transferred model structure

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###Context###
#### Model category theory
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[[!include model category theory - contents]]
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#Contents#
* table of contents
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## Idea

For $C$ a [[category]] with the structure of a [[model category]] and $U:D\to C$ a functor having an [[adjoint]], under certain conditions it is possible to _transfer_ the model structure from $C$ to a model structure on $D$ by declaring the weak equivalences, and the fibrations (if $U$ is a right adjoint) or cofibrations (if $U$ is a left adjoint), in $D$ to be precisely those morphisms whose images under $U$ are such in $C$.  If $U$ is a right adjoint this is called the **right transferred**, **right induced**, or **right lifted** model structure, and dually in the left case.

Typically this arises in situations where $D$ consist of the "same" objects as $C$ but equipped with extra [[stuff, structure, property]], and $U$ is the corresponding [[forgetful functor]] sending objects in $D$ to their underlying objects in $C$.  Then a left adjoint $F$ of $U$ is the corresponding [[free functor]], while a right adjoint $G$ is a [[cofree functor]].

## Definition

### Right transfer
 {#RightTransfer}

\begin{definition}\label{RightTransferredModelStructure}
**(right transferred model structure)**
\linebreak
Let $C$ be a [[model category]] and

$$
  (F \dashv U )
  \;\colon\;
  D
  \underoverset
    {\underset{U}{\longrightarrow}}
    {\overset{F}{\longleftarrow}}
    {\;\;\;\bot\;\;\;}
  C
$$

a pair of [[adjoint functors]] with [[right adjoint]] $U$.  Define a morphism in $D$ to be

* a *fibration* or *weak equivalence* precisely if its image under $U$ is so in $C$, respectively

  (by [adjointness](adjoint+functor#InTermsOfHomIsomorphism) this means that the fibrations form equivalently the class with the [[right lifting property]] against the image under $F$ of the [[acyclic cofibrations]] in $C$);

* a *cofibration* precisely if it has the [[left lifting property]] with respect to the fibrations that are also weak equivalences, according to the previous item.

If these classes of maps define a [[model category]] [[mathematical structure|structure]] on $D$, then it is called the *right transferred model structure* ([Crans 1995, §3](#Crans95))
or sometimes *right induced model structure* or *right lifted model structure* from $C$ along $R$.
\end{definition}

In the above situation of Def. \ref{RightTransferredModelStructure} it is clear what one means by an [[acyclic fibration]] in $D$, but the notion of [[acyclic cofibration]] in $D$ is, *a priori*, ambiguous. For the purpose of stating the following Thm. \ref{NeccSuff} we declare that:

1. an *[[acyclic fibration]]* in $D$ is (of course) a morphism that is both a fibration and a weak equivalence, or equivalently whose image under $U$ is an acyclic fibration in $C$.

1. a morphism is $D$ is

   1. a **cofibration weak equivalence** if it is both a cofibration and a weak equivalence in the sense of Def. \ref{RightTransferredModelStructure};

   1. an **anodyne map** if it has the [[left lifting property]] with respect to all fibrations.

   Of course, if the transferred model structure exists, then these two classes of maps coincide.

Conversely we have:

\begin{theorem}\label{NeccSuff}
A necessary and sufficient condition for the existence of the right transferred model structure (Def. \ref{RightTransferredModelStructure}) is that:

1. Every morphism in $D$ factors as

   1. a cofibration followed by a trivial fibration,

   1. an anodyne map followed by a fibration;

1. every anodyne map is a weak equivalence.

\end{theorem}

\begin{proof}
Clearly the conditions are necessary.  For the converse, the weak equivalences have the [[2-out-of-3 property]], all the classes of maps are closed under [[retracts]], the lifting properties hold by definition (for the anodyne maps), and we have assumed the factorization properties (for the anodyne maps), so it remains to show that every cofibration weak equivalence is anodyne.  This follows by the standard [[retract argument]]: if $f \colon A \longrightarrow B$ is a cofibration weak equivalence, factor it as $f = p i$ with $i$ anodyne and $p$ a fibration.  Then since anodyne maps are weak equivalences and weak equivalences satisfy [[2-out-of-3]], $p$ is an acyclic fibration.  Thus $f$ has the left lifting property against $p$, hence $f$ is a [[retract]] of $i$ and thus also anodyne.
\end{proof}

### Left transfer
 {#LeftTransfer}

[[formal duality|Dually]], if $C$ is a model category and we have an [[adjoint functor|adjunction]]
$$
  (U \dashv G )
  \;\colon\;
  D
  \underoverset
    {
      \underset{G}{\longleftarrow}
    }
    {
      \overset{U}{\longrightarrow}
    }
    {\;\;\;\bot\;\;\;}
  C
$$
with $U$ [[left adjoint]],  define a morphism in $D$ to be

* a **cofibration** or **weak equivalence** precisely if its image under $U$ is, respectively, in $C$.

* a **fibration** precisely if it has the [[right lifting property]] with respect to the cofibrations that are [[weak equivalences]].

If these classes of maps define a model structure on $D$, it is called the **left transferred model structure** (or sometimes **left induced model structure** or **left lifted model structure**) from $C$.

The above necessary and sufficient conditions dualize directly.

## Existence

### For right transfer
 {#ConstructingFactorizationsForRightTransfer}

#### Constructing factorizations
 {#ConstructingFactorizations}

The most traditional way to obtain the factorization properties in the right-transferred case (Thm. \ref{NeccSuff}) is to assume that $C$ is [[cofibrantly generated]].

\begin{proposition}\label{SufficientConditions}
In the right-transferred situation (Def. \ref{RightTransferredModelStructure}), suppose that

1. $C$ is [[cofibrantly generated]],

1. the functor $F$ preserves [[small objects]]

   (which is the case, in particular, when $U$ preserves [[filtered colimits]]),

then every morphism in $D$ factors as a cofibration followed by a trivial fibration, and as an anodyne map followed by a fibration.

Moreover, if every anodyne map is a weak equivalence so that the transferred model structure on $D$ exists (by Thm. \ref{NeccSuff}), then it is itself cofibrantly generated, and for $I$ (resp. $J$) a set of [[generating cofibrations]] (resp. acyclic cofibrations) in $C$, the image set $F(I)$ (resp. $F(J)$) forms a set of generating cofibrations (resp. acyclic cofibrations) in $D$.
\end{proposition}

\begin{proof}
By the [hom-isomorphism](adjoint+functor#InTermsOfHomIsomorphism) of the [[adjoint pair]] $F \dashv U$, a morphism $f$ is a fibration (resp. trivial fibration) if and only if it has the right lifting property against the maps in $F(I)$ (resp. $F(J)$).  Since $F$ preserves small objects, the sets $F(I)$ and $F(J)$ permit the small object argument in $D$, which therefore yields (cofibration, trivial fibration) and (anodyne, fibration) factorizations.
\end{proof}

Another way to produce these factorizations is to generate them by a [[functorial factorization]]:

+-- {: .num_lemma #AlgebraicLift}
###### Lemma
Suppose $F \colon C \rightleftarrows D \colon U$ is an [[adjoint functor|adjunction]], where $D$ has [[pushouts]], and that

1. $C$ has a [[functorial factorization]] $(L,R)$ with a [[right weak composition law for factorizations]] (e.g. if it is [[awfs|algebraic]]),

1. the induced functorial factorization on $D$ permits Garner's [[small object argument]] (e.g. if $C$ and $D$ are locally presentable and all functors are accessible).

