free groupoid

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#### Category theory
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#### Homotopy theory
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[[!include homotopy - contents]]
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#Contents#
* table of contents
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## Idea

The _free groupoid_ on a [[directed graph]] is the [[groupoid]] whose [[objects]] are the vertices of the graph and whose morphisms are finite concatenations of the edges in the graph and formal inverses to them.

This construction is the [[left adjoint]] [[free construction]] to the [[forgetful functor]] that sends a groupoid to its underlying [[directed graph]].

{#FreeGroupoidOnACategory} More generally, there is a free groupoid construction on ([[small category|small]]) [[strict categories]], given by freely adjoining [[inverse morphisms]] to all existing non-invertible morphisms, and the free groupoid on a directed graph is the pre-composition of that operation with the [[free category]]-construction on the graph:

$$
  Grpd^{strct}_{smll}
  \underoverset
    {\underset{U_{Grpd}}{\longrightarrow}}
    {\overset{F_{Grpd}}{\longleftarrow}}
    {\;\;\bot\;\;\;}
  Cat^{strct}_{smll}
  \underoverset
    {\underset{U_{Cat}}{\longrightarrow}}
    {\overset{F_{Cat}}{\longleftarrow}}
    {\;\;\bot\;\;\;}
  DiGrph
$$

(Incidentally, the [[forgetful functor]] $F \,\colon\,Cat^{strct}_{smll} \longrightarrow Grpd^{strct}_{smll}$ also has a [[right adjoint]], known as the *[[core]]*-construction).

## Definition

Given a [[graph]] $D$, that is, a collection of vertices and of labeled arrows between them, the **free groupoid $G(D)$ on $D$** is the [[groupoid]] that has the vertices of $D$ as [[objects]], and whose [[morphisms]] are constructed recursively by formal composition (i.e., juxtaposition) from identity maps, the arrows of $D$ and formal [[inverses]] for the arrows of $D$.

The only relations between morphisms of $G(D)$ are the necessary ones defining the [[identity]] of each object, the [[inverse]] of each arrow in $D$ and the associativity of composition. This is clearly a groupoid, which comes with an evident morphism $D \to G(D)$ of quivers.

The above sketched construction could be made more precise, but what really matters is the [[universal property]] it enjoys: the free groupoid $G(D)$ is the universal ([[initial object|initial]]) [[groupoid]] mapping out of $D$. By varying $D$, the free groupoid yields a [[functor]] $G$ from [[directed graph]]s to [[groupoid]]s, [[left adjoint]] to the forgetful functor.

This last conceptual characterization is best taken as the definition. Similarly, it is possible to construct the left adjoint to the forgetful functor from groupoids to categories, that is the **free groupoid over a category**.

The construction of free groupoids in "Topology and Groupoids" is by taking a disjoint union of copies of the unit interval groupoid $\mathbf I$ and then identifying the vertices according to the scheme given by the directed graph.

See the paper by Crisp and Paris for an application of free groupoids.

## Properties

### Fundamental group
 {#FundamentalGroup}

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

The [[fundamental group]] of a free groupoid on a [[countable set|countable]] [[directed graph]] (for any basepoint) is a [[free group]].

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For instance ([Cote, theorem 2.3](#Cote)).

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

The [[fundamental group]] of the free groupoid of a graph with a single vertex is the [[free group]] on the set of edges of the graph. A result relevant to the Jordan Curve Theorem and the Phragmen-Brouwer Property  is given in the Corrigendum referenced below. It gives conditions on a pushout of groupoids to contain a free groupoid as a retract.

=--

## Related concepts

* [[free category]], [[core groupoid]]

* [[Nielsen-Schreier theorem]]

* [[Dwyer-Kan simplicial path groupoid]]

## References

On the construction of free groupoids on a graph:

* [[Philip Higgins]], §4 of: *Categories and groupoids*, Van Nostrand Reinhold (1971); Reprints in Theory and Applications of Categories **7** (2005) 1-195 [[tac:tr7] (http://www.tac.mta.ca/tac/reprints/articles/7/tr7abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/7/tr7-2l.pdf)]

* {#MacLane} [[Saunders MacLane]], p. 51 Ex. 3 in: *[[Categories for the Working Mathematician]]*, Springer (1971) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

On the construction of free groupoids on a category ([[localization]] at all morphisms):

* [[Pierre Gabriel]], [[Michel Zisman]], §1.5.4 of: *[[Calculus of fractions and homotopy theory]]*, Ergebnisse der Mathematik und ihrer Grenzgebiete **35**, Springer (1967) [[doi:10.1007/978-3-642-85844-4](https://link.springer.com/book/10.1007/978-3-642-85844-4), [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/GZ.pdf)]

and in the generality of [[sSet-enriched categories]] and [[sSet]]-[[enriched groupoids]] ("[[simplicial groupoids]]"):

* {#DwyerKan80} [[William Dwyer]], [[Daniel Kan]], §5.5 of: *Simplicial localizations of categories*, J. Pure Appl. Algebra **17** 3 (1980), 267-284 [doi:10.1016/0022-4049(80)90049-3]

* [[André Joyal]], [[Myles Tierney]], pp. 75 of: *On the theory of path groupoids*, Journal of Pure and Applied Algebra **149** 1 (2000) 69-100 [doi:10.1016/S0022-4049(98)00164-9]

* [Goerss & Jardine (2009), V.7](Dwyer-Kan+loop+groupoid#GoerssJardine09)

  > (highlighting relation to the [[Milnor construction]])

See also:

* {#Cote} Lauren Cote, _Free groups and graphs: the Hanna Neumann theorem_ (2008) [[pdf](http://www.math.uchicago.edu/~may/VIGRE/VIGRE2008/REUPapers/Cote.pdf)]

* [[Ronald Brown|Ronnie Brown]], _Topology and Groupoids_ (2006)  [[web] (http://pages.bangor.ac.uk/~mas010/topgpds.html)]

* Omar Antolin Camarena and [[Ronnie Brown]], "Corrigendum to "Groupoids, the Phragmen-Brouwer Property, and the Jordan Curve Theorem", J. Homotopy and Related Structures 1 (2006) 175-183." J. Homotopy and Related Structures ([pdf] (http://pages.bangor.ac.uk/~mas010/pdffiles/brouwer-cor-fin.pdf))

* J. Crisp, L. Paris, *The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group*, Invent. math. 145, 19–36 (2001).

[[!redirects free groupoids]]


free Hopf algebra

There is no [[adjoint functor|left adjoint]] to the forgetful functor from [[Hopf algebra]]s to [[vector space]]s. However, Takeuchi in

* M. Takeuchi, Free Hopf algebras generated by coalgebras, J. Math. Soc. Japan 23 (1971), No.4, pp. 561--582.

constructed a left adjoint to the forgetful functor from Hopf algebras to [[coalgebra]]s (his purpose was to construct the historically first example of a Hopf algebra with a noninvertible antipode map).

Let $C$ be a coalgebra over a commutative unital ring $k$. Let $C_i = C$
for $i$ an even nonnegative integer, and $C_i = C_i^{{\mathrm {cop}}}$
(the cooposite coalgebra of $C$) for $i$ an odd positive integer. Then define
$V$ to be the external [[direct sum]] (coproduct) of coalgebras $C_i$,
$$V = \coprod C_i. $$
The [[tensor algebra]] $T(V)$ of $V$, as the tensor algebra of any coalgebra, has a unique [[bialgebra]] structure such that the natural inclusion
$i_V : V \rightarrow T(V)$ is a morphism of coalgebras.
Then $T(V^{\mathrm{cop}}) \cong T(V)^{\mathrm{cop}}$.
Define a $k$-linear map
$ S_V : V \rightarrow V^{\mathrm{cop}}$
by

$$S_V(v_0, v_1, v_2,\ldots) =
(0, v_0, v_1, v_2, \ldots).$$

There is a unique bialgebra map
$S : T(V) \rightarrow T(V)^{\mathrm{cop}}$ extending $S_V$.
Let $I_S$ be the 2-sided [[ideal]] in $T(V)$
generated by all elements of the form
$\sum c_{(1)} S(c_{(2)}) - \epsilon(c) 1$ and
$\sum S(c_{(1)}) c_{(2)} - \epsilon(c) 1$, $c \in C_i$, $i = 1,2, \ldots $.
This 2-sided ideal is a biideal and $S(I_S) \subset I_S$, hence it induces a bialgebra map
$$ S : T(V)/I_S \rightarrow (T(V)/I_S)^{\mathrm{cop}}. $$
It follows that $H(C) = T(V)/I_S$ is a Hopf algebra with antipode $S$, the **free Hopf algebra** on $C$.
For any Hopf algebra $H'$ and a coalgebra map
$\phi : C \rightarrow H'$ there is a unique Hopf algebra
map $\phi' : H(C) \rightarrow H'$ such that
$\phi' \circ i = \phi$ where $i : C \rightarrow H(C)$ is the
composition of inclusion
into $T(V)$ and projection $T(V) \rightarrow H(C)$. Takeuchi's free Hopf algebra construction is functorial.

A comparison with Manin's closely related construction of a [[Hopf envelope]] of a bialgebra can be found in section 13.2 of

* Z. Škoda, Localizations for construction of quantum coset spaces, [math.QA/0301090](http://arxiv.org/abs/math/0301090), "Noncommutative geometry and Quantum groups", W.Pusz, P.M. Hajac, eds. Banach Center Publications vol.61, pp. 265--298, Warszawa 2003.


free infinite loop space

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#Contents#
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## Idea

The [[free construction|free]] [[infinite loop space]] on a given [[pointed topological space]].

## Definition

Write

$$
  Q\;\colon\; Top^{\ast/} \longrightarrow Top^{*/}
$$

for the [[endofunctor]] on [[pointed topological spaces]] given as the [[colimit]]

$$
  Q X \coloneqq \underset{\longrightarrow}{\lim}_k \Omega^k \Sigma^k X
$$

over iterated [[suspension]] and [[loop space]] construction.

This is the degree-0 part of [[spectrification]] of [[suspension spectra]].

Equivalently this is

$$
  Q X \simeq \Omega^\infty \Sigma^\infty X
$$

for

$$
 (\Sigma^\infty \dashv \Omega^\infty)
  \;\colon\;
  Ho(Spectra)
   \underoverset{\underset{\Omega^\infty}{\longrightarrow}}{\overset{\Sigma^\infty}{\longleftarrow}}{}
  Ho(Top)
$$

the [[stabilization]] [[adjunction]] between the [[homotopy category]] [[Ho(Top)]] and the [[stable homotopy category]] $Ho(Spectra)$.

## Related concepts

* [[spectrification]]

* [Omega spectrum -- Completion of a suspension spectrum](Omega-spectrum#CompletionOfSuspensionSpectra)

* [[stable homotopy group]]

## References

* {#Malkiewich11} [[Cary Malkiewich]], _Some facts about $Q X$_, 2011 ([pdf](http://math.uiuc.edu/~cmalkiew/Q.pdf))

[[!redirects free infinite loop spaces]]


free Lie algebra

The __[[free object|free]] [[Lie algebra]] [[functor]]__ is the [[left adjoint]] functor $FreeLieAlg$ to the [[forgetful functor]] $LieAlg\to Set$. The proof that
a left adjoint exists relies on a concrete construction. Let $X$ be a set, then
define recursively for $n\geq 1$ sets $X_n = \coprod_{p=1}^{n-1} X_p\times X_{n-p}$
with the basis of recursion $X_1 = X$. If $x\in X_p$ and $y\in X_q$ then denote
$x.y$ the element $(x,y)\in X_{p+q}$; this defines a binary operation on $\coprod_{n=0}^\infty X_n$, which is therefore the free [[magma]] on the set $X$.
Let $k$ be the ground ring (commutative and unital).
As a $k$-module define $Lib_k(X) = k[\coprod_{n=0}^\infty X_n]$, the free $k$-module
with basis $\coprod_{n=0}^\infty X_n$. It is a nonassociative $k$-algebra
with product $(\sum_i a_i x_i).(\sum_j b_j y_j) = \sum_{i,j} a_i b_j (x_i.y_j)$ where both sums are finite and $a_i, b_j\in k$, $x_i,y_i\in X_i$. Define a two-sided
ideal $I$ in this nonassociative $k$-algebra, generated by all elements of $Lib_k(X)$
of the form $a.a$
and all elements of the form $a.(b.c)+b.(c.a)+c.(a.b)$, where $a,b,c\in Lib_k(X)$.
Then $FreeLieAlg(X) = Lib_k(X)/I$.

The __free Lie algebra__ on the [[set]] $X$ is the result $FreeLieAlg(X)$ of evaluating the free Lie algebra functor on object $X$.

The subject of free Lie algebras is combinatorially rich with lots of open problems. By a 1953 theorem of A. I. Širšov (Shirshov) every Lie subalgebra of a free Lie subalgebra is free (an analogue of the [[Nielsen-Schreier theorem]] in combinatorial group theory). The study of bases of a free Lie algebra considered as a vector space is very nontrivial; special attention has been paid to so-called Hall bases.

