type]] (in [[type theory|type-theoretic]] foundations).

## Related concepts

* [[free functor]], [[relative adjoint]], [[universal property]]

* [[projective object]], [[projective presentation]], [[projective cover]], [[projective resolution]]

  * [[projective module]]

* [[injective object]], [[injective presentation]], [[injective envelope]], [[injective resolution]]

  * [[injective module]]

* **free object**, [[free resolution]]

  * [[free module]]

* flat object, [[flat resolution]]

  * [[flat module]]

[[!redirects free object]]
[[!redirects free objects]]

[[!redirects cofree object]]
[[!redirects cofree objects]]
[[!redirects co-free object]]
[[!redirects co-free objects]]
[[!redirects fascist object]]
[[!redirects fascist objects]]

[[!redirects free property]]
[[!redirects free properties]]


free operad

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#### Higher algebra
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#Contents#
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## Idea

A _free [[operad]]_ is [[free functor|free]] on a collection of operations.

Given a collection $k$-ary operations-to-be for each $k \in \mathbb{N}$, the free operad on this collection has as $n$-ary operations the collection of all [[trees]] with $n$ leaves equipped with a labelling of each vertex $v$ with a $k$-ary operation, for $k$ the incoming edges to $v$.

## Definition

Let $V$ be a [[symmetric monoidal category]].

For $G$ a [[discrete group]], write $V^G$ for the category of objects of $V$ equipped with a $G$-[[action]]. For $V$ symmetric monoidal this is again a [[symmetric monoidal category]] and the [[forgetful functor]] $V^G \to V$ is symmetric monoidal.

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

The category of **collections** ([Berger-Moerdijk](#BergerMoerdijk)) or **$\mathbb{S}$-modules** ([Getzler-Kapranov](#GetzlerKapranov)) of $V$, or the category of $V$-[[species]], is

$$
  V Coll := \prod_{n \in \mathbb{N}} V^{S_n}
  \,.
$$

=--

Notice that both $S_0$ and $S_1$ are the trivial group.

So a $V$-operad $P$ is a special $V$-collection with extra structure relating its components. This gives an evident [[forgetful functor]]

$$
  U : V Operad \to V Coll
  \,.
$$

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

The [[free functor]] [[left adjoint]] to this forgetful functor is the  the **free operad functor**

$$
  F : V Coll \stackrel{\leftarrow}{\to} V Operad : U
  \,.
$$

For $C$ a given collection, we call $F(C)$ the operad _free on the collection $C$_.

=--

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

This free/forgetful [[adjunction]] is used to define the [[model structure on operads]] by [[transferred model structure|transfer]].

=--

## Properties

### Explicit construction

The free operad functor may more explcitly be described as follows (see ([Berger-Moerdijk, section 5.8](#BergerMoerdijk))).

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

Let $\mathbb{T} := Core(\Omega_pl)$ be the [[core]] of the category of planar rooted [[trees]] and _non-planar_ morphisms (so the morphisms need not respect the given planar structure).

Write

* $t_n \in \Omega_n$ for the $n$-corolla (the tree with a single vertex, $n$ inputs and its unique output root);

* for $T$ any tree with $n$-ary root vertex let $\{T_i\}_{i=1}^n$ be the sub-trees such that $T = t_n \circ (T_1, \cdots, T_n)$.

Then every collection $K \in V Coll$ defines a functor $\bar K : \mathbb{T}^{op} \to V$ by the inductive formula

$$
  \bar K (|) \coloneqq I
$$

$$
  \bar K(t_n(T_1, \cdots, T_n))
  \coloneqq K(n) \otimes \bar K(T_1) \otimes \cdots \bar K(T_n)
  \,.
$$

Define moreover the functor

$$
  \lambda : \mathbb{T} \to Set
$$

to be the functor that sends a tree to the set of numberings of its leaves, and let $\bar \lambda : \mathbb{T} \to V$ be given by postcomposition with $S \mapsto \coprod_{s \in S} I$, where on the right we have the [[coproduct]] of ${\vert S \vert}$ copies of the tensor unit in the [[monoidal category]] $V$.

=--

So for $T$ a tree with $n$ leaves we have

$$
  \bar \lambda(T) \simeq \coprod_{\sigma \in \Sigma_n} I
  \,,
$$

where the coproduct ranges over the elements of the [[symmetric group]] on $n$ elements.
+-- {: .num_prop}
###### Proposition

The **free operad** on a collection $K$ is isomorphic to the [[coend]]

$$
  \bar K \otimes_{\mathbb{T}} \bar \lambda
  =
  \int^{T \in \mathbb{T}}
   \bar K(T) \otimes \bar \lambda(T)
  \,.
$$

=--

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

The groupoid $\mathbb{T}$ is equivalent to the [[disjoint union]] over isomorphism classes of planar trees of the one-object groupoids with morphisms the given [[automorphism group]]

$$
  \mathbb{T} \simeq \coprod_{[T] \in \pi_0\mathbb{T}}
  \mathbf{B} Aut(T)
  \,.
$$

Therefore the above coend is equivalently

$$
  \bar K \otimes_{\mathbb{T}} \bar \lambda
  =
  \coprod_{[T] \in \pi_0\mathbb{T}}
   \bar K(T) \otimes_{Aut(T)} \bar \lambda(T)
  \,.
$$

=--

## Examples

### Rooted tree operad

Let $K$ be the collection with $K(0) = \emptyset$ and $K(n) = I$ for $n \gt 0$. The corresponding free operad has as $n$-ary operations all rooted trees with $n$ leaves. And composition of operations is given by grafting of trees.

## Related concepts

* [[free category]], [[free monoid]]

Riemann surfaces operad (TO BE EXPANDED)

Deligne-Mumford opeard (TO BE EXPANDED)

[[little discs operad]], [[framed little discs operad]] (TO BE EXPANDED) -- See [[Deligne conjecture]]

## References

A brief remark on free operads is in (1.12) of

* [[Ezra Getzler]], [[Mikhail Kapranov]], _Cyclic operads and cyclic homology_, Conf. Proc. Lect. Notes Geom. Topology IV, Int. Press Camb. (1995), 167–201.
 {#GetzlerKapranov}

A detailed discussion is in Part II, chapter I, section 1.9 of

* [[Martin Markl]], S. Shnider, [[Jim Stasheff]], _Operads in Algebra, Topology and Physics_, Surveys and Monographs of the Amer. Math. Soc. 96 (2002).

and in section 3 of

* [[Clemens Berger]], [[Ieke Moerdijk]], _The Boardman-Vogt resolution of operads in monoidal model categories_ , Topology 45 (2006), 807–849. ([pdf](http://math.unice.fr/~cberger/BV.pdf))
{#BergerMoerdijkResolution}

The [[coend]]-description is given in section 5.8 of

* [[Clemens Berger]], [[Ieke Moerdijk]], _Axiomatic homotopy theory for operads_ ([arXiv:0206094](http://arxiv.org/abs/math/0206094v3))
 {#BergerMoerdijk}

[[!redirects collection]]
[[!redirects collections]]

[[!redirects free operads]]


free probability

Free probability is a noncommutative analogue of the probability theory introduced by [[Dan-Virgil Voiculescu]] around 1990 and having some precursors in earlier works on random matrices by [[Freeman Dyson]] and others.

* [[Dan-Virgil Voiculescu]], _Free probability theory: random matrices and von Neumann algebras_, Proc. ICM 1994 [pdf]()
* D.V. Voiculescu, K.J. Dykema, A. Nica. _Free random variables_,
CRM Monograph Series, vol. 1, Amer. Math. Soc. 1992.
* D.V. Voiculescu, _Limit laws for random matrices and free products_,
Inventiones Mathematicae __104__:201–220, 1991.
* D.V. Voiculescu, _Symmetries of some reduced free product $C^\ast$-algebras_, in Operator algebras and their connections with topology and ergodic theory (Bucsteni, 1983), Lecture Notes in Math. 1132, 556–588. Springer 1985.

* A. Nica, R. Speicher, _Lectures on the combinatorics of free probability_, Cambridge University Press, 2006
* Terrence Tao, [notes on free probability](http://terrytao.wordpress.com/2010/02/10/245a-notes-5-free-probability), blog entry

* wikipedia [free probability](https://en.wikipedia.org/wiki/Free_probability)

* Michael Hartglass, _Free product $C^\ast$-algebras associated to graphs, free differentials, and laws of loops_, [arxiv/1509.02553](http://arxiv.org/abs/1509.02553)
* Roland Friedrich, John McKay, _Formal groups, Witt vectors and free probability_, [arxiv/1204.6522](http://arxiv.org/abs/1204.6522); _Homogeneous Lie groups and quantum probability_, [arxiv/1506.07089](http://arxiv.org/abs/1506.07089)

category: probability, functional analysis


free product of groups

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#### Limits and colimits
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#Contents#
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## Idea

Given a [[pair]] of [[groups]] $G_1, G_2 \,\in\,$ [[Grp]],
or more generally an [[n-tuple|$n$-tuple]] $\big(G_1, G_2, \cdots, G_n\big)$ of groups, what is traditionally called their __free product__ and denoted

$$
  G_1 \star G_2
  \;\;\;
  \in
  \;
  Grp
$$

or generally

$$
  \star_i G_i
  \;\coloneqq\;
  G_1 \star G_2 \star \cdots \star G_n
  \;\;\;
  \in
  \;
  Grp
$$

is really their *[[coproduct]]* in [[Grp]]:

$$
  \star_i G_i \;\simeq\; \amalg_i G_i
  \;\;\;
  \in
  \;
  Grp
  \,.
$$

(Compare the [[formal duality|dual notion]] of "[[direct product of groups]]", which are really [[category theory|category theoretic]] *[[products]]* of groups.)

Informally, the "free product" is the group whose elements are [[free construction|freely generated]] form those of the $G_i$, subject only to the relations given by the group operations in each of these groups.