Then there is an induced ([[awfs|algebraic]]) weak factorization system on $D$ whose right class consists precisely of those maps whose $U$-image admits an $R$-algebra structure.
=--
+-- {: .proof}
###### Proof
Given a morphism $f\colon A\to B$ in $D$, factor it as follows, where the upper commutative square is a pushout:
$$\array{
  F U A & \longrightarrow & A
  \\
  \mathllap{ {}^{F ((U f)_L)} }
  \big\downarrow & & \big\downarrow
  \\
  F E (U f) & & E f
  \\
  \mathllap{ {}^{F ((U f)_R)} }
  \big\downarrow & & \big\downarrow
  \\
  F U B & \longrightarrow & B
}$$
This yields a functorial factorization $(L',R')$ on $D$.  By the universal property of pushout, an $R'$-algebra structure on $f$ is determined by a diagonal lifting in the square
$$
  \array{
    F U A & \longrightarrow & A
    \\
    \mathllap{ {}^{F ((U f)_L)} }
    \big\downarrow & & \big\downarrow \mathrlap{{}^f}
    \\
    F E (U f)
    &
    \underset{F ((U f)_R)}{\longrightarrow}
    &
    B
  }
$$
which corresponds by adjunction to precisely an $R$-algebra structure on $U f$.  Thus, the right weak composition law for $R$-algebras lifts to a right weak composition law for $R'$-algebras.  Garner's small object argument applied to $(L',R')$ then produces a monad over $cod$ with a right strong composition law, which is therefore an [[algebraic weak factorization system]] whose algebraic right maps are the $R'$-algebras, and in particular whose underlying right chlass is as desired those maps whose $U$-image is in $R$.
=--

In particular, therefore, the factorizations of any [[accessible model structure]] right-lift along any adjunction between locally presentable model categories, whether or not they are cofibrantly generated.  But the cofibrantly generated case can also be subsumed, by taking $(L,R)$ to be the one-step factorization produced by the generating left maps.

Left-lifting is generally rather trickier.  But by invoking fancier categorical machinery, one can show more generally:

\begin{lemma}
  \label{TransferFromAnAccessibleFactorizationSystem}
  Suppose given a functor $U \colon D\to C$ between [[locally presentable categories]] and an [[accessible weak factorization system]] $(L,R)$ on $C$.

  1. If $U$ has a left adjoint $F$, then $(L,R)$ right-lifts along $U$ to an accessible wfs on $D$.

  1. If $U$ has a right adjoint $G$, then $(L,R)$ left-lifts along $U$ to an accessible wfs on $D$.

  Moreover, if $(L,R)$ is cofibrantly generated, so are the lifted wfs.

\end{lemma}
\begin{proof}
  See [HKRS15](#HKRS15) and [GKR18](#GKR18) for the lifting of accessible wfs.  Cofibrant generation of a right-lifted wfs follows as in Theorem \ref{SufficientConditions} above, while in the left-lifted case it is Remark 3.8 of [Makkai & Rosický 2014](#MakkaiRosický14).
\end{proof}

This sort of result seems to be necessary for the case of left-lifting; the simpler cofibrantly-generated argument does not dualize as directly.

#### Verifying acyclicity

The "acyclicity condition" in Thm. \ref{NeccSuff}, i.e. that anodyne maps are weak equivalences, is usually the most difficult to check.  Sometimes useful is the observation that for right-lifting in the cofibrantly generated case, it suffices to show that [[sequential colimits]] of [[pushouts]] of [[images]] under $F$ of the generating acyclic cofibrations in $C$ (i.e. forming $F(J)$-[[cell complexes]]) yield weak equivalences in $D$.  This is because the small object actually factors any map as such an $F(J)$-cell complex followed by a fibration; hence by the retract argument every anodyne map is a retract of an $F(J)$-cell complex.

Another useful sufficient condition is the following, going back roughly to section II.4 of ([Quillen](#Quillen)).

\begin{proposition}\label{PathObjects}
In the situation of right transfer (Def. \ref{RightTransferredModelStructure}), where $U \colon D\to C$ is a [[right adjoint]], suppose that

1. $D$ has a [[fibrant replacement functor]] (in fact it suffices for individual objects and morphisms to have fibrant replacements, while functoriality is not necessary),

1. $D$ has [[path objects]] for fibrant objects, i.e. a factorization of the [[diagonal]] $\Delta : A \to A \times A$ as a weak equivalence followed by a fibration $\Delta \colon A \stackrel{\simeq}{\to} P(A) \stackrel{fib}{\to} A \times A$.

Then any anodyne map in $D$ is a weak equivalence.  Thus, if the two factorizations exist in $D$, then the transferred model structure exists.
\end{proposition}

\begin{proof}
The notation in the following is mostly that of [Rezk 02, Lemma 7.6](#Rezk02).
Let $R$ be a fibrant replacement functor, with natural weak equivalence $j \colon Id \to R$, suppose $f \colon X\to Y$ is anodyne, and consider the following [[commuting diagram]]:

\begin{center}
\begin{tikzcd}
X \ar[dr,"i" description] \ar[r,"f"] \ar[d,"{(1,f)}"'] &
Y \ar[r,"{j_Y}"] & R Y \ar[d] \\
X\times Y \ar[dr,"{j_X \times j_Y}"'] &
P R f \ar[r] \ar[d]  & P R Y \ar[d]\\
& R X \times R Y \ar[d,"{\mathrm{pr}_1}"'] \ar[r,"{R f \times 1}"'] & R Y \times R Y \ar[d,"{\mathrm{pr}_1}"'] \\
& R X \ar[r,"R f"'] & R Y
\end{tikzcd}
\end{center}

Both squares are [[pullbacks]], defining the object $P R f$ and the morphism $i$ making the following square commute:

\begin{center}
\begin{tikzcd}
X \ar[d,"f"'] \ar[r,"i"] & P R f \ar[r,"\pi"] \ar[d] & R X\\
Y \ar[r,"{j_Y}"'] \ar[ur,dashed,"g"] & R Y.
\end{tikzcd}
\end{center}

The morphism $\pi$ is the composite $P R f\to R X \times R Y \to R X$ at the top, which is a pullback of $P R Y \to R Y$; but the latter is a fibration (as the composite $P R Y \to R Y \times R Y \to R Y$) and a weak equivalence (as a retraction for $R Y \to P R Y$), so it and hence also $\pi$ are acyclic fibrations.  Moreover, since $\pi i = j_X$, by [[2-out-of-3]] $i$ is also a weak equivalence.

Now the projection $P R f\to P Y$ is a fibration, so since $f$ is anodyne there is a lift $g$ in the second square as shown.  Since $i$ and $j_Y$ are weak equivalences, by [[2-out-of-6]] it follows that $f$ is a weak equivalence.
\end{proof}

Note that this condition also dualizes straightforwardly to the left-transferred case.

### For left transfer
 {#ConstructingFactorizationsForLeftTransfer}

\begin{proposition}
\label{RecognitionOfLeftTransferUnderCofibrantlyGeneration}
Given a pair of [[adjoint functors]]
$
  \mathcal{D}
    \underoverset
      {\underset{R}{\longleftarrow}}
      {\overset{L}{\longrightarrow}}
      {\;\;\bot\;\;}
  \mathcal{C}
$
such that:

1. $\mathcal{C}$ and $\mathcal{D}$ are [[locally presentable categories]],

1. $\mathcal{C}$ is equipped with the [[mathematical structure|structure]] of a [[cofibrantly generated model category]] (hence a [[combinatorial model category]]) with [[class|classes]] of ([[cofibration|co]]-)/[[fibrations]] and [[weak equivalences]] $Cof, Fib, W \,\subset\, Mor(\mathcal{C})$,

1. $RLP\big( L^{-1} Cof \big) \,\subset\, L^{-1}(W)$ (i.e. co-anodyne maps are weak equivalences),

then the left-transferred model category structure on $\mathcal{D}$ exists (i.e. with cofibrations $L^{-1}(Cof)$ and weak equivalences $L^{-1}(W)$) and is itself [[cofibrantly generated model category|cofibrantly generated]].
\end{proposition}
\begin{proof}
\label{ProofOfCombinatorialLeftTransfer}
  As observed in the proof of [BHKKRS 2015, Thm. 2.23](#BHKKRS15), the existence of the required cofibrantly generated [[factorization systems]] follows by [Makkai & Rosický 2014, Rem. 3.8](#MakkaiRosický14) with the comment below Def. 2.3 there, which in turn invokes [Lurie 2009, §A.1.5.12](#Lurie09). In particular, from inside the proof of [Makkai & Rosický 2014, Thm. 3.2](#MakkaiRosický14) one has that a [[small set]] of generating (acyclic) cofibrations of the left transferred model structure is given by those between [[presentable object|$\kappa$-presentable objects]], for some  un-[[countable set|countable]] [[regular cardinal]] $\kappa$.
\end{proof}

## Properties

### General

\begin{lemma}
If $C$ carries the structure of a [[right proper model category]], then also a right-transferred model structure on $D$ (Def. \ref{RightTransferredModelStructure}) is right proper.
\end{lemma}

\begin{proof}
Let

$$
  \array{
     A \times_C B &\stackrel{f}{\to}& B
     \\
     \downarrow && \downarrow^{\mathrlap{\in Fib}}
     \\
     A &\stackrel{\simeq}{\to}& C
  }
$$

be a [[pullback]] diagram in $D$, with the bottom morphism a weak equivalence and the right morphism a fibration. We need to show that then also the top morphism $f$ is a weak equivalence. By definition of transfer, this is equivalent to $U(f)$ being a  weak equivalence in $C$.