* [[N. Bourbaki]], _Lie groups and Lie algebras_, Chap. II: Free Lie Algebras
* Christophe Reutenauer, _Free Lie algebras_, Oxford Univeristy Press 1993

* [[C. Reutenauer]], _Free Lie algebras_, Handbook of Algebra, vol. 3, 2003, 887-903, doi

* [[C. Reutenauer]], _Applications of a noncommutative jacobian matrix_, Journal of Pure and Applied Algebra __77__, n. 2, 1992, p. 169-181, doi

* wikipedia: [free Lie algebra](http://en.wikipedia.org/wiki/Free_Lie_algebra)

* [[Mikhail Kapranov]], _Free Lie algebroids and space of paths_, [math,DG/0702584](http://arxiv.org/abs/math/0702584)

* sbseminar blog: [Tannakian construction of the fundamental group and Kapranov's fundamental Lie algebra](http://sbseminar.wordpress.com/2008/03/17/tannakian-construction-of-the-fundamental-group-and-kapranovs-fundamental-lie-algebra)

* Nantel Bergeron, [[Muriel Livernet]], _A combinatorial basis for the free Lie algebra of the labelled rooted trees_, Journal of Lie Theory __20__ (2010) 3--15, [pdf](https://www.math.univ-paris13.fr/~livernet/publications/BerLiv_JOLT.pdf)

* [[Leila Schneps]], _On the Poisson bracket on the free Lie algebra in two generators_, [pdf](http://www.math.jussieu.fr/~leila/Poisson.pdf)

* A. Murua, _The Hopf algebra of rooted trees, free Lie algebras, and
Lie series_, [pdf](www.ehu.es/ccwmuura/research/MS111rev.pdf)

* F. Chapoton, _Free pre-Lie algebras are free as Lie algebras_,
[math.RA/0704.2153](http://arxiv.org/abs/math/0704.2153), Bulletin Canadien de Mathe'matiques

category: algebra

[[!redirects free Lie algebra]]
[[!redirects free Lie algebras]]
[[!redirects free Lie algebra functor]]


free loop orbifold

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#### Mapping space
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[[!include mapping space - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

For $\mathcal{X}$ an [[orbifold]], (or, more generally, any [[differentiable stack]] or yet more generally a [[smooth ∞-groupoid]]), its *free loop orbifold* (*free loop stack*) is the [[mapping stack]] into it out of the [[circle]] $S^1$ (the latter regarded as a [[smooth manifold]] and hence as an [[orbifold]]/[[differentiable stack]] in the canonical way):

$$
  \mathcal{L} \mathcal{X}
  \;\coloneqq\;
  \big[
    S^1, \, \mathcal{X}
  \big]
  \,.
$$

For $\mathcal{X} = X$ a [[smooth manifold]], this reduces to the [[smooth loop space]] of $X$, which is still a [[Fréchet manifold]].

More generally, for

$$
  \label{GoodOrbifoldQuotientPresentation}
  \mathcal{X} \simeq X \!\sslash\! G
$$

a [[good orbifold]] equivalent to a [[global quotient orbifold]] of $X$ be a [[discrete group|discrete]] [[group action]], the free loop orbifold combines the properties of the [[smooth loop space]] of $X$ with the properties of the *[[inertia orbifold]]* of $\mathcal{X}$ (see Remark \ref{DifferentNotionsOfLoopOrbifolds}).

Concretely, for $\mathcal{U}_{S^1} \xrightarrow{\;\;\simeq\;\;} S^1 $ the [[Cech groupoid]] of a  [[good open cover]] of the circle, the free loop orbifold of a [[good orbifold]] (eq:GoodOrbifoldQuotientPresentation) has [[plots]] of shape $\mathbb{R}^n$ given by the following [[hom-groupoid]] of [[Lie groupoids]]:

\[
  \label{PlotsOfTheFreeLoopOrbifold}
  \mathbf{H}
  \big(
    \mathbb{R}^n,\,
    \mathcal{L}\mathcal{X}
  \big)
  \;\simeq\;
  LieGroupoids
  \big(
    \mathcal{U}_{S^1} \times \mathbb{R}^n,
    \,
    (X \times G \rightrightarrows X)
  \big)
  \,.
\]

Here $\mathbf{H}$ denotes the correct [[(2,1)-category]] of [[differentiable stacks]] (see at [[Smooth∞Groupoid]]), while $LieGroupoids$ is its presentation by [[Lie groupoids]] ([[groupoid objects]] [[internalization|internal]] to [[SmoothManifolds]], where [[Morita morphisms]] are not [[localization|inverted]]).

Notice here how:

* the [[morphisms]] in the [[Cech groupoid]] detect the non-trivial morphisms in $\mathcal{X}$ as for an [[inertia orbifold]],

* while the [[cohesion|cohesive]] smooth structure on the space of objects of the Cech groupoid detects smooth [[paths]] in $X$.

A general plot (eq:PlotsOfTheFreeLoopOrbifold) is a circular sequence of smooth paths in $X$ whose endpoints are cyclically related by the [[group action]] of $G$.

\begin{remark}\label{DifferentNotionsOfLoopOrbifolds}
**(different notions of loop orbifolds)** \linebreak
For the case of [[orbifolds]], the [[cohesion]] of the loops leads to a distinction between various in-equivalent notions of "[[free loop spaces]]" of orbifolds:

Let:

* $\mathcal{X} \,\in\, SmoothGroupoids_\infty$ be an [[orbifold]], regarded as a [[smooth groupoid]], regarded as a [[differentiable stack]].

* $S^1 \,\in\, SmoothManifolds \hookrightarrow SmoothGroupoids_\infty$ be the [[circle]] with its standard [[cohesion|cohesive]] structure as a [[smooth manifold]], and hence as a [[differentiable stack]].

  Notice that the [[shape modality|shape]] of $S^1$ (in the [[cohesive (infinity,1)-topos|cohesive]]) [[(2,1)-topos]] of [[smooth ∞-groupoids]] is the [[delooping groupoid]] of the [[integers]], regarded as a [[discrete object|discrete]] smooth groupoid

  $$
    ʃS^1
    \;\simeq\;
    \mathbf{B}\mathbb{Z}
    \;\;\;
    \in
    Groupoids
    \xrightarrow{ \; Disc\; }
    SmoothGroupoids
     \;
    \,.
  $$

* $[-,-]$ denote the [[mapping stack]]-construction.

Then we have:

1. The *cohesive [[free loop orbifold]]* of $\mathcal{X}$ is

   $$
     \mathcal{L}\mathcal{X}
     \;=\;
     \big[ S^1, \, \mathcal{X} \big]
     \,.
   $$

1. The *[[inertia orbifold]]* of $\mathcal{X}$ is

   $$
     \Lambda \mathcal{X}
     \;\simeq\;
     \big[ ʃS^1, \, \mathcal{X} \big]
     \;\simeq\;
     \big[ \mathbf{B}\mathbb{Z}, \, \mathcal{X} \big]
     \;\simeq\;
     \mathcal{X} \times^h_{\mathcal{X} \times \mathcal{X}} \mathcal{X}
     \,,
   $$

   which is the actual [[free loop space object]] formed in [[smooth groupoids]].

The [[shape modality]]-[[unit of a monad|unit]] $\mathcal{A} \xrightarrow{ \eta_{\mathcal{A}}} ʃ \mathcal{A}$ induces a canonical comparison morphism between the two

$$
  \Lambda \mathcal{X}
  \;=\;
  \big[ ʃS^1, \, \mathcal{X} \big]
    \xrightarrow{ \; [\eta_{S^1},\mathcal{X}]\; }
  \big[ S^1, \, \mathcal{X} \big]
  \;=\;
  \mathcal{L} \mathcal{X}
  \,.
$$

When $\mathcal{X} \simeq X \!\sslash\! G$ is a [[global quotient orbifold]] of a [[smooth manifold]] $X$ (for instance for a [[good orbifold]], but $X$ could more generally be a [[diffeological space]] for the present discussion), then this inclusion is the [[n-truncated object of an (infinity,1)-category|faithful inclusion]] of the *cohesively constant loops*, namely those that map to points in the naive quotient space of $\mathcal{X}$.
\end{remark}

## Related concepts

* [[inertia orbifold]], [[Huan's inertia orbifold]]

* [[cyclic loop stack]]

* [[free loop space]], [[derived loop space]]

* [[free loop space of classifying space]]

## References

* {#LupercioUribe01} [[Ernesto Lupercio]], [[Bernardo Uribe]], _Loop groupoids, gerbes, and twisted sectors on orbifolds_, in: [[Alejandro Adem]], [[Jack Morava]], [[Yongbin Ruan]] (eds.), *[[Orbifolds in Mathematics and Physics]]*, Madison, WI, 2001, in: Contemp. Math. __310__, Amer. Math. Soc., Providence, RI, 2002, pp. 163–184, ([math.AT/0110207](http://arxiv.org/abs/math/0110207), [ISBN:978-0-8218-2990-5](https://bookstore.ams.org/conm-310), [MR2004c:58043](http://www.ams.org/mathscinet-getitem?mr=1950946))

* [[Kai Behrend]], [[Grégory Ginot]], [[Behrang Noohi]], [[Ping Xu]], Section 5 of:  *String topology for stacks*, Astérisque, no. 343 (2012) , 183 p. ([arXiv:0712.3857](https://arxiv.org/abs/0712.3857), [numdam:AST_2012__343__R1_0](http://www.numdam.org/item/AST_2012__343__R1_0))

  > (with an eye towards [[string topology]] of orbidolds)

* [[David Michael Roberts]], [[Raymond Vozzo]], _Smooth loop stacks of differentiable stacks and gerbes_, [[Cahiers de Topologie et Géométrie Différentielle Catégoriques]], Vol LIX no 2 (2018) pp 95-141 ([journal version](http://cahierstgdc.com/index.php/volume-lix-2018), [arXiv:1602.07973](https://arxiv.org/abs/1602.07973))
> (a [[Fréchet–Lie group|Fréchet–Lie]] groupoid presentation)

* {#Huan18} [[Zhen Huan]], Section 2 of: _Quasi-Elliptic Cohomology I_, Advances in Mathematics, Volume 337, 15 October 2018, Pages 107-138 ([arXiv:1805.06305](https://arxiv.org/abs/1805.06305), [doi:10.1016/j.aim.2018.08.007](https://doi.org/10.1016/j.aim.2018.08.007))

  > (in view of [[equivariant elliptic cohomology]] via [[Tate K-theory]])

[[!redirects free loop orbifolds]]

[[!redirects loop orbifold]]
[[!redirects loop orbifolds]]

[[!redirects free loop stack]]
[[!redirects free loop stacks]]

[[!redirects free loop ∞-stack]]
[[!redirects free loop ∞-stacks]]

[[!redirects loop stack]]
[[!redirects loop stacks]]

[[!redirects loop ∞-stack]]
[[!redirects loop ∞-stacks]]


free loop space

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### Context
#### Topology
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[[!include topology - contents]]
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#### Mapping space
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[[!include mapping space - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

The _free loop space_ $\mathcal{L}X$ of a [[topological space]] $X$ (based or not) is the space of *all* [[loops]] in $X$. This is in contrast to the *based* [[loop space]] of a based space $X$ for which the loops are at the fixed [[base point]] $x_0\in X$.

(Regarded as a [[homotopy type]] the concept generalizes to other contexts of [[homotopy theory]], see at _[[free loop space object]]_ for more.)

The free loop space carries a canonical [[action]] ([[infinity-action]]) of the [[circle group]], and furthermore is a [[cyclotomic space]]. The [[homotopy quotient]] by that action $\mathcal{L}(X)/S^1$ (the "[[cyclic loop space]]") contains what is known as the _[[twisted loop space]]_ of $X$.

## Definition

### Explicit description

For $X$ a topological space, the free loop space $\mathcal{L}X$ is the topological space $Map(S^1,X)$ of [[continuous map|continuous maps]] with the [[compact-open topology]].

If we work in a category of [[pointed topological space|based spaces]], then still the topological space $Map(S^1,X)$ is in the non-based sense but has a distinguished point which is the constant map $t\mapsto x_0$ where $x_0$ is the base point of $X$.

### General abstract description

If $X$ is a [[topological space]], the **free loop space** $L X$ of $X$ is defined as the [[free loop space object]] of $X$ formed in the [[(∞,1)-category]] [[Top]].

### Cohomology of $\mathcal{L}X$ and Hochschild homology of $X$
  {#Cohomology}

Let $X$ be a [[simply connected topological space|simply connected]] [[topological space]].