More generally, if each of the given groups is equipped with a [[homomorphism]]

$$
  \iota_i \,\colon\, H \longrightarrow G_i
$$

from a fixed group $H \,\in\, Grp$

> (often taken to be [[monomorphisms]], hence [[injections]], hence [[subgroup]]-inclusions $A \hookrightarrow G_i$ and reducing to the previous situation when $H = 1$ is the [[trivial group]])

then what is traditionally called their **amalgamated free product** (or similar), and denoted

$$
  G_1 \star_A G_2
  \;\;\;
  \in
  \;
  Grp
$$

or generally

$$
  \underset{A}{\star_i} G_i
  \;\coloneqq\;
  G_1 \star_A G_2 \star_A \cdots \star_A G_n
  \;\;\;
  \in
  \;
  Grp
$$

is the corresponding [[pushout]]

$$
  G_1 \star_A G_2
  \;\;
  \simeq
  \;\;
  G_1 \overset{H}{\amalg} G_2
  \;\;
  \in
  \;
  Grp
$$

(or generally the [[colimit]]) in the [[category]] [[Grp]], hence the unique group, up to [[isomorphism]], which

1. receives [[homomorphisms]] $q_i$ from the $G_i$ such that their [[precompositions]] with the respective $\iota_i$ all agree,

2. is [[universal property|universal with this property]] in that for any other group $Q$ receiving such homomorphisms $f_i$ these factor through the respective $q_i$ via a single and unique comparison homomorphism, shown as a dashed arrow in the following diagram:

\begin{tikzcd}[sep=25pt]
  &
  G_1
  \ar[dr, "{ q_1 }"{description}]
  \ar[drr, bend left=20, "{ f_1 }"{description}]
  \\
  H
  \ar[ur, "{ \iota_1 }"{description}]
  \ar[dr, "{ \iota_2 }"{description}]
  \ar[rr, phantom, "{ \scalebox{.6}{(po)} }"]
    &&
  G_1 \star_{{}_H} G_2
  \ar[r, dashed]
  &
  Q
  \\
  &
  G_2
  \ar[ur, "{ q_2 }"{description}]
  \ar[urr, bend right=20, "{ f_2 }"{description}]
\end{tikzcd}

## Examples

\begin{example}
  For $S \in $ [[Sets]], the [[free group]] on $S$ is equivalently the [[free product]] of $S$ copies of the [[additive group of integers]]:
$$
  F(S) \;\simeq\; \star_S \mathbb{Z}
  \,.
$$
An abstract way to understand this elementary fact is to notice that $F$ is the [[free construction]] [[left adjoint]] to the [[forgetful functor]] which sends a [[group]] $G$ to its [[underlying]] [[set]] $U(G)$:
$$
  F \,\colon\,Set \rightleftarrows Grp \,\colon\, U
  \,.
$$
But since

1. [[left adjoints preserve colimits]] and hence in particular [[preserved colimit|preserve]] [[coproducts]],

1. every set is the [[coproduct]] of copies of the [[singleton set]] $\ast \,\in\,$ [[Set]] indexed by its [[elements]] $s \in S$,

1. the [[free group]] on the [[singleton set]] is the additive group of [[integers]]

we have
$$
  \begin{array}{l}
    F(S)
    \\
    \;\simeq\;
    F\Big(\underset{s \in S}{\sqcup} \ast\Big)
    \\
    \;\simeq\;
    \underset{s \in S}{\coprod} F(\ast)
    \\
    \;\simeq\;
    \underset{s \in S}{\coprod} \mathbb{Z}
    \\
    \;\simeq\;
    \underset{s \in S}{\star} \mathbb{Z}
    \,.
  \end{array}
$$
\end{example}

## Properties

### Presentation
 {#Presentation}

If all the $G_i$ have [[group presentations]], then their free product $\star_i G_i$ has a [[group presentation]] as follows:

Assume $G_i = \langle S_i|R_i\rangle = F_i/N_i$, where

1. each $F_i=\langle S_i\rangle$ is the [[free group]] on a [[set]] $S_i$

1. $N_i\subset F_i$ is the [[normal subgroup]] of $F_i$ generated by the subset $R_i\subset F_i$,

then the free product

$$
  \star_i G_i
    \;\coloneqq\;
  \left\langle
    \coprod_i S_i | \coprod_i R_i
  \right\rangle
  \;=\;
  (\star_i F_i)/\langle\cup_i N_i\rangle
$$
is [[group presentation|presented]] by the [[disjoint unions]] of the $S_i$ and the $R_i$.

(As with anything satisfying a [[universal property]], the result does not depend on the presentations, up to unique [[isomorphism]].)

### Existence

Free products of groups always exist.

With the [above](#Presentation) discussion of presentations this follows from the fact that any group $G_i$ has a presentation: we can always take $S_i$ to be the [[underlying set]] of $G_i$ and take $R_i$ to be the set of all [[words]] in $F_i$ that become [[equality|equal]] to the [[neutral element]] in $G_i$.

The value of the more general construction above is that one often has much smaller $S_i$ and $R_i$ to work with.  Even if $G_i$ is infinite, the $S_i$ and $R_i$ might be [[finite set|finite]] (in the strictest sense), making this part of [[finite mathematics]] and directly subject to the methods of [[combinatorial group theory]].

## Related concepts

* [[free group]]

* [[pushout]]

* [[amalgamation]]

* [[direct product of groups]]

* [[semidirect product group]]

* [[central product of groups]]

## References

The notion of amalgamated free products of groups seems to be due to

* [[Otto Schreier]], §1 in: *Die Untergruppen der freien Gruppen*, Abh. Math. Semin. Univ. Hambg. **5** (1927) 161–183 [[doi:10.1007/BF02952517](https://doi.org/10.1007/BF02952517)]

where it is discussed (though not under any name) complete with the [[universal property]] of what today we recognize as a [[colimit]] (long before this [[category theory]]-notion was formulated in generality).

The actual terminology "amalgamated free product" seems to be due to:

* [[Hanna Neumann]], *Generalized Free Products with Amalgamated Subgroups*, American Journal of Mathematics, **71** 3 (1949) 491-540 [[jstor:2372346](https://www.jstor.org/stable/2372346)]

* [[B. Hanna Neumann]], *An Essay on Free Products of Groups with Amalgamations*, Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences **246** 919 (1954) 503-554 [[jstor:91573](https://www.jstor.org/stable/91573)]

Textbook accounts of free products of groups:

* [[Anthony Knapp]], Section VII.3 of: *Basic Algebra*, Springer (2006) [[doi:10.1007/978-0-8176-4529-8](https://doi.org/10.1007/978-0-8176-4529-8), [pdf](https://www.math.mcgill.ca/darmon/courses/19-20/algebra2/knapp.pdf)]

Expository review in the generality of amalgamated free products and making explicit their nature as [[pushouts]] in [[Grp]]:

* Abhay Chandel, *Free groups and amalgamated product*, BSc thesis (2013) [[pdf](http://home.iiserb.ac.in/~kashyap/Group/thesis_abhay.pdf), [[Chandel-AmalgamatedProducts.pdf:file]]]

See also

* [[eom]], *[Amalgam of groups](https://encyclopediaofmath.org/wiki/Amalgam_of_groups)*

Discussion of [[subgroups]] of amalgamated from products:

* A. Karrass, D. Solitar, *The Subgroups of a Free Product of Two Groups with an Amalgamated Subgroup*, Transactions of the American Mathematical Society **150** 1 (1970) 227-255 [[doi:10.2307/1995492](https://doi.org/10.2307/1995492)]

[[!redirects free product]]
[[!redirects free products]]

[[!redirects amalgamated free product]]
[[!redirects amalgamated free products]]

[[!redirects amalgamated free product of groups]]
[[!redirects amalgamated free products of groups]]

[[!redirects amalgamation of groups]]
[[!redirects amalgamations of groups]]


free resolution

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

#Contents#
* table of contents
{:toc}

## Idea

The [[homological resolution]] of a [[chain complex]] by a complex of [[free modules]].

## Properties

### Existence
 {#Existence}

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

For $R$ a [[ring]], every $R$-[[module]] admits a free resolution.

=--

## Examples

+-- {: .num_prop #AbelianGroupHasFreeResolutionOfLength2}
###### Proposition

Assuming the [[axiom of choice]], then every [[abelian group]] $A$ admits a free resolution of just length 2, hence with trivial [[syzygies]], hence there exists a [[short exact sequence]] of abelian groups of the form

$$
  0 \to F_2 \longrightarrow F_1 \longrightarrow A \to 0
$$

where both $F_1$ and $F_2$ are [[free abelian groups]].

=--

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

Let $F_1 = \mathbb{Z}[A]$ be the [[free abelian group]] on the underlying [[set]] of $A$, and let $F_1 \to A$ be the canonical morphism that sends a generator to itself (the [[adjunction counit]] of the [[free-forgetful adjunction]]).

Then let $F_2 \to F_1$ be the [[kernel]] of that map, hence $F_2$ a [[subgroup]] of $F_1$. Assuming the [[axiom of choice]] then every [[subgroup]] of a [[free abelian group]] is itself free abelian, hence $F_2$ is free abelian ([prop.](free+abelian+group#SubgroupsOfFreeAbelianGroupsAreFree)).

=--

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

Prop. \ref{AbelianGroupHasFreeResolutionOfLength2} implies that all [[Ext]]-groups between abelian groups are concentrated in degrees 0 and 1.

(By the discussion at _[[derived functor in homological algebra]]_.)

=--

+-- {: .num_example}
###### Example
**([[Koszul complex]] is free resolution of [[quotient ring]])

For $R$ a [[commutative ring]] and $(x_1, \cdots, x_d)$ a [[regular sequence]] of elements, then the [[Koszul complex]] $K(x_1, \cdots, x_d)$ is a free resolutions of the [[quotient ring]] $R/(x_1, \cdots, x_d)$ by free $R$-modules.

=--

## Related concepts

* [[Adams resolution]]

* [[projective object]], [[projective presentation]], [[projective cover]], [[projective resolution]]

  * [[projective module]]

* [[injective object]], [[injective presentation]], [[injective envelope]], [[injective resolution]]

  * [[injective module]]

* [[free object]], **free resolution**

  * [[free module]]

* flat object, [[flat resolution]]

  * [[flat module]]

## References

Lecture notes include for instance

around page 5 of

* [[Andrew Baker]], _Notes on chain complexes_ ([pdf](http://www.maths.gla.ac.uk/~ajb/dvi-ps/chaincomplexes.pdf))

[[!redirects free resolutions]]


free simplicial abelian group

#Contents#
* table of contents
{:toc}

## Definition

The __free simplicial abelian group functor__
$$\mathbf{Z}[-]\colon sSet \to sAb$$
from [[SimplicialSets]] to [[SimplicialAbelianGroups]]
is given by the functor
$$sSet = Fun(\Delta^{op}, Set) \to Fun(\Delta^{op}, Ab) = sAb,$$
where the middle functor applies the [[free abelian group functor]]
$$\mathbf{Z}[-]\colon Set \to Ab.$$

## Properties

\begin{proposition}\label{FreeSimplicialAbelianGroupAdjunction}
**([[free simplicial abelian group]]-[[adjunction]])** \linebreak
  There is a [[pair]] of [[adjoint functors]]
  (a [[free functor|free]]$\dashv$[[forgetful]]-[[adjunction]])
  $$
    sAb
      \underoverset
        {\underset{ frgt }{\longrightarrow}}
        {\overset{\mathbb{Z}(-)}{\longleftarrow}}
        {\;\;\;\;\;\bot\;\;\;\;\;}
    sSet
  $$

  between [[SimplicialAbelianGroups]] and [[SimplicialSets]], where

  * the [[left adjoint]] is the [[free simplicial abelian group]]-[[functor]];

  * the [[right adjoint]] is the [[functor]] that assigns [[underlying]] [[simplicial sets]].