Since $U$ is a [[right adjoint]] it preserves pullbacks, so that also

$$
  \array{
     U(A \times_{C} B) &\stackrel{U(f)}{\to}& U(B)
     \\
     \downarrow && \downarrow^{\mathrlap{\in Fib}}
     \\
     U(A) &\stackrel{\simeq}{\to}& U(C)
  }
$$

is a pullback diagram in $C$. Since by definition of the transferred model strucure this is still the pullback of a weak equivalence along a fibration, and since $C$ is assumed to be right proper, it follows that $U(f)$ is a weak equivalence in $C$, hence that $f$ is a weak equivalence in $D$.
\end{proof}

### Enrichment {#Enrichment}

Often the underlying model category $C$ is an [[enriched model category]] over some [[monoidal model category]] $S$ and one wishes to transfer also the model enrichment.

\begin{proposition}
Assume the adjunction

$$
  (F \dashv U )\; : \; D \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C
$$

satisfies the conditions of the above proposition for right transfer, so that the model structure on $C$ is transferred to $D$. Consider the case that $C$ is moreover an $S$-[[enriched model category]] and that $D$ can be equipped with the structure of a $S$-[[enriched category]] that is also  $S$-[[power]]ed and [[copower]]ed.

Assume now that the $S$-powering of $D$ is taken by $U$ to the $S$-powering of $C$, in that $U(d^{(s_1 \to s_2)}) = U(d)^{(s_1 \to s_2)}$.

Then the transferred model structure and the $S$-enrichment on $D$ are compatible and make $D$ an $S$-enriched model category.
\end{proposition}

\begin{proof}
By the axioms of [[enriched model category]] one sufficient condition to be checked is that for $s \to t$ any cofibration in $S$ and for $X \to Y$ any fibration in $D$, we have that the induced morphism

$$
  X^t \to X^s \times_{Y^s} Y^{t}
$$

is a fibration, which is a weak equivalence if at least one of the two input morphisms is. By the induced model structure, this is checked by applying $U$. But by assumption $U$ commutes with the powering, and since $U$ is a [[right adjoint]] it commutes with taking the pullback, so that under $U$ the morphism is

$$
  U(X)^t \to U(X)^s \times_{U(Y)^s} U(Y)^{t}
$$

which is the morphism induced from $U(X) \to U(Y)$. That this is indeed an (acyclic) fibration follows now from the fact that $C$ is an $S$-enriched model category.
\end{proof}

## Examples

\begin{example}\label{RightTransferAlongAdjointEquivalence}
  The right-transfer along an [[adjoint equivalence]] $L \dashv R$ always exists, and here also the [[cofibrations]] are given by the [[preimage]] under $R$.
\end{example}
This is straightforward to check directly, even without the above existence theorems (e.g. [math.se:a/2220718](https://math.stackexchange.com/a/2220718/58526)).

\begin{example}
The [[projective model structure on functors]] $M^D$ is right-transferred from the product model structure on $M^{ob D}$.  Dually, the [[injective model structure on functors]], when it exists, is left-transferred from $M^{ob D}$.
\end{example}

\begin{example}
The [[model structure on algebraic fibrant objects]] is right-transferred from the underlying model category by forgetting the choice of fillers.
\end{example}

\begin{example}
If $T$ is an [[accessible functor|accessible]] [[strict 2-monad]] on a [[locally finitely presentable category|locally finitely presentable 2-category]] $K$. then the category $T Alg_s$ of strict $T$-[[algebra over a monad|algebras]] admits a right-transferred model structure from the [[2-trivial model structure]] on $K$.  The acyclicity condition is proved by using [[pseudolimits]] of arrows for path objects, and the (cofibration, trivial fibration) factorization is constructed using a version of the construction of Lemma \ref{AlgebraicLift} (since the 2-trivial model structure on $K$ may not be cofibrantly generated).
\end{example}

\begin{example}
A *[[counterexample]]* is provided as Example 3.7 of ([Goerss & Schemmerhorn 2007](#GoerssShemmerhorn07)). Let $k$ be a field of characteristic 2 and consider the adjunction
$$
 (S \dashv U )\; : \; CGA_k \stackrel{\overset{S}{\leftarrow}}{\underset{U}{\to}} Ch_*k
$$
of the symmetric algebra functor and the forgetful functor between graded commutative DGAs and chain complexes. One sees that $S$ does not preserve the weak equivalence between 0 and the complex with one copy of $k$ in degrees $n$ and $n-1$. Since all chain complexes are cofibrant this means that $(S \dashv U )$ cannot be upgraded to a Quillen adjunction.
\end{example}

\begin{example}
A more category-theoretic *[[counterexample]]* is that the [[Reedy model structure]] on [[directed graphs]] [[internalization|internal]] to [[Cat]] does not right-transfer to the category [[DblCat]] of [[double categories]] along the free-forgetful adjunction.  One of the generatic trivial cofibrations for the Reedy model structure "transports" a horizontal arrow (in double-category terminology) along two vertical isomorphisms, which is a levelwise equivalence.  But upon pushing out the free double category generated by this along a map into some other double category, new composite horizontal arrows may be generated in the codomain that are not isomorphic to the image of anything in the domain; thus the pushout is no longer a levelwise equivalence.  So, the acyclicity condition fails.
\end{example}

## References
 {#References}

The explicit notion of right transfer is due to:

* {#Crans95} [[Sjoerd Crans]], §3-§4 in: *Quillen closed model structure for sheaves*, Journal of Pure and Applied Algebra **101** 1 (1995) 35-57 [<a href="https://doi.org/10.1016/0022-4049(94)00033-F">doi:10.1016/0022-4049(94)00033-F</a>, [web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.48.7459)]

and for the case of [[cofibrantly generated model categories]] due to:

* {#DwyerHirschhornKan97} [[William G. Dwyer]], [[Philip S. Hirschhorn]], [[Daniel M. Kan]], §9.1 in: *[[Model Categories and More General Abstract Homotopy Theory]]* (1997) [[pdf](https://people.math.rochester.edu/faculty/doug/otherpapers/dhk.pdf), [[DwyerHirschhornKan-ModelCategories.pdf:file]]]

{#FurtherReferencesForRightTransfer} The existence result of right transfer due to [Crans 1995, Thm. 3.3](#Crans95) and [Dwyer, Hirschhorn & Kan 1997, Thm. 91](#DwyerHirschhornKan97) ("Crans-Kan transfer theorem") is reviewed/reworked, with variations, in:

* {#GoerssJardine99} [[Paul Goerss]], [[J. F. Jardine]], Section II.4 of: _[[Simplicial homotopy theory]]_, Progress Mathematics **174**, Birkhäuser (1999) [[doi:10.1007/978-3-0346-0189-4](https://link.springer.com/book/10.1007/978-3-0346-0189-4)]

* {#Hirschhorn02} [[Philip Hirschhorn]], Thm. 11.3.2 in: *[[Model Categories and Their Localizations]]*, Math. Survey and Monographs **99**, AMS (2002) [[ISBN:978-0-8218-4917-0](https://bookstore.ams.org/surv-99-s/), [pdf toc](http://www.gbv.de/dms/goettingen/360115845.pdf), [pdf](http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf)]

* {#Cisinski06} [[Denis-Charles Cisinski]], Prop. 1.4.23 of: *Les préfaisceaux comme types d'homotopie|Les préfaisceaux comme modèles des types d'homotopie*, Astérisque **308** Soc. Math. France (2006), 392 pages [[numdam:AST_2006__308__R1_0](http://www.numdam.org/item/?id=AST_2006__308__R1_0), [pdf](http://www.math.univ-toulouse.fr/~dcisinsk/ast.pdf)]

* {#GoerssShemmerhorn07} [[Paul Goerss]], [[Kristen Schemmerhorn]], [§3.2](http://arxiv.org/PS_cache/math/pdf/0609/0609537v2.pdf#page=20) of: *Model Categories and Simplicial Methods*, Notes from lectures given at the University of Chicago, August 2004, in: *Interactions between Homotopy Theory and Algebra*, Contemporary Mathematics **436** AMS (2007) [[arXiv:math.AT/0609537](http://arxiv.org/abs/math.AT/0609537), [doi:10.1090/conm/436](http://dx.doi.org/10.1090/conm/436)]