The [[ordinary cohomology]] $H^\bullet$ of its free loop space is the [[Hochschild homology]] $HH_\bullet$ of its [[singular cohomology|singular chains]] $C^\bullet(X)$:

$$
  H^\bullet(\mathcal{L}X)
    \simeq
  HH_\bullet( C^\bullet(X) )
  \,.
$$

Moreover the $S^1$-equivariant cohomology of the loop space, hence the ordinary cohomology of the [[cyclic loop space]] $\mathcal{L}X/^h S^1$ is the [[cyclic homology]] $HC_\bullet$ of the singular chains:

$$
  H^\bullet(\mathcal{L}X/^h S^1)
    \simeq
  HC_\bullet( C^\bullet(X) )
$$

([Loday 11](#Loday11))

If the [[coefficients]] are [[rational numbers|rational]], and $X$ is of [[finite type]] then this may be computed by the _[[Sullivan model for free loop spaces]]_, see there the section on _[Relation to Hochschild homology](Sullivan+model+of+free+loop+space#RelationToHochschildHomology)_.

In the special case that the [[topological space]] $X$ carries the structure of a [[smooth manifold]], then the singular cochains on $X$ are equivalent to the [[dgc-algebra]] of [[differential forms]] (the [[de Rham algebra]]) and hence in this case the statement becomes that

$$
  H^\bullet(\mathcal{L}X)
    \simeq
  HH_\bullet( \Omega^\bullet(X) )
  \,.
$$

$$
  H^\bullet(\mathcal{L}X/^h S^1)
    \simeq
  HC_\bullet( \Omega^\bullet(X) )
  \,.
$$

This is known as _[[Jones' theorem]]_ ([Jones 87](#Jones87))

An [[(infinity,1)-category theory|infinity-category theoretic]] proof of this fact is indicated at _[Hochschild cohomology -- Jones' theorem](Hochschild+cohomology#JonesTheorem)_.

### In rational homotopy theory

See at _[[Sullivan model of free loop space]]_.

## Examples

* [[free loop space of classifying space]]

## Related entries

* [[loop space object]], [[free loop space object]],

  * [[delooping]]

  * [[loop space]], **free loop space**, [[derived loop space]]

  * [[Sullivan model of free loop space]]

* [[Hochschild homology]], [[cyclic homology]], [[dihedral homology]]

* [[smooth loop space]],

* [[free loop orbifold]], [[inertia orbifold]]

* [[cyclic loop space]], [[cyclic loop stack]]

* [[caloron correspondence]]

* [[suspension object]]

  * [[suspension]]

* [[fusion bundle]]

## References

* [[David Chataur]], Alexandru Oancea, *Basics on free loop spaces*, Chapter I in: Janko Latchev, Alexandru Oancea (eds.): *Free Loop Spaces in Geometry and Topology*, IRMA Lectures in Mathematics and Theoretical Physics **24**, EMS 2015 [[ISBN:978-3-03719-153-8](https://bookstore.ams.org/emsilmtp-24/)]

* [[Kathryn Hess]], *Free loop spaces in topology and physics*, talk at *Meeting of Edinburgh Math. Soc. Glasgow* (14 Nov 2008) [[pdf](http://sma.epfl.ch/~hessbell/Pub_Edinburgh.pdf), [[Hess-FreeLoopSpaces.pdf:file]]]

In the context of [[Hochschild homology]] (and [[cyclic homology]]):

* {#Jones87} [[John D.S. Jones]], _Cyclic homology and equivariant homology_, Invent. Math. __87__, 403-423 (1987) ([pdf](https://math.berkeley.edu/~nadler/jones.pdf), [doi:10.1007/BF01389424](https://doi.org/10.1007/BF01389424))

reviewed in:

* {#Loday92} [[Jean-Louis Loday]], *Cyclic Spaces and $S^1$-Equivariant Homology* ([doi:10.1007/978-3-662-21739-9_7](https://link.springer.com/chapter/10.1007/978-3-662-21739-9_7))

  Chapter 7 in: *Cyclic Homology*, Grundlehren **301**, Springer 1992 ([doi:10.1007/978-3-662-21739-9](https://link.springer.com/book/10.1007/978-3-662-21739-9))

* {#Loday11} [[Jean-Louis Loday]], Section 4 of: _Free loop space and homology_, Chapter 4 in: Janko Latchev, Alexandru Oancea (eds.): *Free Loop Spaces in Geometry and Topology*, IRMA Lectures in Mathematics and Theoretical Physics **24**, EMS 2015 ([arXiv:1110.0405](https://arxiv.org/abs/1110.0405), [ISBN:978-3-03719-153-8](https://bookstore.ams.org/emsilmtp-24/))

[[!redirects free loop space]]
[[!redirects free loop spaces]]


free loop space object

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $(\infty,1)$-Category theory
+--{: .hide}
[[!include quasi-category theory contents]]
=--
#### Stable homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Mapping space
+--{: .hide}
[[!include mapping space - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In an [[(∞,1)-category]] $C$ with $(\infty,1)$-[[pullbacks]], the _free loop space object_ $\mathcal{L}X$ of any object $X$ -- also called the _[[inertia groupoid]]_ -- is an object that behaves as if its [[generalized elements]] are loops in $X$, morphisms between generalized elements [[homotopy|homotopies]] of loops, and so on.

For the case that $C = $ [[Top]] this reproduces the ordinary notion of  [[free loop spaces]] of [[topological spaces]].

Over each fixed element $x \in X$, the free loop space object $\mathcal{L}X$ looks like the based [[loop space object]] $\Omega_x X$ of $X$.

Free loop space objects come naturally equipped with various structures of interest, such as a categorical circle action. The [[cohomology]] of $\mathcal{L}X$ is [[Hochschild cohomology]] or [[cyclic cohomology]] of function algebras $C(X)$ on $X$. The categorical circle action induces [[differential]]s on these cohomologies, identifying them, in suitable cases, with algebras of [[Kähler differential|Kähler]] [[differential form]]s on $X$.

## Definition

### In higher category theory

There are various equivalent ways to define the free loop space object.

Let $C$ be an [[(∞,1)-category]].  Recall that every $(\infty,1)$-category is [[enriched category|enriched]] over [[?Gpd]], in that there is a hom-space $Map(X,Y) \in \infty Gpd$ for any $X,Y\in C$.  This enables us to define the [[power]] of an object $X\in C$ by any $\infty$-groupoid $K$, as an object $X^K \in C$ together with a map $K \to Map(X^K,X)$ inducing equivalences $Map(Y,X)^K \simeq Map(Y,X^K)$ for all $Y\in C$ (where $Map(Y,X)^K$ denotes the mapping-space from $K$ to $Map(Y,X)$ in $\infty Gpd$), if such exists.

+-- {: .num_defn}
###### Definition
The **free loop space object** of $X\in C$ is the power $\mathcal{L}X = X^{S^1}$, if it exists, where $S^1$ denotes the homotopical [[circle]].
=--

This can also be written in terms of "conical" limits in $C$.  Most commonly, if we note that $S^1$ is the pushout of two copies of $\ast$ under $\ast\sqcup \ast$, and that $Map(\ast,X) \simeq X$ while $Map(\ast\sqcup \ast ,X) \simeq X\times X$, we find that $X^{S^1}$ is the pullback of $X$ and $X$ over $X\times X$:

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

In an [[(∞,1)-category]] $C$ with [[(∞,1)-pullbacks]], for $X \in C$ an [[object]], its **free loop space object** $\mathcal{L}X$ is the $(\infty,1)$-pullback of the [[diagonal]] along itself

$$
  \array{
    \mathcal{L} X &\to& X
    \\
    \downarrow && \downarrow^{\mathrlap{(Id,Id)}}
    \\
    X &\stackrel{(Id,Id)}{\to}& X \times X
  }
  \,.
$$

=--

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

This is the $(\infty,1)$-categorical [[span trace]] of the identity-[[span]]

$$
  \mathcal{L}X =
  Tr
  \left(
    \array{
       && X
       \\
       & {}^{\mathllap{Id}}\swarrow && \searrow^{\mathrlap{Id}}
       \\
       X &&&& X
    }
  \right)
  \,.
$$

=--

We can also use the fact that $S^1$ is the [[homotopy coequalizer]] of $\ast \rightrightarrows \ast$:

+-- {: .num_defn}
###### Definition
The **free loop space object** of $X\in C$ is the homotopy equalizer of two copies of the identity map $X \rightrightarrows X$.
=--

### In homotopy type theory
 {#InHomotopyTypeTheory}

In the literature (see [below](#References)) when the free loop space object $\mathcal{L}X$ is defined as the homotopy pullback $X\times_{X\times X} X$, it is sometimes described heuristically as: "a point of $\mathcal{L}X$ is a choice of making two points of $X$ equal in two ways." In terms of [[homotopy type theory]] this heuristics becomes a theorem. In that higher categorical logic we have the expression

$$
  \begin{aligned}
    \mathcal{L}X
    & \coloneqq
    \left\{
       x,y : X \;|\; (x = y) \, and\, (x = y)
    \right\}
    \\
    & =
    \left\{
       x,y : X \;|\; (x,x) = (y,y)
    \right\}
  \end{aligned}
  \,.
$$

Here on the right we have

1. the [[dependent sum]];

1. over the [[identity type]] $Id (X \times X)$;

1. of the [[product type]] $X \times X$.

See the discussion at _[[homotopy pullback]]_ in the section _[Construction in homotopy type theory](http://ncatlab.org/nlab/show/homotopy+pullback#ConstructionInHomotopyTypeTheory)_ for how this is equivalent to the previous definition.

Now since $\sum_{y:X} (x=y)$ is [[contractible type|contractible]], the above type is equivalent to

$$ \sum_{x:X} (x=x) $$

i.e. the type of points that are equal to themselves (in a specified, not necessarily reflexivity, way).  This corresponds to the other definitions, as a homotopy equalizer or a powering by $S^1$.

## Properties

### Models by homotopy pullbacks {#HomPullModel}

To see what the definition of a free loop space object amounts to in more detail, assume that the [[(∞,1)-category]] is modeled by a [[homotopical category]], say for simplicity a [[category of fibrant objects]], for instance the full subcategory on fibrant objects of a [[model category]].

Then following the discussion at [[homotopy pullback]] and [[generalized universal bundle]] we can compute the about $(\infty,1)$-pullback $X\times_{X\times X}^h X$ as the ordinary [[limit]]

$$
  \array{
    \mathcal{L}X &\to& &\to& X
     \\
    \downarrow && && \downarrow^{\mathrlap{(Id,Id)}}
    \\
    && (X \times X)^I &\stackrel{(d_0,d_0)}{\to}& X \times X
    \\
    \downarrow &&  {}^{\mathllap{(d_1,d_1)}}\downarrow
    \\
    X &\stackrel{(Id,Id)}{\to} &  X \times X
    \,,
  }
$$

where $(X\times X)^I$ is a [[path space object]] for $X \times X$.  At least if we have the structure of a [[model category]] we may take $(X \times X)^I = X^I \times X^I$ for a [[path space object]] $X^I$ of $X$.

From this description one sees that $\mathcal{L}X$ is built from _pairs of paths_ in $X$ with coinciding endpoints, that are _glued at their coinciding endpoint_ . So the loops here are all built from two semi-ciricle paths.

### Relation to based loop space object
 {#RelationToBasedLoops}

The fiber of $\mathcal{L}X$ over a [[point]] $x : {*} \to X$ is the corresponding (based) [[loop space object]] $\Omega_x X$ of $X$: we have an $(\infty,1)$-[[pullback]] diagram

$$
  \array{
    \Omega_x X &\to& \mathcal{L}X &\to& X
    \\
    \downarrow && \downarrow && \downarrow^{\mathrlap{(Id,Id)}}
    \\
    {*} &\stackrel{x}{\to} & X &\stackrel{(Id,Id)}{\to}& X\times X
  }
  \,.
$$

To see this, use that [[homotopy pullback]]s paste to homotopy pullbacks, so that the outer pullback is modeled by the ordinary [[limit]]

$$
  \array{
    \Omega_x^{I \vee I}X &\to& &\to& X
     \\
    \downarrow && && \downarrow^{\mathrlap{(Id,Id)}}
    \\
    && (X \times X)^I &\stackrel{(d_0,d_0)}{\to}& X \times X
    \\
    \downarrow &&  {}^{\mathllap{(d_1,d_1)}}\downarrow
    \\
    {*} &\stackrel{(x,x))}{\to} &  X \times X
    \,,
  }
$$

which builds based loops on $X$ from two consecutive paths, the first starting at the basepoint $x$, the second ending there. This is weakly equivalent $\Omega_x X = \Omega^I_x X \stackrel{\simeq}{\to} \Omega^{I \vee I} X$ to the based [[loop space object]] $\Omega_x X$ built from just the path space object $X^I$ with a single copy of $I$, by standard arguments as for instance form page 12 on in

* [[Kenneth Brown]], _[[BrownAHT|Abstract Homotopy Theory and Generalized Sheaf Cohomology]]_ .