  This is a [[Quillen adjunction]] with respect to the [[classical model structure on simplicial sets]] and the projective [[model structure on simplicial abelian groups]].
\end{proposition}

## Applications

Free simplicial abelian groups
are the crucial ingredient of [[simplicial chains]] and [[simplicial cochains]],
and such also [[simplicial homology]] and [[simplicial cohomology]],
in particular, [[singular homology]] and [[singular cohomology]].
See these articles for more information.

## Related entries

* [[free abelian group]]

* [[simplicial chains]]

* [[simplicial cochains]]

[[!redirects free simplicial abelian group functor]]
[[!redirects free simplicial abelian group functors]]
[[!redirects free simplicial abelian groups]]

[[!redirects free simplicial abelian group functor]]


free spectrum

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

#Contents#
* table of contents
{:toc}

## Idea

For [[sequential spectra]] and for [[highly structured spectra]] such as [[symmetric spectra]] and [[orthogonal spectra]], the [[functor]] $(-)_n$ which picks their $n$th component space, for any $n \in \mathbb{N}$, has a [[left adjoint]] $F_n$.

A structured spectrum in the image of this [[free functor]] is called a _free symmetric spectrum_ or _free orthogonal spectrum_, respectively ([Hovey-Shipley-Smith 00, def. 2.2.5](HoveyShipleySmith00), [Mandell-May-Schwede-Shipley 01, section 8](#MandellMaySchwedeShipley01), [Schwede 12, example 3.20](#Schwede12)).

For a general abstract account see at _[[Model categories of diagram spectra]]_ the section _[Free spectra](Model+categories+of+diagram+spectra#FreeSpectra)_.

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

Explicitly, these free spectra look as follows:

For [[sequential spectra]]: $(F_n K)_q = K \wedge S^{q-n}$;

for [[orthogonal spectra]]: $(F_n K)_q = O(q)_+ \wedge_{O(q-n)} K \wedge S^{q-n}$;

for [[symmetric spectra]]: $(F_n K)_q = \Sigma(q)_+ \wedge_{\Sigma(q-n)} K \wedge S^{q-n}$.

=--

## Properties

For $n = 0$ the free construction is [[isomorphism|isomorphic]] to the corresponding structured [[suspension spectrum]] construction: $F_0 \simeq \Sigma^\infty$. Generally, the [[stable homotopy type]] of $F_n K$ is that of $\Omega^n (\Sigma^\infty K)$ ([Schwede 12, example 4.35](#Schwede12)).

## References

* {#HoveyShipleySmith00} [[Mark Hovey]], [[Brooke Shipley]], [[Jeff Smith]], _Symmetric spectra_, J. Amer. Math. Soc. 13 (2000), 149-208 ([arXiv:math/9801077](http://arxiv.org/abs/math/9801077))

* {#MandellMaySchwedeShipley01} [[Michael Mandell]], [[Peter May]], [[Stefan Schwede]], [[Brooke Shipley]], _[[Model categories of diagram spectra]]_, Proceedings of the London Mathematical Society, 82 (2001), 441-512 ([pdf](http://www.math.uchicago.edu/~may/PAPERS/mmssLMSDec30.pdf))

* {#Schwede12} [[Stefan Schwede]], section I.3 of _[[Symmetric spectra]]_ (2012)

[[!redirects free spectra]]

[[!redirects free structured spectrum]]
[[!redirects free structured spectra]]

[[!redirects free symmetric spectrum]]
[[!redirects free symmetric spectra]]

[[!redirects free orthogonal spectrum]]
[[!redirects free orthogonal spectra]]


free strict cocompletion

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### Limits and colimits
+--{: .hide}
[[!include infinity-limits - contents]]
=--
#### Yoneda lemma
+--{: .hide}
[[!include Yoneda lemma - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The [[small presheaf]]-construction exhibits the [[free cocompletion]] of a [[locally small category]] (only) up to [[equivalence of categories]]. In other words, it exhibits a [[universal property]] that is [[bicategorical]] in nature. However, it is also possible to give a description of the free *strict* cocompletion, i.e. one satisfying a [[strict 2-category|strict 2-categorical]] universal property, in the sense of $Cat$-[[enriched category theory]], which characterises the free cocompletion up to an [[isomorphism of categories]].

Explicitly, the small presheaf construction exhibits a left [[pseudoadjoint]] to the forgetful 2-functor from the 2-category of [[locally small]] [[cocomplete categories]], whereas a strict free cocompletion exhibits a left [[2-adjoint|strict 2-adjoint]] (or left $Cat$-enriched adjoint).

Abstractly, this follows by the results of [Kelly and Lack (2000)](#KellyLack2000). However, there is also an explicit description, due to [Ehresmann (1981)](#Ehresmann1981), and expanded upon by [Beurier, Pastor & Guitart (2021)](#BPG2021). The idea is to give a much more "naïve" description of the free cocompletion of a category $\mathscr{C}$ whose objects are not presheaves, but simply [[diagrams]] into $\mathscr{C}$. While the objects are, in some sense, simpler than presheaves, the morphisms are considerably more complicated: their dual is called called **atlases** by [Ehresmann (1981)](#Ehresmann1981), and they are called **clusters** by [Ehresmann and Vanbremeersch (1987)](#EV1987). The free strict cocompletion is consequently called the **category of clusters** $Clu(\mathscr{C})$ by [Beurier, Pastor, and Guitart (2021)](#BPG2021).

The morphisms can be seen to satisfy the **limit--colimit formula** ([Beurier–Pastor–Guitart](#BPG2021), Proposition 3.11): given $P \colon \mathscr{P} \to \mathscr{C}$ and $Q \colon \mathscr{Q} \to \mathscr{C}$, we have $Clu(\mathscr{C})(P, Q) \,\cong\, lim_{p \in \mathscr{P}} colim_{q \in \mathscr{Q}} \mathscr{C}\big(P(p), Q(q)\big)$.

The category of clusters embeds [[fully faithfully]] into the [[small presheaf category]]. Its full image is called $LClu(\mathscr{C})$ by [Beurier, Pastor & Guitart (2021)](#BPG2021).

That the category of clusters forms the free strict cocompletion was stated in [Ehresmann (1981)](#Ehresmann1981) and [Ehresmann and Vanbremeersch (1987)](#EV1987), but without detailed proofs. A detailed treatment is given by [Beurier, Pastor &Guitart (2021)](#BPG2021).

[Ehresmann (1981)](#Ehresmann1981) also considers the restriction to the free strict cocompletion under a specified class $\mathscr{J}$ of [[diagram]] shapes: this is given by restricting the objects of the category of clusters.

## Universal property

See [Beurier–Pastor–Guitart](#BPG2021), Theorem 4.4.

Consider a [[locally small category]] $\mathscr{C}$,
its category of clusters $\mathscr{F}=Clu(\mathscr{C})$ (as desribed above),
and the canonical inclusion $I\colon\mathscr{C}\to Clu(\mathscr{C})$ that sends an object of $\mathscr{C}$ to a diagram indexed by the terminal category.
The triple $(\mathscr{C},\mathscr{F},I)$ has the following additional canonical structure: for every [[small diagram]] $P\colon\mathscr{P}\to\mathscr{C}$ we have a canonical colimit cocone $\lambda^P\colon I P\Rightarrow \lambda(I P)$ in $\mathscr{F}$.
Here $\lambda(I P)$ is an object of $\mathscr{F}$ given by the diagram $P$ itself, interpreted as an object of $\mathscr{F}$.

The free strict cocompletion $Clu(\mathscr{C})$ of a [[locally small category]] $\mathscr{C}$ satisfies the following universal property: given another such triple $(\mathscr{F}',I',\lambda')$, where $I'\colon \mathscr{C}\to\mathscr{F}'$ is a functor landing in a locally small cocomplete category $\mathscr{F}'$ and $\lambda'$ is a choice of a colimit cocone for every diagram of the form $I' P'$ ($P'\colon \mathscr{P}'\to \mathscr{C}$), there is a unique functor $J\colon\mathscr{F}\to\mathscr{F}'$ such that $J I = I'$ and $J$ sends cocones in $\lambda$ to the corresponding cocones in $\lambda'$.

In particular, $\mathscr{F}$ is unique up to an isomorphism.

## Properties

The assignment $\mathscr{C}\mapsto Clu(\mathscr{C})$ yields a __strict__ 2-functor from locally small categories to locally small categories that implements the free cocompletion construction.

This stands in contrast to the usual construction of [[small presheaves]], which only yields a pseudofunctor.

By restricting the types of diagrams in the construction of $Clu(\mathscr{C})$, we get strict cocompletion functors for certain types of colimits, e.g., [[ind-completions]].

## References

* {#Ehresmann1981} [[Andrée Ehresmann]], _Pro-objects and atlases_, Comments 199.1...3, p. 368. in [[Charles Ehresmann]], _Œuvres complètes et commentées. Vol. IV-1. Esquisses et complétions,
Amiens 1981, supplément no. 1 au volume XXII (1981) des [[Cahiers de topologie et géométrie différentielle]] ([pdf](https://mes-ehres.fr/ChEh/oeuvres/Ehresmann_C.-Oeuvres_IV_1.pdf#page=390))

* {#EV1987} [[Andrée Ehresmann]] and J-P. Vanbremeersch, _Hierarchical Evolutive Systems: A mathematical model for complex systems_, Bulletin of Mathematical Biology 49.1 (1987): 13-50.

* {#KellyLack2000} [[G. M. Kelly]], [[Stephen Lack]], _On the monadicity of categories with chosen colimits_, Theory and Applications of Categories **7** 7 (2000) 148-170 [[TAC](http://www.tac.mta.ca/tac/volumes/7/n7/7-07abs.html)]

* {#BPG2021} Erwan Beurier, Dominique Pastor, [[René Guitart]] _Presentations of clusters and strict free-cocompletions_, [[Theory and Applications of Categories]] **36** 17 (2021) 492-513 [[tac:36-17](http://www.tac.mta.ca/tac/volumes/36/17/36-17abs.html)]

[[!redirects free strict cocompletions]]
[[!redirects strict free cocompletions]]
[[!redirects category of clusters]]
[[!redirects strict free cocompletion]]


free topos

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

#Contents#
* table of contents
{:toc}

## Idea

Given a [[higher-order logic|higher-order]] [[intuitionistic type theory]] $\mathfrak{L}$, it is possible to construct a [[topos]] $T(\mathfrak{L})$ out of the [[syntax]] of $\mathfrak{L}$. The **free topos** $T(\mathfrak{L}_0)$ is the result of this construction when $\mathfrak{L}_0$ is 'pure' type theory i.e. the only types are 1 ([[unit type]]), $N$ ([[natural numbers type]]), and $\Omega$ ([[type of propositions]]) lacking relations beyond the bare necessities.

As $\mathfrak{L}_0$ is an [[initial object]] in the appropriate category of type theories, the free topos $T(\mathfrak{L}_0)$ is itself initial in the corresponding category of toposes (with a natural numbers object) and suitable [[logical morphisms]] and is, therefore, also known as the **initial topos**[^sga4].