* [[Clemens Berger]], [[Ieke Moerdijk]], Section 2.5 of: *Axiomatic homotopy theory for operads*, Comment. Math. Helv. **78** (2003) 4  [[arXiv:math/0206094](https://arxiv.org/abs/math/0206094), [doi:10.1007/s00014-003-0772-y](https://doi.org/10.1007/s00014-003-0772-y), [pdf](http://www.math.uu.nl/publications/preprints/1242.pdf)]

These arguments all go back (cf. [Goerss & Shemmerhorn 2007, p. 20](#GoerssShemmerhorn07)) to:

* {#Quillen67} [[Daniel Quillen]], §II.4 in: *[[Homotopical Algebra]]*, Lecture Notes in Mathematics **43**, Springer (1967) [[doi:10.1007/BFb0097438](https://doi.org/10.1007/BFb0097438)]

The above proof of acyclicity in the presence of path objects is taken from:

* {#Rezk02} [[Charles Rezk]], *Every homotopy theory of simplicial algebras admits a proper model*, Topology Appl. **119** 1 (2002) 65-94 [[arxiv:math/0003065](https://arxiv.org/abs/math/0003065)]

Discussion of left transfer (left-induced model structures):

* {#HessShipley14} [[Kathryn Hess]], [[Brooke Shipley]], §4 in: *The homotopy theory of coalgebras over a comonad*, Proceedings of the London Mathematical Society **108** 2 (2014) [[arXiv:1205.3979](https://arxiv.org/abs/1205.3979), [doi:10.1112/plms/pdt038](https://doi.org/10.1112/plms/pdt038)]

* {#BHKKRS15} [[Marzieh Bayeh]], [[Kathryn Hess]], [[Varvara Karpova]], [[Magdalena Kedziorek]], [[Emily Riehl]], [[Brooke Shipley]], *Left-induced model structures and diagram categories*, in: *Women in Topology: Collaborations in Homotopy Theory*, Contemporary Mathematics **641** American Mathematical Society (2015) [[arXiv:1401.3651](http://arxiv.org/abs/1401.3651), [ISBN:978-1-4704-2495-4](https://bookstore.ams.org/conm-641)]

* {#HKRS15} [[Kathryn Hess]], [[Magdalena Kedziorek]], [[Emily Riehl]], [[Brooke Shipley]], *A necessary and sufficient condition for induced model structures*, J. Topology **10** 2  (2017) 324-369 [[arXiv:1509.08154](http://arxiv.org/abs/1509.08154), [doi:10.1112/topo.12011](https://doi.org/10.1112/topo.12011)]

Beware that [HKRS15](#HKRS15) contains an error, corrected by:

* {#GKR18} [[Richard Garner]], [[Magdalena Kedziorek]], [[Emily Riehl]], *Lifting accessible model structures*, J. Topology **13** 1 (2020) 59-76 [[arXiv:1802.09889](https://arxiv.org/abs/1802.09889), [doi:10.1112/topo.12123](https://doi.org/10.1112/topo.12123)]

Technical lemmas regarding the underlying [[cellular categories]]:

* {#MakkaiRosický14} [[M. Makkai]], [[J. Rosický]], _Cellular categories_, J. Pure Appl. Alg. **218** (2014) 1652-1664 [[arXiv:1304.7572](https://arxiv.org/abs/1304.7572), [doi:10.1016/j.jpaa.2014.01.005](https://doi.org/10.1016/j.jpaa.2014.01.005)]

* {#Lurie09} [[Jacob Lurie]], Appendix A of: *[[Higher Topos Theory]]* (2009)

The right-transferred model structure on algebras for a 2-monad is from

* {#Lack06} [[Steve Lack]], *Homotopy-theoretic aspects of 2-monads*, [arXiv](http://arxiv.org/abs/math.CT/0607646)

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transfinite arithmetic

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

#Contents#
* table of contents
{:toc}

## Idea

The term _transfinite arithmetic_ refers to [[arithmetic]] of quantities ([[numbers]]) larger than the the finite [[natural numbers]], for instance _[[cardinal arithmetic]]_ is the arithmetic of [[cardinalities]] of [[sets]] possibly larger than that of [[finite sets]]. Similarly there is _[[ordinal arithmetic]]_.

## References

See also

* Wikipedia, _[Transfinite number](https://en.wikipedia.org/wiki/Transfinite_number)_

[[!redirects transfinite number]]
[[!redirects transfinite numbers]]


transfinite composition

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

# Contents
* table of contents
{:toc}

## Idea

The axioms of a [[category]] ensure that every _finite_ number of composable morphisms has a (unique) [[composite]].

_Transfinite composition_ is a means to talk about morphisms in a category that behave as if they were the result of composing infinitely many morphisms.

## Definition

Transfinite composition is indexed by [[ordinals]]. For convenience we first recall the definition of these assuming [[excluded middle]] in the ambient [[set theory]] (for definitions not assuming this see at _[[ordinal]]_ and pointers given there):

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

A _[[partial order]]_ is a [[set]] $S$ equipped with a [[relation]] $\leq$ such that for all elements $a,b,c \in S$

1) ([[reflexive relation|reflexivity]]) $a \leq a$;

2) ([[transitive relation|transitivity]]) if $a \leq b$ and $b \leq c$ then $a \leq c$;

3) ([[antisymmetric relation|antisymmetry]]) if a $a\leq b$ and $\b \leq a$ then $a = b$.

This we may and will equivalently think of as a [[category]] with [[objects]] the elements of $S$ and a unique morphism $a \to b$ precisely if $a\leq b$. In particular an order-preserving function between partially ordered sets is equivalently a [[functor]] between their corresponding categories.

A _[[bottom element]]_ $\bot$ in a partial order is one such that $\bot \leq a$ for all a. A _[[top element]]_ $\top$ is one for wich $a \leq \top$.

A partial order is a _[[total order]]_ if in addition

4) ([[total relation|totality]]) either $a\leq b$ or $b \leq a$.

A total order is a _[[well order]]_ if in addition

5) ([[well-founded relation|well-foundedness]]) every non-empty subset has a least element.

An _[[ordinal]]_ is the [[equivalence class]] of a well-order.

The _[[successor]]_ of an ordinal is the class of the well-order with a [[top element]] freely adjoined.

A _[[limit ordinal]]_ is one that is not a successor.

=--

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

The finite ordinals are labeled by $n \in \mathbb{N}$, corresponding to the well-orders $\{0 \leq 1 \leq 2 \cdots \leq n-1\}$. Here $(n+1)$ is the successor of $n$. The first non-empty limit ordinal is $\omega = [(\mathbb{N}, \leq)]$.

=--

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

Let $\mathcal{C}$ be a [[category]], and let $I \subset Mor(\mathcal{C})$ be a [[class]] of its morphisms.

For $\alpha$ an [[ordinal]] (regarded as a [[category]]), an $\alpha$-indexed _transfinite sequence_ of elements in $I$ is a [[diagram]]

$$
  X_\bullet
    \;\colon\;
  \alpha \longrightarrow \mathcal{C}
$$

such that

1. $X_\bullet$ takes all [[successor]] morphisms $\beta \stackrel{\leq}{\to} \beta + 1$ in $\alpha$ to elements in $I$

   $$
     X_{\beta,\beta + 1} \in I
   $$

1. $X_\bullet$ is _continuous_ in that for every nonzero [[limit ordinal]] $\beta \lt \alpha$, $X_\bullet$ restricted to the [[full subcategory|full-subdiagram]] $\{\gamma \;|\; \gamma \leq \beta\}$ is a [[colimit|colimiting cocone]] in $\mathcal{C}$ for $X_\bullet$ restricted to $\{\gamma \;|\; \gamma \lt \beta\}$:

   $$
     X_\beta \simeq \underset{\longrightarrow}{\lim}_{\gamma \lt \beta} X_\gamma
     \,.
   $$

The corresponding **transfinite composition** is the induced morphism

$$
  X_0 \longrightarrow X_\alpha \coloneqq \underset{\longrightarrow}{\lim}X_\bullet
$$

into the [[colimit]] of the diagram, schematically:

$$
  \array{
    X_0 &\stackrel{X_{0,1}}{\to}& X_1
    &\stackrel{X_{1,2}}{\to}& X_2
    &\to& \cdots
    \\
    & \searrow & \downarrow & \swarrow & \cdots
    \\
    &&
    X_\alpha
  }
  \,.
$$

=--

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

For purposes of [[constructive mathematics]], the continuity condition should be stated as follows:

* For *every* ordinal $\beta \lt \alpha$, $X_\bullet$ restricted to $\{\gamma \;|\; \gamma \leq \beta\}$ is a colimiting cone in $\mathcal{C}$ for the disjoint union of $\{X_0\}$ and the restriction of $X_{\bullet}$ to $\{\gamma + 1 \;|\; \gamma \lt \beta\}$.