### Action on objects of the slice
 {#SliceAction}

The free loop space object $\mathcal{L} X$ is a [[group object]] in the [[slice (∞,1)-category]] $C/X$, and has a canonical [[action]] on all objects of $C/X$.

Intuitively, the group structure comes from composition and inversion of loops.  When the free loop space is expressed as a [[power]] $X^{S^1}$, this group structure comes from the canonical [[cogroup]] structure on $S^1$.  In [[homotopy type theory]], it is literally concatenation of paths.  And when $\mathcal{L}X$ is expressed as the pullback $X\times_{X\times X} X$, the group multiplication can be obtained by considering the following pasting square:

$$\array{
\mathcal{L}X \times_X \mathcal{L}X & \to & \mathcal{L} X & \to & X\\
\downarrow && \downarrow && \downarrow \\
\mathcal{L} X & \to & X & \to & X\times X \\
\downarrow && \downarrow && \downarrow^{id} \\
X & \to & X\times X & \xrightarrow{id} & X\times X
}$$

Here the bottom-left and top-right squares are the pullback defining $\mathcal{L}X$, the top-left square is the pullback defining $\mathcal{L}X \times_X\mathcal{L}X$, and the bottom-right square commutes but is not a pullback.  By the universal property of the pullback square defining $\mathcal{L}X$, this square factors through it uniquely, giving the composition map $\mathcal{L}X \times_X \mathcal{L}X\to \mathcal{L}X$.  Similarly but more simply, the inversion map $\mathcal{L}X\to \mathcal{L}X$ comes from transposing its defining pullback square and factoring it through itself.

Now consider $Y\to X$ an object of $C/X$.  There is a canonical projection $\mathcal{L}X \times_X Y \to Y$, which is not the action of $\mathcal{L}X$ on $Y$, but it almost is.  In fact since this projection is a morphism in the "homotopy" slice $C/X$, it consists not just of a morphism in $C$ but a homotopy witnessing that a certain triangle commutes, which is equivalently the homotopy in the left-hand commutative square below (which is the pullback defining $\mathcal{L}X \times_X Y$):

$$\array{ \mathcal{L}X \times_X Y & \to & \mathcal{L} X & \xrightarrow{id} & \mathcal{L}X \\
\downarrow && \downarrow && \downarrow \\
Y & \to & X & \xrightarrow{id} & X}
$$

Pasting this with the right-hand commutative square, which is the canonical automorphism of the projection $\mathcal{L}X\to X$, we obtain a different homotopy witnessing a  different morphism $\mathcal{L}X \times_X Y \to Y$ in $C/X$ (with the same underlying morphism in $C$), and *this* is the action of $\mathcal{L}X$ on $Y$.

The definitiong of the right-hand commutative square above may not be obvious.  It is clear when we write $\mathcal{L}X = X^{S^1}$; when we write $\mathcal{L}X = X\times_{X\times X} X$ it can be obtained as the following pasting square, where $p$ and $q$ are the two projections in the defining pullback of $\mathcal{L}X$:

$$ \array{
\mathcal{L}X & \xrightarrow{p} & X & \to & X\times X & \xrightarrow{\pi_1} & X\\
^{id}\downarrow && && \downarrow^{id} && \downarrow \\
\mathcal{L}X & \xrightarrow{q} & X & \to & X\times X & \xrightarrow{\pi_1} & X\\
^{id}\downarrow && \downarrow^{id} && && \downarrow \\
\mathcal{L}X & \xrightarrow{q} & X & \to & X\times X & \xrightarrow{\pi_2} & X\\
^{id}\downarrow && && \downarrow^{id} && \downarrow \\
\mathcal{L}X & \xrightarrow{p} & X & \to & X\times X & \xrightarrow{\pi_2} & X\\
^{id}\downarrow && \downarrow^{id} && && \downarrow \\
\mathcal{L}X & \xrightarrow{p} & X & \to & X\times X & \xrightarrow{\pi_1} & X
}$$

To extend these structures to a coherent $\infty$-group structure and a coherent $\infty$-action, see for instance [this MO answer](https://mathoverflow.net/a/281937).

### In an $(\infty,1)$-topos {#InATopos}

We consider now the case that $C = \mathbf{H}$ is an  [[(∞,1)-topos]] (of [[(∞,1)-sheaves]]/[[∞-stack]]s). This comes canonically with its [[terminal object|terminal]] [[global section]]s [[(∞,1)-geometric morphism]]

$$
  (LConst \dashv \Gamma) : \mathbf{H} \stackrel{\overset{LConst}{\leftarrow}}{\underset{\Gamma}{\to}}
  \infty Grpd
  \,.
$$

In this case we can reformulate the power definition $\mathcal{L}X = X^{S^1}$ using a version of $S^1$ that is an object of $\mathbf{H}$ itself.

#### As a mapping space object {#AsMappingSpaceObject}

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

Write

$$
  S^1 \in Top \simeq \infty Grpd \stackrel{LConst}{\hookrightarrow} \mathbf{H}
$$

for the [[circle]]. In [[Top]] this is the usual topological circle. In [[∞Grpd]] this is (the [[homotopy type]] of) the [[fundamental ∞-groupoid]] of the topological circle. We may think of this as the [[(∞,1)-pushout]]

$$
  S^1 \simeq * \coprod_{* \coprod *} *
$$

hence as the universal [[cocone]]

$$
  \array{
    * \coprod * &\to& *
    \\
    \downarrow &\swArrow_{\simeq}& \downarrow
    \\
    * &\to& S^1
  }
$$

in $\infty Grpd$.

In $\mathbf{H}$ we still write $S^1$ for the [[constant ∞-stack]] on this, the image of this under $LConst$. Since $LConst$ is a [[left adjoint]] and hence preserves this poushout, there is no risk of confusion.

=--

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

To see that the given $(\infty,1)$-pushout indeed produces the circle, we use the standard [[model structure on simplicial sets]] $sSet_{Quillen}$ to [[presentable (∞,1)-category|present]] [[∞Grpd]]. In $sSet_{Quillen}$ the $(\infty,1)$-pushout is computed by the [[homotopy pushout]]. By general facts about this, it may be computed as an ordinary [[pushout]] in [[sSet]] once we pass to an equivalent pushout diagram in which at least one morphism is a [[monomorphism]]. This is the case for

$$
  \array{
    * \coprod * &\to& *
    \\
    \downarrow && \downarrow
    \\
    \Delta[1] &\to& \Delta[1]/\partial \Delta[1]
  }
  \,.
$$

=--

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

Informally the $(\infty,1)$-pushout $* \coprod_{* \coprod *} *$ may be thought of as

* the disjoint union of two points $*_1$, $*_2$;

* equipped with _two_ non-equivalent abstract [[homotopies]] between them

  $$
    S^1
    =
   \left\{
    \array{
       & \nearrow \searrow^{\mathrlap{\simeq}}
       \\
       *_1 && *_2
       \\
       & \searrow \nearrow_{\mathllap{\simeq}}
    }
    \right\}
    \,.
  $$

This equivalent way of modelling the circle not as a single point with an automorphism, but as two points with two isomorphisms is what connects directly to the definition of the free loop space object. This we now come to. It is also the fundamenal source of the basic structure of [[Hochschild cohomology|Hochschild (co)homology]] (as discussed there).

=--

\begin{lemma}

Every [[(∞,1)-topos]] is a [[cartesian closed (∞,1)-category]]: we have for every object $X \in \mathbf{H}$ an [[internal hom]]-[[(∞,1)-functor]]

$$
  [X,-] : \mathbf{H} \to \mathbf{H}
  \,.
$$

\end{lemma}

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

This is discussed at [[(∞,1)-topos]] in the section Closed monoidal structure.

=--

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

There is a natural [[equivalence in an (∞,1)-category|equivalence]]

$$
  \mathcal{L}X \simeq [S^1 , X]
  \,.
$$

=--

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

This follows from the above by the fact (see closed monoidal structure on (∞,1)-toposes) that the internal hom in an $(\infty,1)$-topos preserves finite colimits in its first argument and satisfies

$$
  [*,X] \simeq X
  \,.
$$

This yields

$$
  \begin{aligned}
    [S^1 ,X] &\simeq [* \coprod_{* \coprod *} *, X]
    \\
      & \simeq
      [*,X] \times_{[* \coprod *, X]} [*,X]
     \\
      & \simeq X \times_{X \times X} X
     \\
     & \simeq \mathcal{L}X
     \,.
  \end{aligned}
$$

=--

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

We have that the free loop space object of $X \in \mathbf{H}$ is equivalently the powering of $X$ by the $\infty$-groupoid $S^1$:

$$
  \mathcal{L} X \simeq X^{S^1}
$$

=--

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

Follows by the above from the equivalence $[LConst S^1 , X] \simeq X^{S^1}$ discussed at [[(∞,1)-topos]].

=--

#### Intrinsic circle action {#CircleAction}

By precomposition, the [[automorphism 2-group]] of the [[circle]] $S^1$ acts on free loop space of an object $X \in \mathbf{H}$

$$
  Aut(S^1) \times [S^1, X]
  \to
  [S^1, S^1] \times [S^1, X] \stackrel{\circ}{\to} [S^1, X]
  \,.
$$

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

The connected component of $[S^1,S^1]$ on the identity is equivalent to $S^1$

$$
  [S^1 , S^1]_{Id} \simeq S^1
$$

=--

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

We say that

$$
  S^1 \times [S^1, X]
    \simeq
  [S^1, S^1]_{Id} \times [S^1, X]
  \stackrel{\circ}{\to}
  [S^1, X]
$$

is the intrinsic **circle action** on the free loop space object.

=--

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

We spell out in detail what this action looks like.
The reader should thoughout keep the [[homotopy hypothesis]]-equivalence, $(|-| \dashv \Pi) : Top \simeq \infty Grpd$ in mind.

We may realize the [[circle]] $S^1 \in $ [[Top]] under $\Pi : Top \simeq \infty Grpd$ as the [delooping]] [[groupoid]] $\mathbf{B}\mathbb{Z}$ of the additive [[group]] $\mathbb{Z}$ of [[integer]]s

The [[automorphism 2-group]] of this object is the [[functor category|functor groupoid]]

$$
  Aut_{Grpd}(\mathbf{B}\mathbb{Z})
$$

whose [[object]]s are invertible [[functor]]s $\mathbf{B}\mathbb{Z} \to \mathbf{B}\mathbb{Z}$ and whose [[morphism]]s are [[natural transformation]]s between these.

The functors $\mathbf{B}\mathbb{Z} \to \mathbf{B}\mathbb{Z}$ correspond bijectively to group [[homomorphism]]s $\mathbb{Z} \to \mathbb{Z}$, hence to multiplication by $n\in\mathbb{Z}$

$$
  [n] : \mathbf{B}\mathbb{Z} \to \mathbf{B}\mathbb{Z}
$$

$$
  (\bullet \stackrel{k}{\to} \bullet) \mapsto (\bullet \stackrel{n\cdot k}{\to} \bullet).
$$

Natural transformations between two such endomorphisms are given by a component $\ell \in \mathbb{Z}$ such that all diagrams

$$
  \array{
    \bullet &\stackrel{\ell}{\to}& \bullet
    \\
    {}^{\mathllap{n\cdot k}}\downarrow && \downarrow^{\mathrlap{n' \cdot k}}
    \\
    \bullet &\stackrel{\ell}{\to}& \bullet
  }
$$

commute in $\mathbf{B}\mathbb{Z}$. This can happen only for $n = n'$, but then it happens for arbitrary $\ell$.

In other words we have

$$
  Aut(\mathbf{B}\mathbb{Z}) \simeq
  \coprod_{[n] \in \mathbb{Z}^\times}\mathbf{B}\mathbb{Z}
  \,.
$$

and

$$
Aut_{Id}(\mathbf{B}\mathbb{Z}) \simeq \mathbf{B}\mathbb{Z}
  \,.
$$

The object $[n]$ corresponds to the self-mapping of the circle that fixes the basepoint and has winding number $n\in\mathbb{Z}$. The transformation $\ell$ corresponds then to a rigid rotation of the loop by $\ell$ full circles

Notably for $n = 1$ and $k = 1$ we may think of the diagram

$$
  \array{
    \bullet &\stackrel{\ell}{\to}& \bullet
    \\
    {}^{\mathllap{1}}\downarrow && \downarrow^{\mathrlap{1}}
    \\
    \bullet &\stackrel{\ell}{\to}& \bullet
  }
$$

as depicting the unit loop around the circle (on the left, say) and the result of translating its basepoint $\ell$-times around the circle (the rest of the diagram). Of course since we are using a model of $S^1$ with a single object here, every rotation of the loop is a full circle rotation, which is a bit hard to see.