[^sga4]: This is not to be confounded with the gadget of the same name in [[SGA4]] (1972, p.313) i.e. the topos $sh(\emptyset)$ of sheaves on the empty topological space aka the one point category, also called the _empty topos_ there. In this context the appropriate maps are [[geometric morphisms]].

## Properties

The [[internal language]] of the free topos is precisely pure (intuitionistic) higher order type theory. In 1978, [[Jim Lambek]] and [[Phil Scott]] exploited this connection in order to prove properties of intuitionistic type theory by [[proof theory|proof-theoretic]] means. It was observed by [[Peter Freyd]] then that the concept of a [[Freyd cover]] permits to give conceptual proofs of their findings. The following lemma and proposition is a replication of his ideas.

+-- {: .num_lemma}
###### Lemma
For any [[category]] $C$ with a [[terminal object]] $\mathbf{1}$, the terminal object of the [[Freyd cover]] $\widehat{C}$ is [[small-projective]], i.e., the representable $\Gamma = \widehat{C}(1, -) \colon \widehat{C} \to Set$ preserves any colimits that exist.
=--

+-- {: .proof}
###### Proof
To check that $\Gamma^{op} \colon \widehat{C}^{op} \to Set^{op}$ preserves limits, it suffices to check that the composite

$$\widehat{C}^{op} \stackrel{\Gamma^{op}}{\to} Set^{op} \stackrel{2^-}{\to} Set$$

preserves limits, because the contravariant power set functor $P = 2^-$ is monadic. But it is easily checked that this composite is the contravariant representable given by $(2, \mathbf{1}, 2 \to \Gamma(\mathbf{1}))$.
=--

+-- {: .num_theorem}
###### Theorem
The [[terminal object]] in the free topos $\mathcal{T}$ is [[connected object|connected]] and [[projective object|projective]] in the sense that $\Gamma = \hom(1, -) \colon \mathcal{T} \to Set$ preserves finite [[colimits]].
=--

+-- {: .proof}
###### Proof
We divide the argument into three segments:

* The [[hom-functor]] preserves [[finite limits]], so by general properties of [[Artin gluing]], the [[Freyd cover]] $\widehat{\mathcal{T}}$ is also a topos. Observe that $\mathcal{T}$ is equivalent to the [[slice category|slice]] $\widehat{\mathcal{T}}/M$ where $M$ is the object $(\emptyset, \mathbf{1}, \emptyset \to \Gamma(\mathbf{1}))$. Since pulling back to a slice is a logical functor, we have a logical functor
$$\pi \colon \widehat{\mathcal{T}} \to \mathcal{T}$$
Since $\mathcal{T}$ is initial, $\pi$ is a retraction for the unique logical functor $i \colon \mathcal{T} \to \widehat{\mathcal{T}}$.

* We have maps $\mathcal{T}(1, -) \to \widehat{\mathcal{T}}(i 1, i-) \cong \widehat{\mathcal{T}}(1, i-)$ (the isomorphism comes from $i 1 \cong 1$, which is clear since $i$ is logical), and $\widehat{\mathcal{T}}(1, i-) \to \mathcal{T}(\pi 1, \pi i-) \cong \mathcal{T}(1, -)$ since $\pi$ is logical and retracts $i$. Their composite must be the identity on $\mathcal{T}(1, -)$, because there is only one such endomorphism, using the Yoneda lemma and terminality of $1$.

Finally, since $\mathcal{T}(1, -)$ is a retract of a functor $\widehat{\mathcal{T}}(1, i-)$ that preserves finite colimits (by the lemma, and the fact that the logical functor $i$ preserves finite colimits), it must also preserve finite colimits.
=--

This is important because it implies that the [[internal logic]] of the free topos satisfies the following properties:

* The *disjunction property*: if "P [[or]] Q" is provable in the empty [[context]], then either P is so provable, or Q is so provable.  (Note that this clearly fails in the presence of [[excluded middle]].)

* The *existence property*: if "there exists an $x\in A$ such that $P(x)$" is provable in the empty context, then there exists a [[global element]] $x\colon 1\to A$ such that $P(x)$ is provable in the empty context.  (Again, this is clearly a [[constructive mathematics|constructivity]] property.)

* The *negation property*: False is not provable in the empty context.

* All numerals in the free topos are "standard", i.e., the global sections functor $\Gamma = \hom(1, -): \mathcal{T} \to Set$ preserves the [[natural numbers object]] $N$ (because $N$ can be characterized in terms of finite colimits and $1$, by a theorem of Freyd).

### Foundations

In the [[foundations of mathematics]], [[Jim Lambek]] proposed to use the free topos as ambient world to do mathematics in; see ([Lambek 2004](#LambekWorld)). Being syntactically constructed, but universally determined, with higher-order [[intuitionistic type theory]] as [[internal language]] he saw it as a reconciliation of the three classical schools of [[philosophy of mathematics]], namely [[formalism]], [[platonism]], and [[intuitionism]]. His latest views on this variant of categorical foundations can be found in ([Lambek-Scott 2011](#LS11)).

### Remark: Two properties of the free topos

Lambek and Scott mention in their 1986 monograph ([pp.viii, 233, 250](#LambekScott86)) two further remarkable properties of the free topos $\mathcal{T}$:

* All maps $R\to R$ in the free topos represent continuous functions. Note that this is not the same as to say that Brouwer's theorem ("all functions $R\to R$ are continuous") is true in the free topos, which is not the case.

* The natural numbers object $N$ is an (external) [[projective object]].

The first result is attributed to [[André Joyal]], presumably unpublished, and for the second claim they refer to an unpublished manuscript by [[Michael Makkai]] and a manuscript by Friedman and Scedrov, presumably ([1983](#FriedmanScedrov83)).

## Related entries

* [[Freyd cover]]

* [[Artin gluing]]

* [[syntactic category]]

* [[adjoint modality]]

## Links

* [MO Brouwer's theorem in the free topos?](http://mathoverflow.net/questions/264652/brouwers-theorem-in-the-free-topos)

* [nForum discussion](https://nforum.ncatlab.org/discussion/6665/free-topos/#Item_0)

## References

The free topos was first constructed by H. Volger. A second construction appears in

* [[Jim Lambek]], _From Types to Sets_ , JPAA **36** (1980) pp.113-164.

Freyd's proof of the above properties appears first in

* [[Jim Lambek]], [[Phil Scott]], _Intuitionist Type Theory and the Free Topos_ , JPAA **19** (1980) pp. 215-257. {#LambekScott80}

On the relation between the proof and the topos-theoretic techniques in the proof see

* A. Scedrov, [[Phil Scott]], _A note on the Friedman Slash and Freyd Covers_ , pp. 443-452 in _The LEJ Brouwer Centenary Symposium_ , North Holland Amsterdam 1982.

* [[Jim Lambek]], [[Phil Scott]], _New proofs of some intuitionistic principles_ , Zeit. für Math. Logik **29** (1983) pp. 493-504.

For textbook accounts of the free topos see

* {#LambekScott86}[[Jim Lambek]], [[Phil Scott]], _[[Introduction to Higher-Order Categorical Logic]]_ , Cambridge UP 1986. ([pdf](https://synrc.com/publications/cat/Category%20Theory/Categorical%20Logic/Lambek%20J.,%20Scott%20P.J.%20Introduction%20to%20Higher%20Order%20Categorical%20Logic.pdf))

* [[Peter Freyd|P. Freyd]], A. Scedrov, _[[Categories, Allegories]]_ , North-Holland Amsterdam 1990.  (1.(10)31, p.192)

* [[Elephant]] F3, to appear.

For dependent choice in intuitionistic type theory:

* {#FriedmanScedrov83}H. M. Friedman, A. Scedrov, _Set existence property for intuitionistic theories with dependent choice_ , APAL **25** no.2 (1983) pp.129-140.

For J. Lambek's views on the role of the free topos in [[foundations of mathematics]] see:

* J. Couture, [[Jim Lambek]], _Philosophical Reflections on the Foundations of Mathematics_ , Erkenntnis **34** (1991) pp.187-209.

* [[Jim Lambek]], _Are the traditional philosophies of mathematics really incompatible?_ , Math. Intelligencer **16**
(1994) pp.56–62.

* {#LambekWorld}[[Jim Lambek]], _What is the world of mathematics?_ , APAL **126** (2004) pp.149-158. ([draft](http://www.math.mcgill.ca/barr/lambek/pdffiles/WorldMath.pdf))

* {#LS11} [[Jim Lambek]], [[Phil Scott]], _Reflections on Categorical Foundations of Mathematics_ , pp.171-185 in  Sommaruga (ed.), _Foundational Theories of Classical and Constructive Mathematics_,  Springer New York 2011. ([draft](https://www.site.uottawa.ca/~phil/papers/LS11.final.pdf))

[[!redirects initial topos]]


free-forgetful adjunction

[[adjunction]] : [[free functor]] $\dashv$ [[forgetful functor]]

[[!redirects free-forgetful adjunctions]]


Freed-Witten anomaly

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###Context###
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The effective bosonic exponentiated [[action functional]] of the [[type II superstring]] [[sigma-model]] for [[open strings]] ending on [[D-branes]] has three factors:

1. the [[higher holonomy]] of the background [[B-field]] over the string 2-dimensional worldsheet;

1. the ordinary [[holonomy]] of the [[Chan-Paton bundle]] on the D-brane along the boundary of the string;

1. the [[Berezin integral|Berezinian]] [[path integral]] over the fermions.

Each single contribution is in general not a globally well defined function on the space of string configurations, instead each is a [[section]] of a possibly non-trivial [[line bundle]] over the configuration space (the last one for instance of the [[Pfaffian bundle]]). Therefore the total action functional is a section of the [[tensor product]] of these three line bundles.

The non-triviality of this tensor product line bundle (as a line bundle [[connection on a bundle|with connection]]) is the _Freed-Witten-Kapustin [[quantum anomaly]]_. The necessary conditions for this anomaly to vanish, hence for this line bundle to be trivializable, is the _Freed-Witten anomaly cancellation condition_.

More precisely, the naive holonomy of an ordinary Chan-Paton [[principal connection]] would be globally well defined. But in order to cancel the anomaly contribution from the other two factors, one may take the Chan-Paton bundle to be a [[twisted bundle]], the twist being the [[B-field]] restricted to the brane. Then its holonomy becomes anomalous, too, but there are then interesting configurations where the product of all three anomalies cancels. This refined argument has been made precise by Kapustin, and so one should probably speak of the _Freed-Witten-Kapustin anomaly cancellation_.