This actually includes $F(0) = X$ as a special case but says nothing when $\beta$ is a successor (so the successor clause is still required).

=--

## Applications

Transfinite composition plays a role in

* the [[small object argument]];

* [[cofibrantly generated model category|cofibrantly generated model categories]].

* the [[transfinite construction of free algebras]]

## Related concepts

* [[transfinite induction]]

## References

For instance

* [[Tibor Beke]], page 6  of _Sheafifiable homotopy model categories_ ([arXiv](http://arxiv.org/abs/math.CT/0102087))

[[!redirects transfinite composite]]
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transfinite construction of free algebras

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

* table of contents
{: toc}

## Idea

In [[category theory]], there is a common problem of the construction of [[free objects]] in "algebraic" categories.  The idea is that one has to start from some "[[generators and relations|generators]]" and repeatedly throw in the results of applying "operations", subject to some "[[generators and relations|relations]]", over and over again until the result "converges".  In general, this convergence will require a [[ordinal number|transfinite]] iteration, and the fact that it converges at all requires some smallness/accessibility hypotheses.

If this construction is performed in sufficient generality, then it includes a large number of important special cases, such as the following (not mutually exclusive):

* the [[free monad]] generated by an [[endofunctor]]
* the [[free monad]] generated by a [[pointed endofunctor]]
* the free [[monoid]] generated by an object of a [[monoidal category]]
* the [[orthogonal factorization system]] generated by a set of morphisms
* the [[reflective subcategory|reflectiveness]] of a [[orthogonality class|small-orthogonality class]]
* the reflectiveness of [[enriched category|enriched]] [[presheaves]] preserving certain [[limits]]
* the [[cocomplete category|cocompleteness]] of the category of [[algebra over a monad|algebras]] for a [[monad]]
* the existence of [[left adjoints]] to functors between categories of algebras for monads
* the existence of colimits for diagrams of monads
* the existence of colimits in categories of monoids
* the [[algebraic small object argument]] generating [[algebraic weak factorization systems]]
* the existence of [[W-types]] and [[inductive types]] in categories
* the existence of [[higher inductive types]] in categories

In 1980, [[Max Kelly]] [(Kelly)](#Kelly) wrote a very long paper detailing very precise and general conditions under which such constructions can be performed, and showing how they can all be reduced to the case of free algebras for *well-pointed endofunctors*.

In what follows we will summarize Kelly's constructions, restricted to the special case of [[accessible functors]] on [[locally presentable categories]].  Much of the technical apparatus of Kelly's paper (including, among other things, two different ambient orthogonal factorization systems $(E,M)$ and $(E',M')$, and assumptions such as "$T$ preserves the $E$-tightness of $(M',K)$-cones") becomes unnecessary in this case, which also includes many if not most of the important examples.

## Basic concepts

Let $\mathcal{A}$ be a [[locally presentable category]].  All endofunctors of $\mathcal{A}$ which we consider will be [[accessible functor|accessible]], that is, they preserve $\kappa$-[[filtered colimits]] for some sufficiently large [[cardinal number]] $\kappa$.  In fact, we will only need to know that they preserve sufficiently long [[transfinite compositions]].

+-- {: .num_defn}
###### Definition
An endofunctor $S:\mathcal{A}\to \mathcal{A}$ is called **[[pointed endofunctor|pointed]]** if it is equipped with a natural transformation $\sigma:Id_\mathcal{A} \to S$.  It is called **well-pointed** if $S\sigma = \sigma S$ (as natural transformations $S\to S^2$).
=--

Of course, a [[monad]] can be regarded as a pointed endofunctor where $\sigma$ is its unit.  Such an endofunctor is well-pointed precisely when the monad is [[idempotent monad|idempotent]].

+-- {: .num_defn}
###### Definition
If $H$ is an endofunctor, then an $H$-[[algebra for an endofunctor|algebra]] is an object $A$ together with a map $H A\to A$.  If $(S,\sigma)$ is a pointed endofunctor, then an $S$-algebra is an object $A$ with a map $a:S A \to A$ such that $a \sigma_A = 1_A$.
=--

We have obvious categories of algebras of both sorts.  Of course, any [[algebra for a monad]] is also an algebra for that monad *qua* pointed endofunctor.  The converse holds in the well-pointed/idempotent case.  In fact, we have:

+-- {: .num_lemma #WellPointedAlgebras}
###### Lemma
If $(S,\sigma)$ is well-pointed, then an object $A$ admits at most one $S$-algebra structure.  This happens exactly when $\sigma_A$ is an isomorphism, in which case its inverse is that unique algebra structure.
=--
+-- {: .proof}
###### Proof
Commutativity of the diagram
$$
\begin{array}{ccc}
S S A & \overset{S a}\to & S A \\
\mathllap{\scriptsize{\sigma_{S A}=S\sigma_A}}\uparrow && \;\;\;\;\uparrow\mathrlap{\scriptsize{\sigma_A}} \\
S A & \underset{a}\to & A
\end{array}
$$
gives that $\sigma_A\circ a= 1_{S A}$, which together with $a\sigma_A= 1_A$ gives the invertibility of $a\colon S A\to A$ and the uniqueness of the algebra structure.
=--
Thus, when $S$ is well-pointed, the category of $S$-algebras is a [[full subcategory]] of $\mathcal{A}$.  Well-pointed endofunctors are important, so we mention a few other ways to obtain them.

+-- {: .num_lemma #WellPointedQuotient}
###### Lemma
If $(S,\sigma)$ is a well-pointed endofunctor and $\alpha\colon S\to T$ is a natural transformation whose components are [[epimorphisms]], then $T$ is also well-pointed.
=--
+-- {: .proof}
###### Proof
The putative transformation $\eta\colon 1\to T$ is the composition $\alpha\circ \sigma$. First of all notice that $\sigma T\circ \alpha = S\eta$, since
$$\begin{array}{cc}
S\eta_A &= S \alpha_A \circ  S\sigma_A \\
&= S\alpha_A \circ \sigma_{S A}\\
&= \sigma_{T A}\circ \alpha_A
\end{array}$$
by naturality of the transformations involved. Now, it suffices to show that $\eta T \circ \alpha = T\eta \circ \alpha$: this follows, again, from
$$\begin{array}{ccc}
(\eta T \circ \alpha)_A &=& \alpha_{T A}\circ \sigma_{T A}\circ \alpha_A \\
&=& \alpha_{T A} \circ S \eta_A\\
&=& T \eta_A \circ \alpha_A \\
&=& (T\eta \circ\alpha)_A\end{array}$$
Now since $\alpha$ is an epimorphism, we can conclude.
=--
+-- {: .num_lemma #WellPointedCointersection}
###### Lemma
Suppose $(S_i,\sigma_i)$ is a family of (accessible) well-pointed endofunctors and let $\sigma:Id \to S$ be the [[wide pushout]] of all the points $\sigma_i:Id \to S_i$.  Then $(S,\sigma)$ is (accessible and) well-pointed, and $S Alg = \bigcap_i (S_i Alg)$.
=--
+-- {: .proof}
###### Proof
The last statement is almost obvious using Lemma \ref{WellPointedAlgebras} above. Well-pointedness for $S$ follows adapting Lemma \ref{WellPointedQuotient} to the case of a jointly epimorphic family of arrows $\{S_i\to S\}$, like the colimiting cocone for $S$.
=--
+-- {: .num_lemma #WellPointedTransfer}
###### Lemma
Let $F:\mathcal{A} \rightleftarrows \mathcal{B} :G$ be an adjunction and $(S',\sigma')$ an (accessible) well-pointed endofunctor on $\mathcal{A}$.  Let $(S,\sigma)$ be the pushout
$$ \array{ F G & \xrightarrow{F \sigma' G} & F S' G \\
  \downarrow && \downarrow \\
  Id & \xrightarrow{\sigma} & S. } $$
Then $(S,\sigma)$ is (accessible and) well-pointed, and $B\in\mathcal{B}$ is an $S$-algebra exactly when $G B$ is an $S'$-algebra.
=--
+-- {: .proof}
###### Proof

=--
## Free algebras for well-pointed endofunctors

The basic theorem to which all others can be reduced is the following.