=--

> Exercise: spell out the above discussion analogously for the equivalent model given by the [[fundamental groupoid]] $\Pi_1(S^1)$ of the standard circle. The is the groupoid with $S^1_{Top}$ as its set of objects homotopy classes of paths in the circle as morphisms. In this model things look more like one might expect from a circle action. Notice that $\mathbf{B}\mathbb{Z}$ is the [[skeleton]] of $\Pi_1(S^1)$.

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

Consider $\mathbf{H} = $ [[∞Grpd]], $G$ a [[group]] and  $X = \mathbf{B}G$ the [[delooping groupoid]]. Then $\mathcal{L}X = G//_{Ad}G$ (as discussed in detail [below](#LoopsInBG)). A morphism $(g \stackrel{h}{\to} Ad_h a)$ in $G//G$ corresponds to a natural transformation

$$
  \array{
    & \nearrow \searrow^{\mathrlap{g}}
    \\
    \mathbf{B}\mathbb{Z}
    &\Downarrow^{h}&
    \mathbf{B}G
    \\
    & \searrow \nearrow_{\mathrlap{Ad_h g}}
  }
  \,.
$$

Precomposing this with the automorphism $\ell$ of the object $[n]$ in $END(\mathbf{B}\mathbb{Z})$

$$
  \array{
    & \nearrow \searrow^{\mathrlap{n}}
    \\
    \mathbf{B}\mathbb{Z}
    &\Downarrow^{\ell}&
    \mathbf{B}\mathbb{Z}
    \\
    & \searrow \nearrow_{\mathrlap{n}}
  }
$$

produces the new transformation

$$
  \array{
    & \nearrow \searrow^{\mathrlap{n}} && \nearrow \searrow^{\mathrlap{g}}
    \\
    \mathbf{B}\mathbb{Z}
    &\Downarrow^{\ell}&
    \mathbf{B}\mathbb{Z}
    &\Downarrow^{h}&
    \mathbf{B}G
    \\
    & \searrow \nearrow_{\mathrlap{n}} && \searrow \nearrow_{\mathrlap{Ad_h g}}
  }
  \,.
$$

By the rules of horizontal composition of [[natural transformation]]s, this is the transformation whose component naturality square on $(\bullet \stackrel{1}{\to} \bullet)$ in $\mathbf{B}\mathbb{Z}$ is the diagram

$$
  \array{
    \bullet &\stackrel{g^\ell}{\to}& \bullet &\stackrel{h}{\to}&\bullet
    \\
    {}^{\mathllap{g^{n}}}\downarrow && {}^{g^n}\downarrow && \downarrow^{\mathrlap{Ad_h g^n}}
   \\
    \bullet &\underset{g^\ell}{\to}& \bullet &\underset{h}{\to}&\bullet
  }
$$

in $\mathbf{B}\mathbb{Z}$, hence the morphism $(g^n \stackrel{g^{\ell} h}{\to} Ad_h g^n)$ in $G//_{Ad}G$. In particular, the categorical circle action is
$$
\ell:(g \stackrel{h}{\to} Ad_h g)\mapsto (g \stackrel{g^{\ell} h}{\to} Ad_h g).
$$

=--

### Hochschild cohomology and cyclic cohomology

[[quasicoherent ∞-stack]]s on $\mathcal{L}X$ form the [[Hochschild cohomology|Hochschild homology]] object of $X$ (if the axioms of [[geometric function theory]] are met) as described there. The circle acton on $\mathcal{L}X$ induces differentials on these.

> ... details to be written, but see [[Hochschild cohomology]] and [[cyclic cohomology]] for more.

## Examples

### Free topological loop spaces

In [[Top]] the notion of free loop space objects reproduces the standard notion of topological [[free loop space]]s.

### Details for $\mathcal{L} \mathbf{B}G$
  {#LoopsInBG}

Let the ambient [[(∞,1)-category]] be [[∞Grpd]], let $G$ be an ordinary [[group]] and $\mathbf{B}G$ its one-object [[delooping]] [[groupoid]].

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

We have that the [[loop groupoid]]

$$
  \mathcal{L} \mathbf{B}G \simeq G//_{Ad} G
  \,,
$$

the [[action groupoid]] of the adjoint action of $G$ on itself.

=--

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

We spell this out in full pedestrian detail, as a little exercise in computing [[homotopy pullback]]s.

We have that the [[path space object]] is $\mathbf{B}G^I = [I,\mathbf{B}G]$ -- the [[functor category|functor groupoid]], where $I$ is the free groupoid $I = \{a \stackrel{\simeq}{\to} b\}$ on the standard [[interval object]] -- which is (by the definition of [[natural transformation]]) the [[action groupoid]]

$$
  \mathbf{B}G^I = G\backslash \backslash G // G
$$

for the action of $G$ on itself, by _inverse_ left and direct right multiplication separately: the naturality square of a [[natural transformation]] defining a morphism $g \stackrel{h_1,h_2}{\to} h_1^{-1} g h_2$ in this groupoid is the commuting square

$$
  \array{
    \bullet &\stackrel{g}{\to}& \bullet
    \\
    {}^{\mathllap{h_1}}\downarrow && \downarrow^{\mathrlap{h_2}}
    \\
    \bullet &\stackrel{h_1^{-1}g h_2}{\to}& \bullet
  }
$$

in $\mathbf{B}G = {*}//G$.

The [[pullback]] of the top right corner of the above defining limit diagram is

$$
  \array{
    (G\backslash\backslash G \times
    G\backslash\backslash G)//G
    &\to& \mathbf{B}G
    \\
    \downarrow && \downarrow^{\mathrlap{(Id,Id)}}
    \\
    (G\backslash\backslash G//G) \times
    (G\backslash\backslash G//G)
    &\to& \mathbf{B}G \times \mathbf{B}G
  }
$$

identifying the two actions from the right, and then the remaining pullback completing the limit diagram is

$$
  \array{
    G\backslash\backslash (G\times G)//G
    &\to&
    (G\backslash\backslash G \times
    G\backslash\backslash G)//G
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}G &\stackrel{(Id,Id)}{\to}&
    \mathbf{B}G \times \mathbf{B}G
  }
$$

now identifying also the two actions from the left, so that  $G\backslash\backslash (G\times G)//G$ is the [[action groupoid]] of $G$ acting diagonally on $G \times G$ by multiplication from the left and from the right, separately.

To see better what this is, we pass to an equivalent smaller groupoid (the [[homotopy pullback]] is defined, of course, only up to weak equivalence). Notice that every morphism
$(g_1,g_2) \stackrel{h_1,h_2}{\to} (g'_1, g'_2)$
in $G\backslash\backslash (G\times G)//G$ corresponding to a [[natural transformation]]

$$
  \array{
     \bullet &\stackrel{(g_1,g_2)}{\to}& \bullet
     \\
     {}^{\mathllap{(h_1,h_1)}}\downarrow && \downarrow^{\mathrlap{(h_2,h_2)}}
     \\
     \bullet &\stackrel{(h_1^{-1} g_1 h_2, h_1^{-1} g_2 h_2)}{\to}
     & \bullet
  }
$$

between functors $I\times I \to \mathbf{B}G \times \mathbf{B}G$
may always be decomposed as

$$
  \array{
     \bullet &\stackrel{(g_1,g_2)}{\to}& \bullet
     \\
     {}^{\mathllap{(e,e)}}\downarrow
     && \downarrow^{\mathrlap{(g_2^{-1}, g_2^{-1})}}
     \\
     \bullet &\stackrel{(g_1 g_2^{-1}, e)}{\to}& \bullet
     \\
     {}^{\mathllap{(h_1,h_1)}}\downarrow
     && \downarrow^{\mathrlap{(h_1,h_1)}}
     \\
     \bullet &\stackrel{(h_1^{-1}(g_1 g_2^{-1})h_1, e)}{\to}& \bullet
     \\
     {}^{\mathllap{(e,e)}}\downarrow
     && \downarrow^{\mathrlap{(g'_2,g'_2)}}
     \\
     \bullet &\stackrel{(h_1^{-1}(g_1 g_2^{-1})h_1 g'_2, g'_2)}{\to}& \bullet
  }
  \,.
$$

Staring at this for a moment shows that this is a unique factorization of every morphism through one of the form

$$
  \array{
    \bullet & \stackrel{(g,e)}{\to} & \bullet
    \\
    {}^{\mathllap{k}}\downarrow && \downarrow^{\mathrlap{k}}
    \\
    \bullet & \stackrel{(Ad_k g,e)}{\to} & \bullet
  }
  \,,
$$

which is naturally identified with a morphism in the [[action groupoid]] $G//_{Ad} G$ of the adjoint action of $G$ on itself.

This means that the inclusion

$$
  G//_{Ad} G
  \stackrel{}{\hookrightarrow}
  G\backslash\backslash(G \times G)//G
$$

given by this identification
is [[essentially surjective functor|essentially surjective]] and [[full and faithful functor|full and faithful]], and hence an [[equivalence of categories|equivalence of groupoids]].

So in conclusion we have that the free loop space object of the [[delooping]] $\mathbf{B}G$ of a group is

$$
  \mathcal{L} \mathbf{B}G \simeq G//_{Ad}G
  \,.
$$

=--

### Chern character

We describe how the [[Chern character]] of [[vector bundle]]s over $X$ may be realized in terms of the [[cohomology]] of the free loop space object $\mathcal{L}X$.

Assume now $C$ is a nice category of smooth spaces, and let $X$ be an object of $C$.

Consider a [[group object]] $G$ in $C$ and a [[representation]] of $G$ given by a group homomorphism to the [[general linear group]] (in $C$):  $\rho:G\to GL(n;\mathbb{C})$. For instance $G$ could be $GL(n)$ itself and this morphism the identity.

The [[trace]] of the representation $\rho$ is invariant under conjugation in the group and so defnes a map $Tr(\rho): G//_{Ad}G\to \mathbb {C}$ -- a [[class function]]. By the equivalence $\mathcal{L}\mathbf{B}G \simeq G//_{Ad} G$ discussed above, this may be regarded as a [[characteristic class]]

$$
  Tr(\rho(-)) : \mathcal{L}\mathbf{B}G\to \mathbb {C}
$$

on the free loop space of $\mathbf{B}G$.

The [[cocycle]] $g : X\to \mathbf{B}G$ of a $G$-[[principal bundle]] on $X$ transgresses to a cocycle

$$
  \mathcal{L} g : \mathcal{L}X \to \mathcal{L}\mathbf{B}G
$$

on the free loop space, by the functoriality of the free loop space object construction.

The above [[characteristic class]] of this cocycle is the composite morphism

$$
  Tr(\rho(\mathcal{L}g)) : \mathcal{L}X \to \mathcal{L} \mathbf{B}G \to \mathbb{C}
  \,,
$$

which by the $Ad$-invariance of the [[trace]] is now $S^1$-invariant and hence defines an element in the [[cyclic cohomology]]
$C(\mathcal{L}X,\mathbb{C})^{S^1_C}$ of $X$.

The Hom-space $C(\mathcal{L}X,\mathbb{C})$ is a model for the graded commutative algebra of complex-valued [[differential form]]s on $X$, with the categorical circle action corresponding to the [[de Rham complex|de Rham differential]]. Hence $C(\mathcal{L}X,\mathbb{C})^{S^1_C}$ is a model for closed forms and maps to [[de Rham cohomology]] $H_{dR}^\bullet(X)$ of $X$. If the [[de Rham theorem]] holds for $X$ in $C$, then this may be identified with the real cohomology $H^\bullet(X,\mathbb{R})$.

In the case that $G=GL(\infty;\mathbb{C})$, the compatibility of the trace with [[direct sum]]s and [[tensor product]]s of [[vector bundle]]s over $X$ makes the above construction a [[ring]] homomorphism $K(X)\to H_{dR}(X)$ from the [[topological K-theory]] of $X$ to [[de Rham cohomology]], hence a very good candidate to being the [[Chern character]]

( _to be completed..._ )

### Isotropy of a topos

The [[isotropy group of a topos]] is its free loop space object in the 2-category [[Topos]].

## Related concepts

* [[loop space object]], **free loop space object**,

  * [[delooping]]

  * [[loop space]], [[free loop space]], [[derived loop space]]

* [[suspension object]]

  * [[suspension]]

## References
 {#References}

Free loop space objects in the [[(∞,1)-topos]] of [[derived stack]]s on the site of [[differential graded algebra]]s are discussed in

* [[David Ben-Zvi]], [[David Nadler]], _Loop Spaces and Connections_ ([arXiv](http://arxiv.org/abs/1002.3636))

More information in the topological case is given in:

* [[Ronnie Brown]] _Crossed modules and the homotopy 2-type of a free loop space_, ([arXiv](http://arxiv.org/abs/1003.5617))

which gives complete information on the 2-type of $\mathcal{L}X$ for a space $X$ which is the classifying space of a crossed module of groups. This generalises the above example of  $ \mathcal{L} \mathbf{B}G$.