## Details
 {#Details}

We interpret the Freed-Witten-Kapustin mechanism in terms of [[push-forward in generalized cohomology]] in [[topological K-theory]] interpreted in terms of [[KK-theory]] with push-forward maps given by [[dual morphisms]] between [[Poincaré duality C*-algebras]] (based on [Brodzki-Mathai-Rosenberg-Szabo 06, section 7](#BrodzkiMathaiRosenbergSzabo06), [Tu 06](#Tu06)):

Let $i \colon Q \to X$ be a map of [[compact topological space|compact]] [[manifolds]] and let $\chi \colon X \to B^2 U(1)$ modulate a [[circle 2-bundle]] regarded as a [[twisted K-theory|twist for K-theory]]. Then forming [[twisted groupoid convolution algebras]] yields a [[KK-theory]] morphism of the form

$$
  C_{i^\ast \chi}(Q)
  \stackrel{i^\ast}{\longleftarrow}
  C_{\chi}(X)
  \,,
$$

with notation as in [this definition](Poincaré+duality+algebra#CStarAlgebraOf2BundleOnManifold). By [this proposition](Poincaré+duality+algebra#DualOfCompactManifoldWithTwist) the [[dual morphism]] is of the form

$$
  C_{\frac{1}{i^\ast \chi \otimes W_3(T Q)}}(Q)
  \stackrel{i_!}{\longrightarrow}
  C_{\frac{1}{\chi \otimes W_3(T X)}}(X)
  \,.
$$

If we redefine the twist on $X$ to absorb this "quantum correction" as $\chi \mapsto \frac{1}{\chi \otimes W_3(T X)}$ then this is

$$
  C_{i^\ast \chi\frac{W_3(i^\ast T X)}{W_3(T Q)}}(Q)
  \stackrel{i_!}{\longrightarrow}
  C_{\chi}(X)
  \,,
$$

where now we may interpret $\frac{W_3(i^\ast \tau_X)}{W_3(\tau_Q)}$ as the third [[integral Stiefel-Whitney class]] of the [[normal bundle]] $N Q$ of $i$ (see [Nuiten](#Nuiten)).

Postcomposition with this map in [[KK-theory]] now yields a map from the $i^\ast \chi \otimes W_3(N Q)$-[[twisted K-theory]] of $Q$ to the $\chi$-[[twisted K-theory]] of $X$:

$$
  i_! \colon K_{\bullet + W_3(N Q) + i^\ast \chi}(Q) \to K_{\bullet +\chi}(X)
  \,.
$$

If we here think of $i \colon Q \hookrightarrow X$ as being the inclusion of a [[D-brane]] [[worldvolume]], then $\chi$ would be the class of the [[background gauge field|background]] [[B-field]] and an element

$$
  [\xi] \in K_{\bullet + W_3(N Q) + i^\ast \chi}(Q)
$$

is called (the K-class of) a _[[Chan-Paton gauge field]]_ on the D-brane satisfying the **Freed-Witten-Kapustin anomaly cancellation** mechanism. (The orginal **Freed-Witten anomaly cancellation** assumes $\xi$ given by a [[twisted unitary bundle|twisted line bundle]] in which case it exhibits a [[twisted spin^c structure]] on $Q$.) Finally its [[fiber integration|push-forward]]

$$
  [i_! \xi] \in K_{\bullet + \chi}(X)
$$

is called the corresponding _[[D-brane charge]]_.

## Related concepts

* [[D-brane]], [[D-brane charge]], [[fractional D-brane]]

* [[twisted spin^c structure]]

* [[Spin^c Dirac operator]]

* [[Green-Schwarz mechanism]]

[[!include orientations in higher quantization - table]]

## References
 {#References}

### General

The special case where the class of the restriction of the [[B-field]] to the D-brane _equals_ the third [[integral Stiefel-Whitney class]] of the D-brane was discussed in

* [[Daniel Freed]], [[Edward Witten]], _Anomalies in String Theory with D-Branes_ ([arXiv:hep-th/9907189](http://arxiv.org/abs/hep-th/9907189))

The generalization to the case that the two classes differ by a [[torsion]] class was considered in

* {#Kapustin99} [[Anton Kapustin]], _D-branes in a topologically nontrivial B-field_, Adv. Theor. Math. Phys. 4, no. 1, pp. 127–154 (2000), ([arXiv:hep-th/9909089](http://arxiv.org/abs/hep-th/9909089))

Aspects of the interpretation of this by [[push-forward in generalized cohomology]] in [[twisted K-theory]] are formalized in

* [[Alan Carey]], [[Bai-Ling Wang]], _Thom isomorphism and Push-forward map in twisted K-theory_ ([arXiv:math/0507414](http://arxiv.org/abs/math/0507414))
 {#CareyWang05}

and section 10 of

* [[Matthew Ando]], [[Andrew Blumberg]], [[David Gepner]], _Twists of K-theory and TMF_, in Robert S. Doran, Greg Friedman, [[Jonathan Rosenberg]], _Superstrings, Geometry, Topology, and $C^*$-algebras_, Proceedings of Symposia in Pure Mathematics [vol 81](http://www.ams.org/bookstore-getitem/item=PSPUM-81), American Mathematical Society ([arXiv:1002.3004](http://arxiv.org/abs/1002.3004))
 {#AndoBlumbergGepner10}

(which discusses twists as [[(infinity,1)-module bundles]]).

The formulation by postcomposition with [[dual morphisms]] in [[KK-theory]] which we use above is based on the observations in section 7 of

* Jacek Brodzki, [[Varghese Mathai]], [[Jonathan Rosenberg]], [[Richard Szabo]], _D-Branes, RR-Fields and Duality on Noncommutative Manifolds_, Commun. Math. Phys. 277:643-706,2008 ([arXiv:hep-th/0607020](http://arxiv.org/abs/hep-th/0607020))
 {#BrodzkiMathaiRosenbergSzabo06}

and generalized to equivariant [[KK-theory]] in

* [[Jean-Louis Tu]], _Twisted K-theory and Poincaré duality_ ([arXiv:math/0609556](http://arxiv.org/abs/math/0609556))
 {#Tu06}

A clean formulation and review is provided in

* Loriano Bonora, Fabio Ferrari Ruffino, Raffaele Savelli, _Classifying A-field and B-field configurations in the presence of D-branes_ ([arXiv:0810.4291](http://arxiv.org/abs/0810.4291))

* Fabio Ferrari Ruffino, _Topics on topology and superstring theory_ ([arXiv:0910.4524](http://arxiv.org/abs/0910.4524))

* Fabio Ferrari Ruffino, _Classifying A-field and B-field configurations in the presence of D-branes - Part II: Stacks of D-branes_ ([arXiv:1104.2798](http://arxiv.org/abs/1104.2798))

* Raffaele Savelli, _On Freed-Witten Anomaly and Charge/Flux quantization in String/F Theory_, Phd thesis (2011) ([pdf](http://digitallibrary.sissa.it/bitstream/handle/1963/4999/PhDthesisSavelli.pdf?sequence=1))

and

* Kim Laine, _Geometric and topological aspects of Type IIB D-branes_, Master thesis ([arXiv:0912.0460](http://arxiv.org/abs/0912.0460))
 {#Laine}

In ([Laine](#Laine)) the discussion of FW-anomaly cancellation with finite-rank gauge bundles is towards the very end, culminating in equation (3.41).

A discussion from the point of view of [[higher geometric quantization]] or [[extended prequantum field theory]] is at the end of

* [[Domenico Fiorenza]], [[Hisham Sati]], [[Urs Schreiber]], _[[schreiber:A higher stacky perspective on Chern-Simons theory]]_ .

Lecture notes along these lines are in _[Lagrangians and Action functionals -- 3d Chern-Simons theory](#3dCSTheory)_ of

* _[[geometry of physics]]_ .

The [[KK-theory]]-description of the FEK anomaly used above is discussed in

* [[Joost Nuiten]], _[[schreiber:master thesis Nuiten|Cohomological quantization of local prequantum boundary field theory]]_, master thesis, August 2013
 {#Nuiten}

### Relation to shifted C-field flux

On relating the [[Freed-Witten anomaly]] to the [[shifted C-field flux quantization]]:

On [[D4-branes]]:

* [[Edward Witten]], Section 2 of: _Duality Relations Among Topological Effects In String Theory_, JHEP 0005:031, 2000 ([arXiv:hep-th/9912086](https://arxiv.org/abs/hep-th/9912086))

On [[D6-branes]]:

* [[Sergei Gukov]], [[James Sparks]], p. 21 of: _M-Theory on $Spin(7)$ Manifolds_, Nucl. Phys. B625 (2002) 3-69 ([arXiv:hep-th/0109025](https://arxiv.org/abs/hep-th/0109025))

* [[James Sparks]], _Global Worldsheet Anomalies from M-Theory_, JHEP 0408:037, 2004 ([arXiv:hep-th/0310147](https://arxiv.org/abs/hep-th/0310147))

[[!redirects Freed-Witten anomaly cancellation]]

[[!redirects Freed-Witten quantum anomaly]]
[[!redirects Freed-Witten quantum anomaly cancellation]]

[[!redirects Freed-Witten-Kapustin anomaly]]
[[!redirects Freed-Witten-Kapustin anomaly cancellation]]

[[!redirects Freed-Witten-Kapustin quantum anomaly]]
[[!redirects Freed-Witten-Kapustin quantum anomaly cancellation]]


Freek Wiedijk

* [webpage](http://www.cs.ru.nl/~freek/)

## Selected writings

* [[Freek Wiedijk]], _Is ZF a hack? Comparing the complexity of some (formalist interpretations of) foundational systems for mathematics_  ([pdf](http://www.cs.ru.nl/~freek/zfc-etc/zfc-etc.pdf))

  **Abstract** This paper presents [[Automath]] encodings (which also are
valid in [[LF]]/λP) of various kinds of foundations of mathematics. Then it compares these encodings according to their size, to find out which foundation is the simplest.

  The systems analyzed in this way are two kinds of [[set theory]] (ZFC and NF), two systems based on Church's [[higher order logic]] ([[Isabelle]]/Pure and [[HOL]]), three kinds of [[type theory]] (the [[calculus of constructions]], Luo's extended calculus of constructions, and Martin-Löf predicative type theory) and one foundation based on category theory. The conclusions of this paper are that the simplest system is type theory (the calculus of constructions) but that type theories that know about serious mathematics are not simple at all. Set theory is one of the simpler systems too. Higher order logic is the simplest if one looks at the number of concepts (twenty-five) needed to explain the system. On the other side of the scale, category theory is relatively complex, as is Martin-Löf's type theory.