+-- {: .num_theorem #WellPointedFreeAlgebras}
###### Theorem
If $\mathcal{A}$ is [[locally presentable category|locally presentable]] and $S$ is an accessible well-pointed endofunctor, then $S Alg$ is [[reflective subcategory|reflective]] in $\mathcal{A}$.  Therefore, the algebraically-free monad on $S$ exists (and is accessible).
=--
+-- {: .proof}
###### Proof
Given $A\in\mathcal{A}$, define [[induction|inductively]] the following transfinite sequence in $\mathcal{A}$:
$$ A \to S A \to S^2 A \to S^3 A \to \cdots $$
in which each transition map is a component of $\sigma$, and we take colimits at limit stages.  If we continue this out to some $\kappa$-filtered ordinal $\alpha$, where $S$ is $\kappa$-accessible, then $S$ will preserve the relevant colimit, and hence we have an induced action $S(S^\alpha A) \to S^\alpha A$.  It is easy to check that this makes $S^\alpha A$ into a reflection of $A$ into $S Alg$.
=--

Some important constructions are directly examples of this theorem.

+-- {: .num_example}
###### Example
Let $K$ be a set of morphisms in $\mathcal{A}$; recall that an object $A\in\mathcal{A}$ is said to be *orthogonal* to $K$ if $\mathcal{A}(k,A)$ is a bijection for all $k\in K$, which is to say that each functor $\mathcal{A}(k,-):\mathcal{A}\to Set^{\mathbf{2}}$ takes $A$ to an isomorphism, where $Set^{\mathbf{2}}$ is the [[arrow category]] of [[Set]].
Now the subcategory of isomorphisms is reflective in $Set^{\mathbf{2}}$, hence is the algebras for a well-pointed endofunctor $R_k$.  By Lemma \ref{WellPointedTransfer}, there is a well-pointed endofunctor $S_k$ on $\mathcal{A}$ whose algebras are the objects orthogonal to $k$.  And by Lemma \ref{WellPointedCointersection}, there is a well-pointed endofunctor $S$ whose algebras are those objects orthogonal to all $k\in K$.  Thus, by Theorem \ref{WellPointedFreeAlgebras}, the category of objects orthogonal to $K$ is reflective in $\mathcal{A}$.
=--

+-- {: .num_example}
###### Example
Applying the previous example on [[slice categories]], we can construct [[orthogonal factorization systems]].
=--

The rest of the constructions proceed by building, out of other data, a well-pointed endofunctor, and applying Theorem \ref{WellPointedFreeAlgebras}.

## Free algebras for pointed endofunctors

Now let $(T,\tau)$ be a general (accessible) pointed endofunctor, not necessarily well-pointed.  Let $T/\mathcal{A}$ denote the [[comma category]], whose objects are triples $(A,B,a)$ where $a:T A \to B$.  Then $T/\mathcal{A}$ is again locally presentable.

If $\alpha:T'\to T$ is a natural transformation, we have an adjunction $\alpha_! : T'/\mathcal{A} \rightleftarrows T/\mathcal{A}: \alpha^*$ in which $\alpha^*$ sends $(A,B,a)$ to $(A,B,a\circ \alpha_A)$ and $\alpha_!$ is defined by a pushout.

Note that if $\alpha$ is $\tau:Id \to T$, then $\tau^*(A,B,a)$ is the composite $A \xrightarrow{\tau_A} T A \xrightarrow{a} B$ in $\mathcal{A}^{\mathbf{2}} = Id/\mathcal{A}$.

+-- {: .num_lemma #PointedAlgebrasInComma}
###### Lemma
The category $T Alg$ of $T$-algebras is a reflective full subcategory of $T/\mathcal{A}$.
=--
+-- {: .proof}
###### Proof
It is easy to show that it is a full subcategory.  Moreover, $(A,B,a)$ lies in $T Alg$ just when $\tau^*(A,B,a)$ is an isomorphism.  The isomorphisms are a reflective subcategory of $\mathcal{A}^{\mathbf{2}}$, hence the algebras for a well-pointed endofunctor $R$.  Now by Lemma \ref{WellPointedTransfer}, there is a well-pointed endofunctor $S$ of $T/\mathcal{A}$ whose algebras are precisely the $T$-algebras.  Now apply Theorem \ref{WellPointedFreeAlgebras}.
=--

+-- {: .num_theorem #PointedAlgebras}
###### Theorem
The category $T Alg$ is monadic over $\mathcal{A}$, and cocomplete.  In particular, the [[algebraically-free monad]] on $T$ exists and is accessible.
=--
+-- {: .proof}
###### Proof
A left adjoint to the forgetful functor $T Alg \to \mathcal{A}$ is supplied by reflecting the object $(A,T A,\tau_A)$ of $T/\mathcal{A}$ into $T Alg$.  The [[monadicity theorem]] then applies.  Colimits can be constructed in $T/\mathcal{A}$ and then reflected into $T Alg$.
=--

It is possible to [[beta-reduction|beta-reduce]] this proof and describe the resulting transfinite construction more explicitly.  Given $(A,B,a)$, we define inductively a transfinite sequence
$$\array{ && T X_0 & \to & T X_1 & \to & T X_2 & \to & \dots \\
&& \downarrow^{x_0} && \downarrow^{x_1} && \downarrow^{x_2} \\
X_0 & \to & X_1 & \to & X_2 & \to & X_3 & \to & \dots }
$$
such that $X_\beta \to X_{\beta+1}$ is equal to $x_\beta \circ \tau_{X_\beta}$.

We begin with $X_0=A$, $X_1=B$, and $x_0 = a$.  Given $X_\beta$ and $X_{\beta+1}$, we define $x_{\beta+1}:T X_{\beta+1}\to X_{\beta+2}$ to be the coequalizer of the two composites
$$ T X_\beta \;\underoverset{T \tau}{\tau T}{\rightrightarrows}\; T^2 X_\beta \xrightarrow{T x_\beta} T X_{\beta+1}.$$
At limit ordinals $\alpha$ we take colimits, and at successors of limit ordinals we let $x_\alpha : T X_\alpha \to X_{\alpha+1}$ be the coequalizer of the two maps
$$ \array{ && \colim_{\beta\lt\alpha} X_{\beta+1} = X_\alpha \\
  & \nearrow && \searrow^{\tau} \\
  \colim_{\beta\lt\alpha} T X_\beta && \to && T X_\alpha.}
$$

+-- {: .num_example}
###### Example
The [[algebraic weak factorization system|algebraic]] [[small object argument]] consists of building the free monad on a pointed endofunctor.
=--

## Free algebras for unpointed endofunctors

An algebra for an unpointed endofunctor $H$ is the same as an algebra for the pointed endofunctor $T \coloneqq Id + H$.  Thus, Theorem \ref{PointedAlgebras} can be applied to unpointed endofunctors as well.

In this case, the transfinite sequence for the free $H$-algebra on $A\in\mathcal{A}$ is much simpler: we have $X_0 = A$ and $X_{\beta+1} = A + H X_\beta$.

+-- {: .num_example}
###### Example
A [[W-type]] in a category is an initial algebra for a [[polynomial functor]].  Equivalently, it is the free algebra on the initial object for that functor.  Thus, we obtain the existence of W-types, and likewise of more general [[inductive types]].
=--

## Colimits of algebras for a monad

Let $T$ be a monad on $\mathcal{A}$, and let $T Alg$ denote the category of $T$-algebras *qua* monad.