[[!redirects free loop space objects]]


free loop space of classifying space

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

#Contents#
* table of contents
{:toc}

## Idea

For $\mathcal{G}$ a [[topological group]] or [[simplicial group]], or generally an [[∞-group]], the [[homotopy type]] of the [[free loop space]] of its [[classifying space]]/[[delooping]] is [[equivalence in an (infinity,1)-category|equivalent]] to the [[homotopy quotient]] of the [[adjoint action|conjugation]] [[∞-action]] of $\mathcal{G}$ on iself:

$$
  \mathcal{L} \mathbf{B}\mathcal{G}
  \;\;
  \simeq
  \;\;
  \mathcal{G} \sslash_{ad} \mathcal{G}
  \,.
$$

For [[discrete groups]] (in particular for [[finite groups]]) this is seen by elementary inspection (Example \ref{FreeLoopSpaceOfDeloopingOfDiscreteGroup} below) and as such is familiar from many related constructions, such as that of [[inertia groupoids]].

In more generality the statement is [[folklore]] (as witnessed by parts of the [MO discussion](#MODiscussion)) but rarely argued in detail.

For [[topological groups]] of the [[homotopy type]] of a [[CW-complex]] there is a [[point-set topology|point-set]] argument ([Gruher 2007](#Gruher07)) with continuous paths in the [[universal principal bundle]] which, with some care, largely mimics the naive example .

More generally, such as for [[simplicial groups]], there is a more abstract computation of the defining [[homotopy pullback]] by [[fibrant resolution]], which is spelled out as Prop. \ref{FreeLoopSpaceOfSimplicialClassifyingSpaceAsAdQuotient} below, following an analogous argument for topological groups indicated in [Klein, Schochet & Smith 2009](#KleinSchochetSmith09).

Finally, since, in the full generality of [[∞-groups]] [[internalization|internal to]] any [[(∞,1)-topos]], hence for [[∞-group]] [[∞-stacks]] $\mathcal{G}$, the [[homotopy quotient]] of an action, regarded in the [[slice (∞,1)-topos|slice]] over the [[delooping]]/[[moduli stack]] $\mathbf{B}\mathcal{G}$, in fact characterizes and *defines* the action as an [[∞-action]]

$$
  \mathcal{G} \,\in\, Groups(\mathbf{H})
  \;\;\;\;\;\;
  \Rightarrow
  \;\;\;\;\;\;
  \mathcal{G}Actions(\mathbf{H})
  \;\;
  \underoverset
    {\;\;\;\;\simeq\;\;\;\;}
    {(-)\sslash \mathcal{G}}
    {\longrightarrow}
  \;\;
  \mathbf{H}_{/\mathbf{B}\mathcal{G}}
$$

one may turn this around ([NSS 12a, Exp. 4.13 (3)](#NSS12a), [SS 20, Exp. 2.82](#SS20)) and *define* the [[adjoint action]] of an [[∞-group]] [[∞-stack]] to be that whose [[homotopy quotient]] over $\mathbf{B}\mathcal{G}$ is the left vertical morphism (eq:HomotopyPullbaclDefiningFreeLoopSpaceObject) in the defining [[homotopy pullback]]-construction of the [[free loop space object]] of $\mathbf{B}\mathcal{G}$.

## Proof

The statement is [[folklore]], but complete proofs in the literature are rare.

Recall that the [[free loop space object]] of $\mathbf{B}\mathcal{G}$ (in particular) is defined to be [[homotopy pullback]]/[[homotopy fiber product]] of the [[diagonal morphism]] on $\mathbf{B}\mathcal{G}$ along/with itself:

\[
  \label{HomotopyPullbaclDefiningFreeLoopSpaceObject}
  \array{
    \mathcal{L} \mathbf{B}\mathcal{G}
    &\longrightarrow&
    \mathbf{B}\mathcal{G}
    \\
    \big\downarrow
    &\swArrow_{\mathrlap{pb}}&
    \big\downarrow {}^{\mathrlap{diag}}
    \\
    \mathbf{B}\mathcal{G}
    &\xrightarrow{\;\;\;diag\;\;\;}&
    \mathbf{B}\mathcal{G} \times \mathbf{B} \mathcal{G}
  }
\]

### For simplicial sets
 {#ForSimplicialSets}

We discuss the [[presentable (infinity,1)-category|presentation]] of the phenomenon in the [[classical model structure on simplicial sets]].

Let

* $\mathcal{G}$ be a [[simplicial group]],

and write

* $\mathcal{G}\Actions(sSet)$ for the [[category]] of $\mathcal{G}$-[[action objects]] [[internalization|internal to]] [[SimplicialSets]]l

* $W \mathcal{G} \in \mathcal{G}Actions(sSet)$ for its [[universal principal simplicial complex]];

* $\overline{W}\mathcal{G} \,=\, \frac{W \mathcal{G}}{\mathcal{G}} \in sSet$ for the [[simplicial classifying space]];

* $\mathcal{G}_{ad} \in \mathcal{G}Actions(sSet)$ for the [[adjoint action]] of $\mathcal{G}$ on itself:

  \[
    \label{ConjugationAction}
    \array{
      \mathcal{G}_{ad} \times \mathcal{G}
      &\xrightarrow{\;\;\;}&
      \mathcal{G}_{ad}
      \\
      (g_k,h_k) &\mapsto&  h_k \cdot g_k \cdot h_k^{-1}
    }
  \]

  which we may understand as the restriction along the [[diagonal morphism]] $\mathcal{G} \xrightarrow{diag} \mathcal{G} \times \mathcal{G}$ of the following action of the [[direct product group]]:

  $$
    \array{
      \mathcal{G}_{ad}
      \times
      (\mathcal{G} \times \mathcal{G})
      &\xrightarrow{\;\;\;}&
      \mathcal{G}_{ad}
      \\
      (g_k, (h'_k, h_k))
      &\mapsto&
      h'_k \cdot g_k \cdot h^{-1}_k
      \mathrlap{\,.}
    }
  $$

\begin{proposition}\label{FreeLoopSpaceOfSimplicialClassifyingSpaceAsAdQuotient}
  The [[free loop space object]] of the [[simplicial classifying space]] $\overline{W} \mathcal{G}$ is [[isomorphism|isomorphic]] in the [[classical homotopy category]] to the [[Borel construction]] of the [[adjoint action]] (eq:ConjugationAction):

$$
  \mathcal{L}
  \big(
    \overline{W}\mathcal{G}
  \big)
  \;\;
  \simeq
  \;\;
  \mathcal{G}_{ad} \sslash \mathcal{G}
  \;\;\;\;\;\;
  \in
  \;\;
  Ho\big(
    sSet_{Qu}
  \big)
$$
\end{proposition}
\begin{proof}

{#TheDiagram} Consider the following [[commuting diagram]] of [[SimplicialSets]]:

\begin{tikzcd}
    &&
    &[+35pt]
    \frac{W \mathcal{G}}{\mathcal{G}}
    \times W\mathcal{G}
    \ar[
      dr,
      "\mathrm{pr}_1 \in \mathrm{W}"{description}
    ]
    &[-50pt]
    \\
    \frac{
      \mathclap{\phantom{\vert_a}}
      W \mathcal{G} \times \mathcal{G}_{\scalebox{.4}{ad}}
    }{
      \mathcal{G}
    }
    \ar[dd]
    \ar[rr]
    \ar[
      ddrr,
      phantom,
      "\mbox{\tiny\textrm (pb)}"{pos=.14}
    ]
    &&
    \frac{
      \mathclap{\phantom{\vert_a}}
      W \mathcal{G} \times W\mathcal{G} \times \mathcal{G}_{\scalebox{.4}{ad}}
    }
    {
      \mathcal{G} \times \mathcal{G}
    }
    \ar[
      ur,
      shorten <=-3pt,
      shorten >=-4pt,
      "\sim"{sloped, below},
      "{
        \scalebox{.8}{$
        \!\!\!\!
        \big[
          \vec g, (\vec g',h)
        \big]
        \mapsto
        \Big(
          [\vec g],\,
          \big(
            \vec g^{-1}
            \cdot
            (h \cdot \vec g')
          \big)
        \Big)
        $}
      }"{sloped, above}
    ]
    \ar[
      dd,
      "\in \mathrm{Fib}"{right},
      "
       \frac{
         W(\mathcal{G} \times \mathcal{G})
         \times
         \left(
           \!\!\!\!\!\!
           \begin{array}{c}
             \mathcal{G}_{\scalebox{.4}{ad}}
             \\
             \downarrow
             \\
             \ast
           \end{array}
           \!\!\!\!\!\!
         \right)
       }{
         \mathclap{\phantom{\vert^{a}}}
         \mathcal{G} \times \mathcal{G}
       }
      "{left}
    ]
    &
    &
    \frac{
      W\mathcal{G}
    }
    {
     \mathcal{G}
    }
    \ar[
      ddll,
      "\mathrm{diag}"{right, xshift=4pt}
    ]
    \ar[
      ll,
      "\in \mathrm{W}"{below},
      "{
        [ \vec g, (\vec g, e) ]
        \raisebox{5pt}{\rotatebox{180}{$\mapsto$}}
        [\vec g]
      }"{above}
    ]
    \\
    \\
    \frac{
      W\mathcal{G}
    }{
      \mathcal{G}
    }
    \ar[
      rr,
      "\mathrm{diag}"{below}
    ]
    &&
    \frac{
      W \mathcal{G} \times W\mathcal{G}
    }{
      \mathcal{G} \times \mathcal{G}
    }
\end{tikzcd}

Here:

* all [[objects]] are [[fibrant objects]] ([[Kan complexes]]), since [[simplicial classifying spaces]] $\overline{W}\mathcal{G}$ are Kan complexes, by [this Prop.](simplicial+classifying+space#SimplicialClassifyingSpaces), which, recall, comes about as follows:

  * [[simplicial groups]] have underlying [[Kan complexes]] ([this Thm.](simplicial+group#MooreTheorem), which we use below to know that $\mathcal{G}_{ad} \to \ast $ is a [[Kan fibration]]), hence are [[fibrant objects]] in the [[model structure on simplicial groups]];

  * the [[simplicial classifying space]]-functor $W(-)/(-)$ is a [[right Quillen functor]] to the [[model structure on reduced simplicial sets]] ([this Prop.](model+structure+on+reduced+simplicial+sets#QuillenAdjunctionWithSimplicialGroups)), hence takes values in [[fibrant objects]];

  * fibrant reduced simplicial sets are Kan complexes ([this Prop.](model+structure+on+reduced+simplicial+sets#FibrationsInTermsOfKanFibrations));

* the right vertical morphism is a [[fibration]] ([[Kan fibration]]), since it is the image of the Kan fibration $\mathcal{G}_{ad} \to \ast$ under the [[right Quillen functor]] $(W \mathcal{G} \times (-)) /\mathcal{G}$ on the [[Borel model structure]] ([this Prop.](Borel+model+structure#QuillenAdjunctionWithSliceOverSimplicialClassifyingSpace)), see [this Example](Borel+construction#ProjectionToSimplicialClassifyingSpaceIsKanFibration);

* the top right morphism is a [[simplicial weak equivalence]], by [[two-out-of-three]], as it is a [[right inverse]] of the triangular morphisms, which are manifestly a simplicial weak eqivalence since $W \mathcal{G}$ is [[contractible homotopy type|contractible]] (by [this Prop.](simplicial+classifying+space#UniversalPrincipalSimplicialComplexIsContractible)).

One verifies by inspection that the [[commuting square]] shown is an ordinary [[pullback]] in [[SimplicialSets]].

Finally notice that the [[simplicial classifying space]]-construction is indeed a model for the [[delooping]]

$$
  \overline{W}\mathcal{G}
  \;\;
  \simeq
  \;\;
  \mathbf{B}\mathcal{G}
  \;\;\;
  \in
  \;
  Ho(\infty Groupoids)
$$

(essentially by its [[Quillen adjunction]] to the [[simplicial loop space]]-functor ([here](model+structure+on+reduced+simplicial+sets#RelationToSimplicialGroups)) and the [[May recognition theorem]], see [NSS 12a, Cor. 3.33](free+loop+space+of+classifying+space#NSS12b)).

In summary this means (by [this Prop.](homotopy+pullback#HomotopyPullbackByOrdinaryPullback)) that this ordinary [[pullback]]-square represents the [[homotopy pullback]] (eq:HomotopyPullbaclDefiningFreeLoopSpaceObject) that defines the [[free loop space object]].

\end{proof}

### For topological spaces
 {#ForTopologicalSpaces}

A [[point-set topology|point-set argument]] for [[topological spaces]]/[[topological groups]] is spelled out in [Gruher 2007, App. A](#Gruher07). Discussion of the [[presentable (infinity,1)-category|presentation]] of the phenomenon in the [[classical model structure on topological spaces]] in the abstract style [above](#ForSimplicialSets) is indicated in [Klein, Schochet & Smith 2009](#KleinSchochetSmith09).