On [[mathematical structures]] formulated in [[dependent type theory]], specifically in [[Coq]]:

* [[Herman Geuvers]], [[Randy Pollack]], [[Freek Wiedijk]], [[Jan Zwanenburg]], *A Constructive Algebraic Hierarchy in Coq*, Journal of Symbolic Computation **34** 4 (2002) 271-286 [[doi:10.1006/jsco.2002.0552](https://doi.org/10.1006/jsco.2002.0552), [pdf](http://www.cs.ru.nl/~herman/PUBS/JSC2002-GeuversPollackWiedijkZwanenburg-alghier1.pdf)]

On [[constructive analysis]] with exact [[real numbers]] via [[type theory]]:

* [[Herman Geuvers]], [[Milad Niqui]], [[Bas Spitters]], [[Freek Wiedijk]], _Constructive analysis, types and exact real numbers_, Mathematical Structures in Computer Science **17** 01 (2007) 3-36 [[doi:10.1017/S0960129506005834](https://doi.org/10.1017/S0960129506005834)]

## Related entries

* [[foundations of mathematics]]

category: people


Freeman Dyson

Physicist, co-founder of [[perturbative quantum field theory]] (see [[Schwinger-Tomonaga-Feynman-Dyson]])

* [webpage](http://www.sns.ias.edu/dyson)

## selected writings

* {#MissedOpportunities} Freeman Dyson, _Missed opportunities_, Bulletin of the AMS, Volume 78, Number 5, September 1972 ([pdf](https://www.math.uh.edu/~tomforde/Articles/Missed-Opportunities-Dyson.pdf))

  containing the now famous quote:

  > the  marriage   between mathematics and  physics, which was so enormously fruitful in past centuries, has recently ended in divorce.

  which is in reference to

  > The first clear sign of a breakdown in communication between physics and mathematics was the extraordinary lack of interest among mathematicians in [[James Clerk Maxwell]]'s discovery  of the laws of [[electromagnetism]]

  For more quotes from this text see also at _[Ausdehnungslehre](Ausdehnungslehre#Dyson72)_ and at _[AQFT on curved spacetime](AQFT+on+curved+spacetimes#Dyson72)_.

* {#Dyson52} [[Freeman Dyson]], _Divergence of perturbation theory in quantum electrodynamics_, Phys. Rev. 85, 631 (1952)

  which is the origin of mathematical [[perturbative quantum field theory]], see [[Schwinger-Tomonaga-Feynman-Dyson]].

* {#Dyson07} [[Freeman Dyson]], _Why is Maxwell’s Theory so hard to understand?_, Proceedings of [The Second European Conference on Antennas and Propagation, EuCAP 2007](https://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=4446147) ([doi: 10.1049/ic.2007.1146](https://ieeexplore.ieee.org/xpl/mostRecentIssue.jsp?punumber=4446147))

On whether/how [[experiments]] can detect [[gravitons]]:

* [[Freeman Dyson]], *Is the graviton detectable?*, International Journal of Modern Physics A **28** 25 (2013) 1330041 [[doi:10.1142/S0217751X1330041X](https://doi.org/10.1142/S0217751X1330041X)]

## Related entries

* [[perturbative quantum field theory]]

* [[Schwinger-Tomonaga-Feynman-Dyson]]

* [[Dyson series]]

* [[stability of matter]]

* [[perturbation theory]]

* [[Maxwell's equations]]

category: people

[[!redirects Dyson]]


Frenet-Serret formulas

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

## Definition

Given a [[curve]] in $n$-[[dimension|dimensional]] [[Euclidean space]] $E^n$, with a $C^{n+1}$-[[differentiable function|differentiable]] parametrization by the [[arc length]] from a given point on the curve

$$
  \array{
    F &\colon& I & \longrightarrow & E^n
    \\
    && s &\mapsto& F(s)
  }
$$

(hence a [[naturally parametrized curve]] of smoothness class $C^{n+1}$), __Frenet-Serret formulas__ express the [[derivatives]] with respect to the [[arc length]] of the Frenet moving frame vectors as a [[linear combination]] of the Frenet moving frame vectors, where the coefficients form an antisymmetric matrix whose nonzero coefficients are (up to sign) the curvature of the curve, its torsion and their higher analogues.

This works under the assumption that the [[derivatives]]

$$
  F^{(k)}(s)
    \;=\;
  d^k F/d s^k
    \;\in\;
  T^k_{F(s)} E^n
    \;\cong\;
  \mathbf{R}^n,
  \,\,\,\,\,
  k = 1,\ldots,n
$$

are [[linear independence|linearly independent]] at each point (using the canonical identification with $\mathbf{R}^n$ with any of its [[tangent spaces]]).

The [[Gram-Schmidt orthogonalization]] procedure then transforms the [[n-tuple]] $(F^{(1)}(s),\ldots,F^{(n)}(s))$ into an [[orthogonality|orthogonal]] $n$-frame $(V_1(s),\ldots,V_n(s))$ called the __Frenet $n$-frame__.

Writing down the $n$-tuple of the derivatives with respect to $s$ of the Frenet frame in a $C^0(\mathbf{R})$-basis formed by the Frenet frame reveals an antisymmetric matrix of coefficients over $C^0(\mathbf{R})$. Its nonzero coefficients are, up to a sign, the _curvature_ of the curve, the _torsion_ and their higher analogues.

In dimension $n=3$, the Frenet frame vectors are called the _tangent_, _normal_ and _binormal_ unit vectors of the curve. In dimension 4 the 4th vector is called the _trinormal_ unit vector. The plane spanned by the tangent and normal unit vectors is called the _osculating plane_ at $F(s)$.

## References

* [[Herman Gluck]], _Higher curvatures of curves in Euclidean space_, The American Mathematical Monthly, 73:7 (1966) 699-704, [doi](https://doi.org/10.1080/00029890.1966.11970818)

See also:

* en.wikipedia/[Frenet-Serret_formulas](https://en.wikipedia.org/wiki/Frenet%E2%80%93Serret_formulas)


frequency

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#### Harmonic analysis
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=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Frequency is a [[number]] of occurences of a periodic [[event]] in [[time]].

For [[waves]]: number of wave crests at a given space point per time.

## Definition

For $p \in \mathbb{N}$ and $\mathbb{R}^{p,1} \simeq_{\mathbb{R}} \mathbb{R}^{p+1}$ the [[Cartesian space]] $\mathbb{R}^{p+1}$ regarded as [[Minkowski spacetime]], then the _frequency_ of a [[plane wave]]

$$
  x \mapsto e^{ - 2 \pi i k_\mu x^\mu}
$$

with respect to the chosen [[coordinate system]] is the 0-component of the given [[wave vector]] $k$:

$$
  \nu \coloneqq k_0
  \,.
$$

The combination

$$
  \omega \coloneqq 2 \pi \nu = 2\pi k_0
$$

is called the _angular frequency_.

[[!include plane waves -- table]]

## Related concepts

* [[wavelength]]

* [[wave vector]]

* [[wave equation]], [[Klein-Gordon equation]]

[[!redirects frequencies]]

[[!redirects angular frequency]]
[[!redirects angular frequencies]]


freshman's dream

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* table of contents
{:toc}

## Idea

Over a [[ring]] whose [[characteristic]] is a [[prime number]] $p$ it is true that

$$
  (a+b)^p = a^p + b^p
  \,,
$$

because all the non-trivial [[binomial theorem|binomial coefficients]] are multiples of $p$.

## Primality

+-- {: .num_lemma #primality}
###### Lemma
An integer $n \geq 2$ is prime if and only if the [[binomial coefficients]] satisfy

$$\binom{n}{k} \equiv 0 \pmod{n}$$

whenever $0 \lt k \lt n$.
=--

+-- {: .proof}
###### Proof
If $p$ is prime, then $p$ divides neither $k!$ nor $(p-k)!$ if $0 \lt k \lt p$. So, since $p$ divides

$$p! = \binom{p}{k}k! \cdot (p-k)!,$$

it must divide $\binom{p}{k}$.

If $n$ is composite, write $n = p q$ where $p$ is any prime dividing $n$, so that $0 \lt p \lt n$. If the factor $p$ appears with multiplicity $j$ in $n$, then it appears with multiplicity $j$ in $n(n - 1) \ldots (n - p + 1)$ and with multiplicity $1$ in $p! = p(p-1)\ldots 1$, and so it appears with multiplicity $j-1$ in $\binom{n}{p}$. It follows that $n$ cannot divide $\binom{n}{p}$, i.e., $\binom{n}{p} \nequiv 0 \pmod{n}$.
=--

## Related concepts

* [[Frobenius morphism]]

## References

* Wikipedia, _[Freshman's dream](http://en.wikipedia.org/wiki/Freshman's_dream)_

[[!redirects freshman dream arithmetic]]
[[!redirects freshman's dream arithmetic]]


Freudenthal magic square

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+-- {: .hide}
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=--
=--
=--

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* table of contents
{:toc}

## Idea

A special square array of [[Lie algebras]]/[[Lie groups]] labeled by pairs of [[normed division algebras]] and including all the [[exceptional Lie groups]] except [[G2]]. Three constructions give this square, namely, the Tits construction, the Vinberg construction and the triality construction (see [BaezMagic](#BaezMagic)).

[[Michael Atiyah]] ([AtiyahMagic](#AtiyahMagic)) has proposed using the magic square to explain the [[Kervaire invariant problem]] similarly to how the existence of the four normed division algebras explains the [[Hopf invariant one]] problem.

## Extensions

* Several mathematicians have constructed a larger triangle of Lie groups, known as the [[magic triangle]], which contains the magic square.

* [[Michael Duff]] and colleagues ([ABDHN](#ABDHN)) have proposed a magic _pyramid_ whose base is the magic square.

* [[Bruce Westbury]] ([Westbury](#Westbury)) has proposed an extension to a $5 \times 5$-square with the new row and column indexed by the [[sextonion|sextonions]].

## Related concepts

* [[magic triangle]]

* [[magic pyramid]]

* [[magic supergravity]]

## References

* {#Freudenthal65} [[Hans Freudenthal]], _Lie groups in the foundations of geometry_, Advances in Mathematics, volume 1, (1965) pp. 145 - 190 ([dspace](http://dspace.library.uu.nl/handle/1874/17442))

* {#SpringerVeltkamp00} [[Tonny Springer]], [[Ferdinand Veldkamp]], chapter 8.5 of _Octonions, Jordan Algebras, and Exceptional Groups_, Springer Monographs in Mathematics, 2000

* {#BaezMagic} [[John Baez]], _4.3 The Magic Square_, [webpage](http://math.ucr.edu/home/baez/octonions/node16.html)

* {#AtiyahMagic} [[Michael Atiyah]], _The Geometry and Topology of the Freudenthal Magic Square_, [videos](http://webhost1.ust.hk/~ias/en/IAS_0910_MATH_seminar_abstract.pdf)

* {#Westbury} [[Bruce Westbury]], _Sextonions and the magic square_, ([pdf](http://homepages.warwick.ac.uk/~masdbn/sex2.pdf))

See also

* Wikipedia, _[Freudenthal magic square](https://en.wikipedia.org/wiki/Freudenthal_magic_square)_

Discussion in relation to [[super Yang-Mills theory]], [[supergravity]] and [[U-duality]] includes

* [[Leron Borsten]], [[Michael Duff]], L. J. Hughes, S. Nagy, _A magic square from Yang-Mills squared_ ([arXiv:1301.4176](http://arxiv.org/abs/1301.4176))

* {#ABDHN} A. Anastasiou, [[Leron Borsten]], [[Michael Duff]], L. J. Hughes, S. Nagy, _A magic pyramid of supergravities_, [arXiv:1312.6523](http://arxiv.org/abs/1312.6523)

See also at _[[supersymmetry and division algebras]]_.