+-- {: .num_theorem #MonadAlgebras}
###### Theorem
The category $T Alg$ is a reflective subcategory of $T/\mathcal{A}$.
=--
+-- {: .proof}
###### Proof
For $(A,B,a)\in T/\mathcal{A}$, consider the pushout
$$ \array{ T^2 A & \xrightarrow{\mu} & T A \\
  ^{T a}\downarrow && \downarrow\\
  T B & \xrightarrow{c} & D }
$$
Define $L:T/\mathcal{A}\to T/\mathcal{A}$ to take $(A,B,a)$ to $(B,D,c)$ in this pushout.  Then $L$-algebras (*qua* pointed endofunctor) are precisely $T$-algebras (*qua* monad), and we have a pointwise epimorphism $S\to L$, where $S$ is the well-pointed endofunctor from the proof of Theorem \ref{PointedAlgebras} whose algebras are the $T$-algebras (*qua* pointed endofunctor).  Thus, by Lemma \ref{WellPointedQuotient}, $L$ is well-pointed; now apply Theorem \ref{WellPointedFreeAlgebras}.
=--

+-- {: .num_cor}
###### Corollary
$T Alg$ is locally presentable.  Moreover, any functor $T Alg \to T' Alg$ induced by a monad morphism $T'\to T$ has a left adjoint.
=--
+-- {: .proof}
###### Proof
Since $T Alg$ is reflective in the cocomplete $T/\mathcal{A}$, it is cocomplete.  Since it is [[accessible category|accessible]] by the [limit theorem](/nlab/show/accessible+category#Limits), it is locally presentable.  For the second statement, the functor $T/\mathcal{A} \to T'/\mathcal{A}$ has an easily defined left adjoint; now compose it with the reflection into $T/\mathcal{A}$.
=--

## Colimits of monads
{#ColimitsOfMonads}

When speaking of colimits of monads, we are not interested merely in colimits in the category of monads, but *algebraic colimits* in the following sense.  Let $V:K\to Monad (\mathcal{A})$ be a diagram in the category of monads on $\mathcal{A}$, and define a **$V$-algebra** to be an object $A$ together with $V_k$-algebra structures which commute with the images of morphisms in $K$.

+-- {: .num_defn}
###### Definition
An **algebraic colimit** of $V$ is a monad $P$ whose usual [[Eilenberg-Moore category]] $P Alg$ is equivalent, over $\mathcal{A}$, to the category of $V$-algebras.
=--

It follows, by arguments similar to those for [[algebraically-free monads]], that an algebraic colimit is also a colimit in the category of monads.

+-- {: .num_theorem #ColimitsOfMonads}
###### Theorem
Algebraic colimits of (accessible) monads exist (and are accessible).
=--
+-- {: .proof}
###### Proof
WLOG we may assume $K$ has an initial object; otherwise, give it a new initial object $0$ and define $V_0 = Id$.  Let $T$ be the (pointwise) colimit of $V$ in the category of endofunctors of $\mathcal{A}$, with coprojections $r_k:V_k \to T$.  Our assumption on $K$ gives $T$ a unique point $Id \to V_0 \to T$ such that each $r_k$ is a map of pointed endofunctors.  Moreover, a $T$-algebra (*qua* pointed endofunctor) is an object $A$ together with compatible pointed-endofunctor-algebra structures for each $V_k$.  This clearly includes the $V$-algebras, as defined above, as a full subcategory.

We will show the category of $V$-algebras to be reflective in $T/\mathcal{A}$.  Given $(A,B,a)$, let $c:T B \to D$ be the joint coequalizer of all the parallel pairs
$$ \array{ && V_k A \\
  & ^{\mu_k}\nearrow && \searrow^{V_k \tau_k}\\
  V_k^2 A && \xrightarrow{id} && V_k^2 A &
  \xrightarrow{V_k (a r_k)} & V_k B & \xrightarrow{r_k } T B. }
$$
Let $L(A,B,a) \coloneqq (B,D,c)$, defining an endofunctor of $T/\mathcal{A}$.  Then $L$ is accessible, and an $L$-algebra is precisely a $V$-algebra.

Moreover, $L$ admits an epimorphic transformation from the well-pointed endofunctor $S$ defined as in Theorem \ref{PointedAlgebras} whose algebras are the $T$-algebras (*qua* pointed endofunctor).  Thus, by Lemma \ref{WellPointedQuotient}, $L$ is well-pointed; now apply Theorem \ref{WellPointedFreeAlgebras}.
=--

+-- {: .num_example}
###### Example
The categorical semantics of [[higher inductive types]] can be described using certain "presentations" of monads.  See [Lumsdaine-Shulman](#LShits).
=--

## Algebraic small object argument

The [[algebraic small object argument]] is an enhancement of the [[small object argument]], based on a free monad construction, that produces [[algebraic weak factorization systems]].

[[!include algebraic model structures - table]]

## References

* [[Max Kelly]], _A unified treatment of transfinite constructions for free algebras, free monoids, colimits, associated sheaves, and so on._  Bull. Austral. Math. Soc. 22 (1980), 1--83. doi:[10.1017/S0004972700006353](http://dx.doi.org/10.1017/S0004972700006353)
 {#Kelly}

* {#LShits} [[Peter LeFanu Lumsdaine]] and [[Mike Shulman]], *Semantics of higher inductive types*, [arxiv](https://arxiv.org/abs/1705.07088)


transfinite induction

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

#Contents#
* table of contents
{:toc}

## Idea

[[induction]] over arbitrary [[well-ordered sets]].

## Related concepts

* [[transfinite composition]]

## References

* Wikipedia, _[Transfinite induction](https://en.wikipedia.org/wiki/Transfinite_induction)_

[[!redirects transfinite inductions]]


transfor

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

#Contents#
* automatic table of contents goes here
{:toc}

## Idea

A **$k$-transfor** is an operation from one $n$-[[n-category|category]] $C$ to another $D$ (for some value of $n$) that takes [[object|objects]] of $C$ to $k$-[[k-morphism|morphisms]] of $D$ (and more generally $j$-morphisms in $C$ to $(j+k)$-morphisms in $D$) in a coherent way.  Equivalently, a $k$-transfor is a $k$-cell in an [[internal-hom]] $n$-category.  Transfors are a common generalisation of:

* $n$-[[n-functor|functor]]s, which are 0-transfors
* $n$-[[natural transformation]]s, which are 1-transfors
* [[modifications]], which are 2-transfors,
* perturbations, which are 3-transfors,
* and so on.

The word "transfor" was coined by Sjoerd Crans in [this paper](http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.26.506&rep=rep1&type=pdf); it is a _portmanteau_ of "functor" and "transformation."  A collection of components which forms a transfor is said to be *transforial*, as a generalization of "functorial" and "natural."

## Terminology

Once upon a time, there were [[categories]], [[functor|functors]] between them, and natural transformations between them.  Then when $n$-categories came along, people called the arrows between them '$n$-functors' even though one could just as easily say 'functors'.  In the same vein, people said '$n$-transformations' for natural transformations (that is, 2-transfors) between $n$-categories.  At the same time, we saw that we needed modifications between $n$-transformations, and that there would have to be things between higher modifications, and so on.  However, due to the prior use of "$n$-transformation" for a 2-transfor between $n$-categories, the natural choice "$k$-transformation" is unavailable to mean a $k$-transfor.

Here are some other possible terms for a $k$-transfor between $n$-categories, which additionally notate the value of $n$ (although this is, strictly speaking, unnecessary).

* $(n,k)$-transformation
* $n$-$k$-transfor
* $n$-dimensional $k$-transfor
* $n$-categorical $k$-transfor
* $n$-natural $k$-transformation

## Definitions

We haven\'t gotten around to saying anything precise yet, but you can see something in the discussion below, or in Crans\'s paper.

## Special cases

See this [[periodic table]] of $k$-transfors between $n$-categories for common names for low values of $n$ and $k$.  On the $n$-Lab, we tend to omit the prefix $n$- whenever possible (as ironic as that may be).

$k$↓\$n$→	$-1$	$0$	$1$	$2$	$3$	...

$0$	[[implication]]	[[function]]	[[functor]]	$2$-[[2-functor|functor]]	$3$-[[n-functor|functor]]	...

$1$	trivial	[[equality]] of functions	[[natural transformation]]	$2$-transformation	$3$-transformation	...

$2$	"	trivial	equality of natural transformations	[[modification]]	$3$-modification	...

$3$	"	"	trivial	equality of modifications	perturbation	...

$4$	"	"	"	trivial	equality of perturbations	...

$5$	"	"	"	"	trivial	...

⋮	"	"	"	"	"	⋱

Note that the [[source]] and [[target]] of a $k$-transfor (between $n$-categories) are $(k-1)$-transfors (between the same $n$-categories).  Given two fixed source and target $(k-1)$-transfors, the $k$-transfors between them (and the $(k+1)$-transfors between those, and so on) form an $(n-k)$-category.