### In homotopy type theory
  {#InHomotopyTypeTheory}

Recall that in [[homotopy type theory]], an [[∞-group]] $G$ is represented by its [[delooping]] type, a pointed connected type $\mathbf{B} G$.

(The [categorical semantics](relation+between+type+theory+and+category+theory#HomotopyWithUnivalence) of this is a groupal [[A-∞ algebra|A-∞]] [[∞-stack]], in some [[(∞,1)-topos]], so that the following is actually about free loop spaces in the generality of [[inertia ∞-stacks]], not though in the yet further generality of [[free loop ∞-stacks]].)

An [[∞-action]] of $G$ on a [[term]] $a$ of [[type]] $A$ is given by a [[group homomorphism]] $G \to Aut_A(a)$, represented by a morphism of [[pointed homotopy types]] $\mathbf{B} G \to_* (A,a)$. (Since $\mathbf{B} G$ is [[connected homotopy type|connected]], it doesn't matter whether we restrict the [[codomain]] to the [[connected component]] of $A$ at $a$.)

We thus see that the type of all [[∞-actions]] of $G$ on objects of $A$ is the [[function type]] $\mathbf{B} G \to A$. In particular, the type of $U$-small $G$-types, where $U$ is a [[type universe]], is $\mathbf{B} G \to U$.

By [[adjoint (∞,1)-functor|adjointness]], the [[homotopy quotient|homotopy orbit]] type of a $G$-type $X \colon \mathbf{B} G \to U$ is given by the [[dependent sum]]-type,

$$
  X \!\sslash\! G
   \;\coloneqq\;
  \sum_{t : \mathbf{B} G} X(t)
$$

(The [[homotopy fixed points]] are given by the [[dependent product]]-type, as discussed at *[[∞-action]]*.)

Now, the [[adjoint action]] of $G$ on itself is given by the morphism

$$
  \array{
    G^{ad} \colon & \mathbf{B} G &\to& U
    \\
    & t &\mapsto& (t =_{\mathbf{B} G} t)
    \,.
  }
$$

Indeed, given any path $p : t =_{\mathbf{B} G} u$ in $\mathbf{B} G$ and any element $q : G^{ad}(t)$, the [[homotopy lifting property|transport]] of $q$ along $p$ in the family $G^{ad}$ is equal to the conjugate $p^{-1} \cdot q \cdot p : G^{ad}(u)$, as proven by [[identity type|path induction]]. (Here, we write path composition in diagrammatic order.)

Putting this together, we get that the [[homotopy quotient|homotopy orbits]] of the [[adjoint action]] are

\[
  \begin{aligned}
    G^{ad} \!\sslash\! G
    &
     \;=\;
    \sum_{t: \mathbf{B} G}
      (t =_{\mathbf{B} G} t)
    \\
    & \;\simeq\;
    (ʃS^1 \to \mathbf{B} G),
  \end{aligned}
\]

where in the last step we used the [[universal property]] of the [[homotopy type]] ([[shape modality|shape]]) of the [[circle type]], $ʃS^1 \simeq \mathbf{B}\mathbb{Z}$, defined as a [[higher inductive type]] ([here](higher+inductive+type#ExamplesTheCircle)) with a point [[term introduction rule|constructor]] $base : ʃS^1$ and a path constructor $loop : base =_{ʃS^1} base$.

The function type $ʃS^1 \to B G$ is the representation of the free loop type $\mathcal{L}(\mathbf{B} G)$ of $\mathbf{B} G$, completing the argument.

## Examples

### For discrete (finite) groups
 {#ExampleDiscreteGroups}

\begin{example}\label{FreeLoopSpaceOfDeloopingOfDiscreteGroup}
**([[free loop space]] of classifying space of [[discrete groups]])** \linebreak

The archetypical example has $\mathcal{G} = G \in Groups(Sets)$ a [[discrete group]], such as (but not necessarily) a [[finite group]]. In this case the [[classifying space]] $B \mathcal{G} \simeq K(G,1)$ is an [[Eilenberg-MacLane space]] whose [[homotopy type]] is represented in the [[classical homotopy category]] simply by the [[delooping groupoid]] $G \rightrightarrows \ast$ of $G$. With this, the analysis of its [[free loop space]] follows from elementary inspection:

The [[hom-groupoid]]

$$
  \big[
    \mathbf{B}\mathbb{Z}
    ,\,
    \mathbf{B}G
  \big]
  \;\simeq\;
  \big(
     G \times G
     \underoverset
       {pr_2}
       { Ad }
       {\rightrightarrows}
     G
  \big)
$$

has

* as [[objects]] the [[functors]] $\mathbf{B}\mathbb{Z} \longrightarrow \mathbf{B}G$, hence equivalently [[group homomorphisms]] $\mathbb{Z} \to G$, hence equivalently elements $g \in G$;

* as [[morphisms]] the [[natural transformations]] between these, whose single component $h \in G$ must make the naturality [[commuting square|square commute]]

$$
  \array{
    \mathbf{B}\mathbb{Z}
    &\xrightarrow{\;\;\;}&
    \mathbf{B}G
    \\
    \\
    \bullet
    &&
    \bullet
    &\xrightarrow{\;\;\;h\;\;\;}&
    \bullet
    \\
    \big\downarrow {}^{\mathrlap{1}}
    &&
    \big\downarrow {}^{\mathrlap{g}}
    &&
    \big\downarrow {}^{\mathrlap{g'}}
    \\
    \bullet
    &&
    \bullet
    &\xrightarrow{\;\;\;h\;\;\;}&
    \bullet
  }
$$

The condition that this commutes means equivalently that $g'$ is the image of $g$ under the [[conjugation action]] by $h$:

$$
  h \cdot g' \;=\; g \cdot h
  \;\;\;\;\;\;\;\;\;
  \Leftrightarrow
  \;\;\;\;\;\;\;\;\;
  g' = h^{-1} \cdot g \cdot h \;=\; Ad_h(g)
  \,.
$$

This shows that the [[hom-groupoid]] is the [[action groupoid]] of the [[conjugation action]]

$$
  \mathcal{L} \mathbf{B}G
  \;\simeq\;
  \big[
    \mathbf{B}\mathbb{Z}
    ,\,
    \mathbf{B}G
  \big]
  \;\simeq\;
  G \sslash_{ad} G
  \,.
$$

\end{example}

This simple example essentially re-appears in the discussion of [[inertia groupoids]].

### For simplicial abelian groups
 {#ExampleForSimplicialAbelianGroups}

As a direct corollary of the general statement we have:

\begin{prop}\label{FreeLoopSpaceOfClassifyingSpaceOfSimplicialAbelianGroup}
  For $\mathcal{A} \in AbGroup(sSet)$
  a [[simplicial group|simplicial]] [[abelian group]],
  the
  [[homotopy type]] of the
  [[free loop space]] of its [[simplicial classifying space]]
  is the [[homotopy product]] ([[Cartesian product]]) of $\mathcal{A}$ with its [[delooping]]:
  $$
    \big[
      \mathbf{B}\mathbb{Z},
      \,
      \mathbf{B}\mathcal{A}
    \big]
    \;\;
    \simeq
    \;\;
    \mathcal{A} \,\times\, \mathbf{B}\mathcal{A}
    \,.
  $$
\end{prop}
\begin{proof}
This is the composite of the following sequence of equivalences ([[isomorphisms]] in the [[classical homotopy category]]):
$$
  \begin{aligned}
    \big[
      \mathbf{B}\mathbb{Z},
      \,
      \mathbf{B}\mathcal{A}
    \big]
    &
    \;\simeq\;
    \frac{
      \mathcal{A}_{\mathrm{ad}}
        \times
      W\mathcal{A}
    }{ \mathcal{A} }
    \\
    & \;\simeq\;
    \frac{
      \mathcal{A}_{\mathrm{triv}}
      \times
      W\mathcal{A}
    }{ \mathcal{A} }
    \\
    & \;\simeq\;
    \mathcal{A}
      \times
    \frac{
      W\mathcal{A}
    }{ \mathcal{A} }
    \\
    & \;\simeq\;
    \mathcal{A}
    \times
    \overline{W}\mathcal{A}
    \\
    & \;\simeq\;
    \mathcal{A} \times \mathbf{B}\mathcal{A}
  \end{aligned}
$$

Here:

* the first step is Prop. \ref{FreeLoopSpaceOfSimplicialClassifyingSpaceAsAdQuotient};

* the second step observes that for [[abelian groups]] the [[adjoint action]] (eq:ConjugationAction) is actually the [[trivial action]];

* the third step is the evident consequence for the [[quotient]];

* the fourth step is the definition of the [[simplicial classifying space]];

* the last step is the identification of the simplicial classifying space with the [[delooping]] as [[∞-groups]].

\end{proof}

As an immediate special cases of Prop. \ref{FreeLoopSpaceOfClassifyingSpaceOfSimplicialAbelianGroup} we have:

\begin{example}\label{FreeLoopSpaceOfEilenbergMacLaneSpaces}
**([[free loop space]] of [[Eilenberg-MacLane spaces]])** \linebreak

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

$$
  \mathbf{B}^{n+1}
  \mathbb{Z}
  \;\simeq\;
  ʃ \mathbf{B}^n S^1
$$

the [[shape modality|shape]] of the [[circle n-group|circle (n+1)-group]], we have

$$
  \mathcal{L} \mathbf{B}^{n+1}\mathbb{Z}
    \;\simeq\;
  \mathbf{B}^n\mathbb{Z}
    \,\times\,
  \mathbf{B}^{n+1} \mathbb{Z}
  \,.
$$

More generally, for $A \in AbGroups(Set)$ any [[discrete group|discrete]] [[abelian group]] we have

$$
  \mathcal{L} \mathbf{B}^{n+1}A
  \;\simeq\;
  \mathbf{B}^n A
    \,\times\,
  \mathbf{B}^{n+1}A
  \,,
$$

where  the $n$-fold [[delooping]]

$$
  \mathbf{B}^n A
  \;\simeq\;
  K(n,A)
$$

is equivalently the [[homotopy type]] of the [[Eilenberg-MacLane space]] for $A$ in degree $n$ (classifying [[ordinary cohomology]] in degree $n$ with [[coefficients]] in $A$).
\end{example}

\begin{remark}
  Example \ref{FreeLoopSpaceOfEilenbergMacLaneSpaces} pertains to the discussion of [[double dimensional reduction]] of brane charges in [[ordinary cohomology]] by the discussion [here](geometry+of+physics+--+fundamental+super+p-branes#DoubleDimensionalReductionViaCyclification) at *[[geometry of physics -- fundamental super p-branes]]*.
\end{remark}

## Related concepts

* [[free loop space]], [[free loop stack]]

* [[transchromatic character]]

* [[double dimensional reduction]]

* [[classifying space]], [[simplicial classifying space]],

* [[Sullivan model of classifying space]]

## References

A point-set proof in [[TopologicalSpaces]] is given in

* {#Gruher07} [[Kate Gruher]], *A proof of $L B G \simeq Ad(E G)$*, Appendix A in: *String Topology of Classifying Spaces*, 2007 ([[Gruher_FreeLoopSpaceOfClassifyingSpace.pdf:file]], [proquest:304826261](https://www.proquest.com/docview/304826261))

An abstract proof in the [above](#ForSimplicialSets) style, for [[topological groups]]/[[topological spaces]], is indicated in:

* {#KleinSchochetSmith09} John R. Klein, [[Claude Schochet]], Samuel B. Smith, Lemma 9.1 of: *Continuous trace $C^\ast$-algebras, gauge groups and rationalization*,  Journal of Topology and Analysis Vol. 01, No. 03, pp. 261-288 (2009)  ([arXiv:0811.0771](https://arxiv.org/abs/0811.0771), [doi:10.1142/S179352530900014X](https://doi.org/10.1142/S179352530900014X))

See also:

* {#MODiscussion} MathOverflow discussion [MO/q/20671](https://mathoverflow.net/q/20671/381)

Discussion in the generality of [[∞-groups]] in [[(∞,1)-toposes]] and defining the [[adjoint action]] in this generality:

* {#NSS12a} [[Thomas Nikolaus]], [[Urs Schreiber]], [[Danny Stevenson]], Example 4.13 (3) in: *[[schreiber:Principal ∞-bundles -- theory, presentations and applications|Principal ∞-bundles -- General theory]]*, Journal of Homotopy and Related Structures, Volume 10, Issue 4 (2015), pages 749-801
([doi:10.1007/s40062-014-0083-6](http://link.springer.com/article/10.1007/s40062-014-0083-6), [arXiv:1207.0248](http://arxiv.org/abs/1207.0248))

* {#SS20} [[Hisham Sati]], [[Urs Schreiber]], Example 2.82 in: *[[schreiber:Proper Orbifold Cohomology]]* ([arXiv:2008.01101](https://arxiv.org/abs/2008.01101))

and formulated more in the language of [[homotopy type theory]]:

* {#BuchholtzDoornRijke18} [[Ulrik Buchholtz]], [[Floris van Doorn]], [[Egbert Rijke]], p. 11 of: _Higher Groups in Homotopy Type Theory_, LICS '18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science  ([arXiv:1802.04315](https://arxiv.org/abs/1802.04315), [doi:10.1145/3209108.3209150](https://doi.org/10.1145/3209108.3209150))

[[!redirects free loop spaces of classifying spaces]]

[[!redirects free loop space of a classifying space]]


free module

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###Context###
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
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#### Homological algebra
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[[!include homological algebra - contents]]
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#Contents#
* table of contents
{:toc}

## Idea

A _free module_ over some [[ring]] $R$ is [[free construction|freely generated]] on a [[set]] of [[basis]] elements.