[[!redirects Freudenthal-Tits magic square]]

[[!redirects magic square]]
[[!redirects magic squares]]


Freudenthal suspension theorem

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=--
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=--

#Contents#
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{:toc}

## Statement

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

Let $X$ be an [[n-connected topological space]]. Then the [[adjunction unit]] of the ([[suspension]] $\dashv$ [[loop space]])-[[adjunction]]

$$
  X \longrightarrow \Omega \Sigma X
$$

is [[n-connected continuous function|(2n+1)-connected]].

=--

(e.g. [Kochman 96, prop. 3.2.2](#Kochman96))

The **Freudenthal suspension theorem** ([Freudenthal 37](#Freudenthal37)) is the following theorem about [[homotopy groups]] of [[n-spheres]]:

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

The [[suspension]] [[homomorphism]] on [[homotopy groups of spheres]]

$$
  \pi_{n+k}(S^n) \longrightarrow \pi_{n+k+1}(S^{n+1})
$$

is an [[isomorphism]] for $n\gt k+1$.

More generally, for $X$ an [[n-connected]] [[CW-complex]], then the suspension homomorphism on [[homotopy groups]]

$$
  \pi_k(X) \longrightarrow \pi_{k+1}(\Sigma X)
$$

is an [[isomorphism]] for $k \leq 2n$.

The suspension isomorphism is equivalently given by the map in lemma \ref{SigmaOmegaAdjunctionUnitOnNConnectedSpaceIsHigherConnected}.

=--

e.g. ([Switzer 75, 6.26](#Switzer75))

An alternative proof proceeds from the [[Blakers-Massey theorem]] (e.g. [Kochman 96, p. 70](#Kochman96)).

The following more general statement is also often referred to as the Freudenthal suspension theorem:

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

For $X$ an [[n-connected]] [[CW-complex]] and $Y$ a [[CW-complex]] of [[dimension]] $\leq 2n$, then the maps of [[homotopy classes]] of [[continuous functions]]

$$
  [Y,X]\stackrel{}{\longrightarrow} [\Sigma Y, \Sigma X] \stackrel{}{\longrightarrow}[\Sigma^2 Y, \Sigma^2 X]
$$

are [[isomorphisms]]. In particular $[Y,X]$ canonically has the structure of an [[abelian group]].

=--

(e.g. [Kochman 96, corollary 3.2.3](#Kochman96))

## Properties

### As motivation for stable homotopy theory

The Freudenthal suspension theorem motivated introducing the [[stable homotopy groups of spheres]] $\pi_k(S):=\pi_{n+k}(S^n)$, more generally the [[stable homotopy groups]] $\pi_k^S(Y) = \pi_{n+k}(\Sigma^n Y)$, both independent of $n$ where $n\gt k+1$, and still more generally the [[Spanier-Whitehead category]], then the [[stable homotopy category]] and eventually the [[stable (infinity,1)-category of spectra]].

## Related concepts

* [[suspension isomorphism]]

* [[stem of homotopy groups of spheres]]

## References

Due to

* {#Freudenthal37} [[Hans Freudenthal]],  _Über die Klassen der Sphärenabbildungen_, Compositio Math., 5:299-314, 1937 ([numdam:CM_1938__5__299_0](http://www.numdam.org/item/?id=CM_1938__5__299_0))

Textbook accounts include

* {#Switzer75} [[Robert Switzer]], theorem 6.26 in _Algebraic Topology - Homotopy and Homology_, Die  Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

* {#Kochman96} [[Stanley Kochman]], section 3.2 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

* [[Allen Hatcher]], _[Algebraic Topology](http://www.math.cornell.edu/~hatcher/AT/ATpage.html)_, chapter 4 ([pdf](http://www.math.cornell.edu/~hatcher/AT/ATch4.pdf))

A nice expanded version of the latter is in

* Tengren Zhang, _Freudenthal suspension theorem_  ([pdf](http://www.math.uchicago.edu/~may/VIGRE/VIGRE2009/REUPapers/ZhangTengren.pdf))

A formalization in [[homotopy type theory]] in [[Agda]] is in

* [[Peter Lumsdaine]], [[Dan Licata]], _[Freudenthal.agda](https://github.com/dlicata335/hott-agda/blob/master/homotopy/Freudenthal.agda)_

Discussion in [[equivariant homotopy theory]] includes

* {#GreenleesMay} [[John Greenlees]], [[Peter May]], p. 7,8 of _Equivariant stable homotopy theory_ ([pdf](http://www.math.uchicago.edu/~may/PAPERS/Newthird.pdf))


Freund-Rubin compactification

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###Context###
#### Gravity
+--{: .hide}
[[!include gravity contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In general a _Freund-Rubin compactification_ ([Freund-Rubin 80](#FreundRubin80)) is a [[Kaluza-Klein compactification]] of a theory of [[gravity]] coupled to [[gauge fields]] or more generally [[higher gauge fields]] with [[flux]] ([[field strength]]) on the compact [[fiber]] spaces such that the result is stable (an example of [[moduli stabilization]] via [[flux compactification]]).

One example are [[Kaluza-Klein compactifications]] of 6d [[Einstein-Maxwell theory]] with [[magnetic flux]] on a 2-dimensional [[fiber]] space ([[sphere]] or [[torus]]) ([RDSS 83](#RDSS83)). This serves these days as a toy example for [[flux compactifications]] and [[moduli stabilization]] in [[string theory]].

In the [[string theory]] literature often _the Freund-Rubin compactification_ refers by default to a [[Kaluza-Klein mechanism|Kaluza-Klein compactification]] of [[11-dimensional supergravity]] on a [[manifold]] $X_7$  of [[dimension]] 7 (in the original [[model (in theoretical physics)|model]] a round [[7-sphere]]) with non-vanishing constant 4-form [[field strength]] ("flux") of the [[supergravity C-field]] in the remaining four dimensional [[anti-de Sitter spacetimes]] $AdS_4$.

If $X_7$ has [[weak G2 holonomy]] with weakness parameter/[[cosmological constant]] $\lambda$ the scale of the flux, then this yields $N = 1$ [[supersymmetry]] in the [[effective QFT]] in four dimensions, discussed at _[[M-theory on G2-manifolds]]_. The [[KK-reduction]] on the circle fiber of these solutions to [[type IIA supergravity]] yields type IIA sugra on [[complex projective space]] $\mathbb{C}P^3$ ([Nilsson-Pope 84](#NilssonPope84), [ABJM 08](#ABJM08))

If $X_7 = S^7/G_{ADE}$ is an [[orbifold]] of the round [[7-sphere]] by an finite group $G_{ADE} \subset SU(2)$ in the [[ADE-classification]], then Freund-Rubin describes the [[near horizon geometry]] of coincident black [[M2-branes]] at an [[ADE-singularity]], see at _[M2-brane -- As a black brane](M2-brane#AsABlackBrane)_.

## Related concepts

* [[Kaluza-Klein mechanism]]

* [[M-theory on G2-manifolds]], [[G2-MSSM]]

* [[AdS-CFT]]

## References

The original article is

* {#FreundRubin80} [[Peter Freund]], M. A. Rubin, _Dynamics Of Dimensional Reduction_, Phys. Lett. B 97, 233 (1980). (doi:10.1016/0370-2693(80)90590-0)

Early developments:

* [[Francois Englert]], _Spontaneous Compactification of Eleven-Dimensional Supergravity_, Phys. Lett. B119 (1982) 339 ([spire:180130](http://inspirehep.net/record/180130))

* [[Riccardo D'Auria]], [[Pietro Fre]], [[Peter van Nieuwenhuizen]], _Symmetry Breaking in $d=11$ Supergravity on the Parallelized Seven Sphere_, Phys. Lett. B 122 (1983) 225 ([spire:181634](https://inspirehep.net/literature/181634), doi:10.1016/0370-2693(83)90689-5)

Identification as [[near horizon geometries]] of [[black brane|black]] [[M2-branes]]:

* {#Page83} [[Don Page]], _Classical stability of round and squashed seven-spheres in eleven-dimensional supergravity_, Phys. Rev. D 28, 2976 (1983) ([spire:14480](http://inspirehep.net/record/14480) [doi:10.1103/PhysRevD.28.2976](https://doi.org/10.1103/PhysRevD.28.2976))

* {#DuffStelle91} [[Mike Duff]], [[Kellogg Stelle]], _Multi-membrane solutions of $D = 11$ supergravity_, Phys. Lett. B 253, 113 (1991) ([spire:299386](http://inspirehep.net/record/299386), doi:10.1016/0370-2693(91)91371-2)

See also

* [[Hermann Nicolai]], [[Krzysztof Pilch]], _Consistent truncation of $d = 11$ supergravity on $AdS_4 \times S^7$_, JHEP 03 (2012) 099 ([arXiv:1112.6131](http://arxiv.org/abs/1112.6131))

A classification of symmetric solutions is discussed in

* [[José Figueroa-O'Farrill]], _Symmetric M-Theory Backgrounds_ ([arXiv:1112.4967](http://arxiv.org/abs/1112.4967))

* Linus Wulff, _All symmetric space solutions of eleven-dimensional supergravity_ ([arXiv:1611.06139](https://arxiv.org/abs/1611.06139))

The class of Freund-Rubin compactifications of 6d [[Einstein-Maxwell theory]] down to 4d is due to

*  {#RDSS83} S. Randjbar-Daemi, [[Abdus Salam]] and J. A. Strathdee, _Spontaneous Compactification In Six-Dimensional Einstein-Maxwell Theory_, Nucl. Phys. B 214, 491 (1983) ([spire](https://inspirehep.net/record/182427/))

now a popoular toy example for [[flux compactifications]] and [[moduli stabilization]] in [[string theory]].

Textbook account (in [[D'Auria-Fre formulation of supergravity|D'Auria-Fré formulation]]):

* {#CastellaniDAuriaFre} [[Leonardo Castellani]], [[Riccardo D'Auria]], [[Pietro Fré]], volume 2, chapters V.4-V.8 of _[[Supergravity and Superstrings - A Geometric Perspective]]_, World Scientific (1991) [[doi:10.1142/0224](https://doi.org/10.1142/0224), ch V.4: [[CastellaniDAuriaFre-ChV4.pdf:file]]]

Discussion of compactification along the fibration $S^1 \to S^7 \to \mathbb{C}P^3$ is in

* {#NilssonPope84} [[Bengt Nilsson]], [[Christopher Pope]], _Hopf Fibration of Eleven-dimensional Supergravity_, Class.Quant.Grav. 1 (1984) 499 ([Spire](http://inspirehep.net/record/202535/))

* {#ABJM08} [[Ofer Aharony]], Oren Bergman, Daniel Louis Jafferis, [[Juan Maldacena]], _$N=6$ superconformal Chern-Simons-matter theories, M2-branes and their gravity duals_ ([arXiv:0806.1218](http://arxiv.org/abs/0806.1218), [[ABJM model]])

Discussion of the case that $X_7$ is an [[orbifold]] or has other [[singularities]] (the case of interest for realistic [[phenomenology]] in [[M-theory on G2-manifolds]]) includes

* [[Bobby Acharya]], F. Denef, C. Hofman, [[Neil Lambert]], _Freund-Rubin Revisited_ ([arXiv:hep-th/0308046](http://arxiv.org/abs/hep-th/0308046))

Specifically, discussion of an [[ADE classification]] of 1/2 [[BPS state|BPS]]-compactifications on $S^7/\Gamma$ for a [[finite group]] $\Gamma$ is in

* [[Paul de Medeiros]], [[José Figueroa-O'Farrill]], [[Sunil Gadhia]], [[Elena Méndez-Escobar]], _Half-BPS quotients in M-theory: ADE with a twist_, JHEP 0910:038,2009 ([arXiv:0909.0163](http://arxiv.org/abs/0909.0163), [pdf slides](http://www.maths.ed.ac.uk/~jmf/CV/Seminars/YRM2010.pdf))

Discussion of [[weak G2 holonomy]] on $X_7$ is in

* [[Adel Bilal]], J.-P. Derendinger, K. Sfetsos, _(Weak) $G_2$ Holonomy from Self-duality, Flux and Supersymmetry_, Nucl.Phys. B628 (2002) 112-132 ([arXiv:hep-th/0111274](http://arxiv.org/abs/hep-th/0111274))

See also:

* [[Ergin Sezgin]], _11D Supergravity on $AdS_4 \times S^7$ versus $AdS_7 \times S^4$_ ([arXiv:2003.01135](https://arxiv.org/abs/2003.01135))

[[!redirects Freund-Rubin compactifications]]


Freyd category

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

#Contents#
* table of contents
{:toc}

## Idea

A **Freyd category** is one way to axiomatize models of [[call-by-value]] [[programming languages]].
It abstracts the structure of the [[Kleisli category]] of a [[monad in computer science|monad]], consisting of a [[category]] $\mathbb{V}$ that models values and another category with the same [[objects]] $\mathbb{C}$ that models computations.

Freyd categories thus provide a [[categorical semantics]] for typed [[programming languages]] with [[side effects]] such as memory access or printing. This is in the spirit of the discussion at [[relation between type theory and category theory]], except that page is for non-side-effecting languages. The types of the programming language are interpreted as objects of the categories. A [[type|typed]] term (i.e., program expression) $\Gamma\vdash t \colon X$ is interpreted as a morphism $\Gamma \to X$ in $\mathbb{C}$. Among the terms, it is often very important to identify those that are "values", which in particular have no side effects, and these are interpreted as morphisms in $\mathbb{V}$. Freyd categories are based around premonoidal categories, which captures the fact that reordering the execution of side effects is often subtle.

Freyd categories resolve the following complaint about using [[monads]] and [[Kleisli category|Kleisli categories]] to model impure effects in programming languages. The Kleisli category for a monad presumes the existence of some slightly higher-order types, since if $X$ is an object then so is $T(X)$, and yet it is surely possible to understand the nature of effectful computation without also assuming the existence of certain types. Freyd categories make sense even for purely first order programming languages, and the object $T(X)$, if it exists, has a universal property, thus decoupling this relationship.

## Definition

A **Freyd category** (following [Levy 04](#Levy04)) may be defined as

* a [[category]] $\mathbb{V}$ with [[finite products]];
* a category $\mathbb{C}$, that [[categories with the same collection of objects|has the same objects]] as $\mathbb{V}$[^sameObjs];
* an [[action]] of $\mathbb{V}$ on $\mathbb{C}$ (with the finite products providing a [[symmetric monoidal category|symmetric monoidal structure]] for $\mathbb{V}$)
* an [[identity-on-objects functor]] $J: \mathbb{V} \to \mathbb{C}$ that preserves the actions.

[^sameObjs]: [[categories with the same collection of objects|Having the same objects]] is required or implied by having an identity-on-objects functor from $\mathbb{V}$ to $\mathbb{C}$.

An alternative but equivalent definition is as follows (following [Power, Thielecke](#PH99)):

A Freyd category is given by

* a category $\mathbb{V}$ with finite products;
* a symmetric [[premonoidal category]] $\mathbb{C}$, that [[categories with the same collection of objects|has the same objects]] as $\mathbb{V}$;
* an identity-on-objects functor $J: \mathbb{V} \to \mathbb{C}$ strictly preserving symmetric premonoidal structure, whose image lies in the centre of $\mathbb{C}$.

These definitions arguably violate the [[principle of equivalence]], because they use an [[identity-on-objects functor]] in the definitions.  That is, if $\mathbb{V}$ is [[equivalent]] to some $\mathbb{V}'$, and $\mathbb{C}$ is equivalent to some $\mathbb{C}'$, while $J: \mathbb{V} \to \mathbb{C}$ is a Freyd category, there may be no way to make a Freyd category $\mathbb{V}' \to \mathbb{C}'$. A resolution of this is to regard a Freyd category as a single structure, with one set of objects, two sets of morphisms, and so on; see [[identity-on-objects functor]] and [[M-category]] for further discussion.

## Properties

### Relation to strong monads.

If $T$ is a [[strong monad]] on a category $\mathbb{V}$ with finite products, then the Kleisli category of $T$ forms a Freyd category and $J$ has a right adjoint.

Conversely, if $(\mathbb{V},\mathbb{C},J)$ is a Freyd category and $J$ has a right adjoint $R$, then the Freyd category arises as the [[Kleisli category]] of the monad $RJ$.

In more detail, the passage from a strong monad to a Freyd category is as follows. If $T$ is a strong monad on a category $\mathbb{V}$ with finite products, then the Kleisli category $\mathbb{C}$ has a premonoidal structure given by $X\otimes Y=X\times Y$ on objects and where for a morphism $f:Y\to Z$ in $\mathbb{C}$ (i.e. $f:Y\to T(Z)$ in $\mathbb{V}$) then the right tensor $(X\rtimes f):X\otimes Y\to X\otimes Z$ is given by the composite
$X\times Y\xrightarrow f X\times T(Z) \xrightarrow {\mathrm{str}} T(X\times Z)$ in $\mathbb{V}$. The left tensor $(f\ltimes X):Y\otimes X\to Z\otimes X$ is given by the composite
$Y\times X\xrightarrow f T(Z)\times X \xrightarrow {\mathrm{str}} T(Z\times X)$ in $\mathbb{V}$. This is a premonoidal structure, and it is monoidal if and only if the monad $T$ is a [[commutative monad]].

### Relation to Lawvere theories and PROPs

To give a small Freyd category is to give an [[enriched Lawvere theory]] relative to the empty [[sound doctrine]] where the enriching category is [[cartesian closed category|cartesian closed]].

For example, if ($\mathrm{FinSet}^{\mathrm{op}}\to \mathbb{T}$) is an ordinary [[Lawvere theory]], then its dual ($\mathrm{FinSet}\to \mathbb{T}^{\mathrm{op}}$) is a Freyd category.

The Lawvere theory is [[commutative algebraic theory|commutative]] if and only if the premonoidal category $\mathbb{T}^{\mathrm{op}}$ is in fact monoidal.

A minor generalization of Freyd category allows $\mathbb{V}$ to be symmetric monoidal rather than cartesian monoidal. Then a commutative Freyd category
$\mathrm{FinSet}_{\mathrm{bij}}\to \mathbb{C}$ is the same thing as a [[PROP]].

### Relation to monads on presheaf categories

This relates to the situation with monads as follows. If $T$ is a strong monad on the [[presheaf]] category $\hat{\mathbb{V}}$ then the
Kleisli category $\hat{\mathbb{V}}\to (\hat{\mathbb{V}})^T$ is a Freyd category.
But also the identity-on-objects/full-and-faithful factorization of the composite $\mathbb{V}\to \hat{\mathbb{V}}\to (\hat{\mathbb{V}})^T$
yields a Freyd category over $\mathbb{V}$.

Every Freyd category arises in this way, giving a correspondence between colimit-preserving strong monads on $\hat{\mathbb{V}}$ and Freyd categories over a category $\mathbb{V}$ with products.

On the other hand, a colimit-preserving monad on $\hat{\mathbb{V}}$ is the [same thing as](profunctor#FuncsOnPresheaves) a monad in the category of profunctors whose carrier is $\mathbb{V}$. But this is the [same thing as](https://ncatlab.org/nlab/show/monad#other_examples) an identity on objects functor $\mathbb{V}\to\mathbb{C}$. And this is a Freyd category if and only if the monad is strong.

## Related Concepts

* [[monad in computer science]]
* [[Kleisli category]]
* [[thunk-force category]]
* [[premonoidal category]]
* [[duploid]]
* [[Freyd multicategory]]

## References

Freyd categories were first used in

* [[John Power]] and Hayo Thielecke "Environments, Continuation Semantics and Indexed Categories",  Theoretical Aspects of Computer Software, Third International Symposium, TACS '97

and named Freyd categories in

* {#PH99} John Power and Hayo Thielecke, "Closed Freyd- and $\kappa$-Categories", Automata, Languages and Programming, 26th International Colloquium, ICALP'99

and the longer journal version of that paper has more discussion:

* Paul Blain Levy, John Power and Hayo Thielecke, "Modelling environments in call-by-value programming languages",  Inf. Comput. 185(2): 182-210, 2003 [preprint pdf](https://www.cs.bham.ac.uk/~hxt/research/envcbv.pdf).

The definition with monoidal actions appears in

* {#Levy04} Paul Blain Levy, "Call-by-Push-Value. A Functional/Imperative Synthesis," Semantic Structures in Computation 2, Springer, 2004; appendix B*

The idea of extending Freyd categories to monads on presheaf categories appears in

* [[John Power]], "Generic models for computational effects",  Theor. Comput. Sci. 364(2): 254-269 (2006)

This also contains the observation that enriched Lawvere theories are examples of Freyd categories, but the precise correspondence between the concepts is given in

* {#Staton14} [[Sam Staton]], 2014. *Freyd categories are Enriched Lawvere Theories*. Electronic Notes in Theoretical Computer Science, Proceedings of the Workshop on Algebra, Coalgebra and Topology (WACT 2013) 303, 197–206. [doi:10.1016/j.entcs.2014.02.010](https://doi.org/10.1016/j.entcs.2014.02.010) (free).

The connection with monads in the category of profunctors is discussed in

* [[Bart Jacobs]], [[Chris Heunen]], Ichiro Hasuo, "Categorical semantics for arrows". J. Funct. Program. 19(3-4): 403-438. 2009.
[preprint pdf](http://homepages.inf.ed.ac.uk/cheunen/publications/2008/arrows/arrows.pdf)

and

* Kazuyuki Asada. "Arrows are strong monads". Proceedings of MSFP '10. [preprint pdf](http://www-kb.is.s.u-tokyo.ac.jp/~asada/papers/arrStrMnd.pdf)

[[!redirects Freyd categories]]
[[!redirects Freyd-categories]]
[[!redirects Freyd-category]]