Under the [interpretation of modules as generalized vector bundles](module#RelationToVectorBundlesInIntroduction) a _free module_ corresponds to a _trivial_ bundle.

## Definition

### General

Let $C$ be a [[monoidal category]], and $Alg(C)$ the [[category of monoids]] in $C$; and for $A \in Alg(C)$ let $A$[[Mod]]$(C)$ be the category of $A$-[[modules]] in $C$.

There is the evident [[forgetful functor]] $U : A Mod(C) \to C$ that sends each module $(N,\rho)$ to its underlying object $N \in C$.

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

The [[left adjoint]] $C \to A Mod(C)$ is the corresponding [[free construction]]. The modules in the image of this functor are _free modules_.

=--

### Over rings

Let $R$ be a ring. We discuss free [[modules]] over $R$.

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

For $R \in $ [[Ring]] a [[ring]] and $S \in $ [[Set]], the free $R$-module on $S$ is isomorphic to the ${\vert S\vert}$-fold [[direct sum]] of $R$ with itself

$$
  R^{(S)}\simeq \oplus_{s \in S} R
  \,.
$$

=--

## Properties

### As a monoidal functor

Let $R$ be a commutative ring, and let $R\{X\}$ denote the free $R$-module on a set $X$.

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

The free $R$-module functor is strong monoidal with respect to the Cartesian monoidal structure on sets, and the tensor product of $R$-modules.

=--

In other words, the free module construction turns set-theoretic products into tensor products. Thus, it preserves algebraic objects (such as [[monoid object]]s, [[Hopf monoid]] objects, etc.) and their homomorphisms. In particular, if $M$ is a monoid in the category of sets (and hence a bimonoid with the canonical comonoid structure) then $R\{M\}$ is a bimonoid object in $R \mathsf{Mod}$, which is precisely a $K$-bialgebra. A group $G$ in the category of sets is a Hopf monoid, and hence $R\{G\}$ is a Hopf algebra --- this is precisely the [[group algebra]] of $G$.

### Submodules of free modules
 {#SubmodulesOfFreeModules}

Let $R$ be a commutative [[ring]].

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

Assuming the [[axiom of choice]], the following are equivalent

1. every [[submodule]] of a free $R$-module is itself free;

1. every [[ideal]] in $R$ is a free $R$-module;

1. $R$ is a [[principal ideal domain]].

=--

+-- {: .proof}
###### Proof
(See also [Rotman, pages 650-651](#Rotman).) Condition 1. immediately implies condition 2., since ideals of $R$ are the same as submodules of $R$ seen as an $R$-module. Now assume condition 2. holds, and suppose $x \in R$ is any nonzero element. Let $\lambda_x$ denote multiplication by $x$ (as an $R$-module map). We have a sequence of surjective $R$-module maps

$$R \stackrel{\lambda_x}{\to} (x) \cong \oplus_J R \stackrel{\nabla}{\to} R$$

(where $\nabla$ is the [[codiagonal]] map); by the [[Yoneda lemma]], the composite map $R \to R$ is of the form $\lambda_r$, where $r \in R$ is the value of the composite at $1 \in R$. Since $\lambda_r$ is surjective, we have $\lambda_r(s) = r s = 1$ for some $s$, so that $r$ is invertible. Hence $\lambda_r$ is invertible, and this implies $\lambda_x$ is monic. Therefore $R$ is a domain. From that, we infer that if $f$ and $g$ belong to a basis of an ideal $I$, then

$$0 \neq f g \in R\cdot f \cap R \cdot g$$

whence $f$ and $g$ are not linearly independent, so $f = g$ and $I$ as an $R$-module is generated by a single element, i.e., $R$ is a principal ideal domain.

That condition 3. implies condition 1. is proved [here](http://ncatlab.org/nlab/show/principal+ideal+domain#free).
=--

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

Assuming the [[axiom of choice]], over a ring $R$ which is a [[principal ideal domain]], every [[module]] has a [[projective resolution]] of length 1.

=--

See at [projective resolution -- Resolutions of length 1](projective+resolution#Lenght1ResolutionsOfAbelianGroups) for more.

### Over a field: vector spaces

\begin{proposition}
\label{EveryModuleOverAFieldIsFree}
Assuming the [[axiom of choice]], if $R = k$ is a [[field]] then every $R$-[[module]] is free: it is $k$-[[vector space]] and by the _[[basis theorem]]_ every such has a [[basis of a vector space|basis]].
\end{proposition}

## Related concepts

* [[locally free module]]

* [[finitely generated module]], [[finitely presented module]]

* **free module** $\Rightarrow$ [[projective module]] $\Rightarrow$ [[flat module]] $\Rightarrow$ [[torsion-free module]]

* [[basis of a free module]]

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

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

  * [[injective module]]

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

* flat object, [[flat resolution]]

  * [[flat module]]

## References

Textbooks:

* {#Rotman} J. Rotman, pp. 650--651 in: _Advanced Modern Algebra_, AMS 2017  ([ISBN:978-1-4704-2311-7](https://bookstore.ams.org/gsm-180/))

[[!redirects free modules]]


free monad

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# Free monads
* table of contents
{: toc}

## Definition

A **free monad** is a [[free object]] relative to a [[forgetful functor]] whose domain is a [[category]] of [[monads]].

This general concept has many different specific incarnations, since there are potentially many different such forgetful functors.  Suppose $C$ is a [[category]], and write $Mnd(C)$ for the category whose objects are monads on $C$ and whose morphisms are natural transformations commuting with the monad structure maps; i.e. it is the category of [[monoids]] in the [[monoidal category]] of endofunctors with composition.  Then we have a string of forgetful functors:
$$ Mnd(C) \to PtEnd(C) \to End(C) \to [ob(C),C] $$
where $End(C)$ denotes the category of [[endofunctors]] of $C$, and $PtEnd(C)$ denotes the category of [[pointed endofunctors]], i.e. endofunctors $F$ equipped with a natural transformation $Id\to F$.  A *free monad* can then be considered as a free object relative to any one of these forgetful functors.

## Free finitary monads

In general, these forgetful functors cannot be expected to have [[left adjoints]], i.e. there will not be a [[free functor|"free monad functor"]], but individual objects can often be shown to generate free monads.  One general case in which this is true is when $C$ is [[locally presentable category|locally presentable]] and we consider monads and endofunctors which are [[accessible functor|accessible]], i.e. preserve sufficiently highly [[filtered colimits]].  Suppose for the sake of argument that $C$ is locally *finitely* presentable (the higher-ary case is analogous).  Then we can restrict the above string of forgetful functors to the [[finitary monads]], i.e. those preserving filtered colimits, to obtain:
$$ Mnd_f(C) \to PtEnd_f(C) \to End_f(C) \to [ob(C)_f,C] $$
where the subscript $f$ denotes restriction to finitary things, and $ob(C)_f$ is the set of [[compact objects]] of $C$.  In this case, all these forgetful functors do have left adjoints, and moreover at least the functors $Mnd_f(C) \to End_f(C)$ and $Mnd_f(C) \to [ob(C)_f,C]$ are monadic.  (This is shown in the papers cited below.)  The construction is by a convergent [[transfinite composition]].

For example, the left adjoint to $Mnd_f(C) \to End_f(C)$, shows that there exists a "free finitary monad" on any finitary endofunctor.  Note, though, that this does not *automatically* imply that the "free finitary monad" on a finitary endofunctor is also a "free monad" on that endofunctor, i.e. that as a free object it satisfies the requisite universal property relative to all objects of $Mnd(C)$, not merely those lying in $Mnd_f(C)$.  It is, however, true: free finitary monads are also free monads.

## Algebraically-free monads

We say that a monad $T$ is **algebraically-free** on an endofunctor $F$ if the category $T Alg_{mnd}$ of $T$-algebras (in the sense of [[algebras for a monad]]) is equivalent to the category $F Alg_{endo}$ of $F$-algebras (in the sense of [[algebras for an endofunctor]]). **N.B.**: Any such equivalence must be an isomorphism $T Alg_{mnd} \cong F Alg_{endo}$, because the underlying functors from the categories of algebras in each case are [[amnestic functor|amnestic]] [[isofibrations]]. See remarks at the article [[monadicity theorem]] on monadicity _vis-à-vis_ strict monadicity.

*A priori*, being algebraically free is different from being free.  However, one can show the following.

+-- {: .num_theorem}
###### Theorem
Any algebraically-free monad is free.
=--
+-- {: .proof}
###### Proof
First observe that for a (perhaps pointed) endofunctor $F$ and a monad $T$, to give a functor $T Alg_{mnd} \to F Alg_{endo}$ over $C$ is equivalent to giving a (pointed) transformation $F\to T$, and if $F$ is a monad then such a functor takes values in $F Alg_{mnd}$ iff the transformation $F\to T$ is a monad morphism.  Thus, if $F Alg_{endo} \cong T Alg_{mnd}$, then for any other monad $S$, (pointed) transformations $F\to S$ correspond to maps $S Alg_{mnd} \to F Alg_{endo} \cong T Alg_{mnd}$ and hence to monad morphisms $T\to S$, i.e. $T$ is free on $F$.
=--

+-- {: .num_theorem}
###### Theorem
If $C$ is [[locally small category|locally small]] and [[complete category|complete]], then any free monad is algebraically-free.
=--
+-- {: .proof}
###### Proof
For any object $x\in C$, the assumptions ensure that the [[codensity monad]] of $x$ exists.  This is the right [[Kan extension]] of $x\colon 1\to C$ along itself, which we write as $\langle x,x\rangle$.  The universal property of Kan extensions means that for any endofunctor $F$, to give a map $F x \to x$ (i.e. to make $x$ into an $F$-algebra) is the same as to give a natural transformation $F\to \langle x,x\rangle$.  Moreover, one can check that if $F$ is a pointed endofunctor (resp. a monad), then the map $F x \to x$ is a pointed (resp. monad) algebra iff the corresponding transformation $F\to \langle x,x\rangle$ is a morphism of pointed endofunctors (resp. of monads).  Therefore, if $T$ is the free monad on $F$, then applying its universal property in $Mnd(C)$ to the monad $\langle x,x\rangle$, we see that it is also algebraically-free.
=--

Notice that this second proof relies crucially on the fact that free monads have a universal property relative to a forgetful functor whose domain is all of $Mnd(C)$, not just some subcategory of finitary or accessible monads, since $\langle x,x\rangle$ will not in general be finitary or accessible.

## Constructions

Perhaps the most general [[set theory|set-theoretically]] based construction of (algebraically) free monads is the [[transfinite construction of free algebras]].  (In [[type theory]], it is natural to use instead [[higher inductive types]].)

## Examples

* The monadicity of the above adjunctions can be used to give [[presentation]]s of monads in terms of [[generators and relations]].  This has close connections with [[Lawvere theories]] and related ideas.

* Free monads on pointed endofunctors play an important role in the construction of cofibrantly generated [[algebraic weak factorization systems]].

## Related concepts

* The [[algebra over a monad]] over a free monad on an endofunctor is an _[[algebra over an endofunctor]]_.

## References

* [[Max Kelly]], ["A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on"](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=4759448) (1980)

* [[Max Kelly]] and [[John Power]], ["Adjunctions whose counits are coequalizers, and presentations of finitary enriched monads"](http://www.sciencedirect.com/science/article/pii/0022404993900928)

* [[Steve Lack]], ["On the monadicity of finitary monads"](http://www.maths.usyd.edu.au/res/Catecomb/Lack/1997-29.html)

* {#GambinoHyland04} [[Nicola Gambino]], [[Martin Hyland]], section 6 of _Wellfounded trees and dependent polynomial functors_. In Types for proofs and programs, volume 3085 of Lecture Notes in Comput. Sci., pages 210-225 Springer (2004) [[web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.98.4534)]

* [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Coequalizers and free triples, II*, Theory Appl. Categories **34** (2019) 662–683 [[tac:34-23](http://www.tac.mta.ca/tac/volumes/34/23/34-23abs.html)]

[[!redirects free monads]]
[[!redirects algebraically-free monad]]
[[!redirects algebraically-free monads]]
[[!redirects algebraically free monad]]
[[!redirects algebraically free monads]]