### For n-posets

A similar table [[periodic table]] of $k$-transfors between $n$-posets exists for common names for low values of $n$ and $k$.

$k$↓\$n$→	$-1$	$0$	$1$	$2$	$3$	...

$0$	[[implication]]	[[monotonic function]]	[[functor]]	$2$-[[2-functor|functor]]	$3$-[[n-functor|functor]]	...

$1$	trivial	[[partial order]] of monotonic functions	[[natural transformation]]	$2$-transformation	$3$-transformation	...

$2$	"	trivial	partial order of natural transformations	[[modification]]	$3$-modification	...

$3$	"	"	trivial	partial order of modifications	perturbation	...

$4$	"	"	"	trivial	partial order of perturbations	...

$5$	"	"	"	"	trivial	...

⋮	"	"	"	"	"	⋱

## Discussion ##

This discussion was originally at [[modification]].  It discusses both terminology and definitions.

+-- {: .query}
[[Finn Lawler|Finn]]: There is a pattern here: functors
are indexed collections of objects, natural
transformations are i.c.s of 1-cells, modifications i.c.s
of 2-cells; and these are what make the collection of all
$n$-categories into an $n+1$-category, for $0 \leq n \leq
2$ anyway.  Any references for the pattern in higher
dimensions?

_Toby_:  Do you mean for the terminology or for the appropriate coherence laws? (the details that you\'ve been leaving out).  Not that I have either ...

Incidentally, I corrected 'function' to 'functor' in you question above; I hope that\'s OK.

[[Finn Lawler|Finn]]:  I meant terminology and/or an explanation for arbitrary $n$ (which Urs gives below).

Actually I was thinking of functions rather than functors, as they are the 1-cells in $0-Cat$.  But of course functions are just functors between discrete categories, and thinking of them as the latter probably makes more sense when moving to higher dimensions.

_Toby_:  Now, I would either have said 'functors are indexed collections of objects' or 'functions are indexed collections of elements'; your mixture confused me!  (^_^)

_Finn_:  Ah!  Point taken.  In any case, I should have said '0-cell' instead of 'object'.  But I think 'functor' is better anyway, as I said.

[[Urs Schreiber|Urs]]: the pattern that Finn is looking for is that embodied in the nature of the [[internal hom]] of the [[closed monoidal structure on presheaves]].

In its most general form, consider an [[infinity-category]] modeled as a [[simplicial set]] with certain properties. Being a simplicial set, this is a presheaf on the [[simplex category]]. Hence for $X$ and $Y$ such $\infty$-categories, the $\infty$-category of morphisms between them corresponds to the internal hom simplicial set
$$
  [X,Y] = Hom_{SSet}(X \times \Delta^\bullet, Y)
  \,.
$$

This simple formula encodes that pattern that Finn observed. It says that:

* [[functor]]s (the 0-cells in $[X,Y]$) are just maps $X \to Y$ from cells to cells;

* [[natural transformation]]s (the 1-cells in $[X,Y]$) are maps $X \times \Delta^1 \to Y$. Notice that $\Delta^1$ is the [[interval object]] in $SSet$ (or at least its Kanification is, but never mind that for the moment). Such maps send $n$-cells in $X$ to $(n+1)$-cells in $Y$.

* modifications are maps $X \times \Delta^2 \to Y$, that map $n$-cells in $X$ to $(n+2)$-cells in $Y$.

It may be helpful to realize the same pattern in the globular context of, for instance, [[strict omega-category]]. These are certain presheaves not on the [[simplex category]] but on the [[globe category]], but the pattern is the same: the [[internal hom]] strict $\omega$-category of morphisms between strict $\omega$-categories $X$ and $Y$ is
$$
  [X,Y] = Hom_{\omega Cat}(X \otimes G^\bullet , Y)
  \,,
$$
where now the tensor product appearing is no longer the cartesian one but the [[Crans-Gray tensor product]] and where $G^n$ is the standard globular $n$-globe. Again $G^1$ is a model for the [[interval object]] and we see that

* functors are morphisms $X \to Y$;

* transformations are morphisms $X \otimes G^1 \to Y$

* modifications are morphisms $X \otimes G^2 \to Y$

etc. Same logic as before.

When thinking about this, it may be useful to explicitly apply the hom-adjunction everywhere and think for instance of a natural transformation as a morphism

$$
  X \to [I,Y]
$$

from $X$ into the "category of cylinders in $Y$". This is maybe the most intuitive way: if for instance $Y$ happens to be just a 2-category, then this says that a transformation between functors between 2-categories is a 1-functor from the 1-category underlying $X$ to the category of cylinders in $Y$ (satisfying some property). Which is exactly what it is, in components.

When in a certain mood, I like to think of this basic fact, that $n$-fold transformations between $k$-functors are essentially (in components) $(k-n)$-functors with values in $n$-cylinders as the "holographic principle" in category theory. That may sound a bit silly, but it is true that in the case the $k$-functors in questions are $k$-functors on $Bord_k$ respresenting $k$-dimensional [[quantum field theory]], then teir transformations, being $(k-1)$-functors, represent $(k-1)$-dim QFT, and this relation between higher and lower dim QFT is called "holography" in phyiscs.

[[Finn Lawler|Finn]]:  Cool!  Thanks, Urs.  I might move this section to an article on $n$-[[n-transformation|transformations]] (if that's what they're called) once I get my head around it properly.

_Toby_:  Unfortunately, '$n$-transformation' already (following '$n$-functor') means a transformation between functors between $n$-categories.  See [Cheng--Gurski](http://cheng.staff.shef.ac.uk/degeneracy/cheng-gurski-degeneracy.pdf) for this, along with '$n$-modification' and even '$n$-perturbation' (gee, that doesn\'t conflict with anything else, does it?), along with the claim that there is 'no existing terminology' thereafter.

I would probably say '$n$-morphism in $n Cat$' (possibly with two different values of $n$); you can use '$n$-cell' in place of '$n$-morphism' if you like.  But it would be nice to find something more specific that\'s not already taken.  Or we could just throw out the Cheng--Gurski meaning of '$n$-transformation'; although it\'s not unique to them, it may not be too entrenched yet.

(But please let a transformation be a $1$-transformation, even though it is a $2$-morphism.)

[[Todd Trimble|Todd]]: I think what Urs and Crans both may be suggesting is that, at least in the context of strict $n$- and $\omega$-categories, there is a uniform notion of "transformation of depth k between n-functors", or just $(n, k)$-transformations, where $(n, 1)$-transformations are usual transformations between $n$-functors, $(n, 2)$-transformations are modifications, and so on. Surely this usage won't conflict with Cheng-Gurski.

_Toby_:  Yeah, that would work, so we could write [[(n,k)-transformation]].  My only disgruntlement is that the $n$ is superfluous; the problem is all those other people that are already using it and preventing us from unambiguously saying simply '$k$-transformation'!

_Finn_: Probably tiros like me shouldn't have a say in this sort of thing, but I would tend to agree with Toby here, that the $k$ is at least more interesting than the $n$, in that you're more likely to vary the values of $k$ than those of $n$.  However, typing the few extra characters does seem a small price to pay to avoid horrible confusion.  I slightly reluctantly vote for $(n,k)$.

_Todd_: I'm not crazy about it either, but I agree it's a small price. I'll note (in case it helps) that in the general theory of Crans-Gray tensor products, both variations in $n$ and $k$ come up, about equally often (e.g., the tensor of a 1-category and an $n$-category is an $(n+1)$-category).

[[Urs Schreiber|Urs]]: yes, so to summarize what I think the main points are

* there is a systematic notion of "transformation of depth k between n-functors" for [[geometric definition of higher category]] in terms of simplicial sets;

* the corresponding notion in the (strict) globular context is formalized by Crans' construction;

* unwrapping what this says, it yields in particular that a transformation of depth $k$ between strict globular $n$-categories $X$ and $Y$ is an $(n-k)$-functor from the truncation $X{\leq k}$ of $X$ to an $(n-k)$-category (throwing all higher cells away) to the $(n-k)$-category of $k$-globes in $Y$ (also truncated)
$$
  \eta : X_{\leq k} \to [G^k, Y]|_{\leq k}
$$
satisfying certain naturality conditions (which ensure precisely that $\eta$ extends uniquely to an $n$-functor
$
  \eta : X \to [G^k,Y]
$).

JCMcKeown: not meaning to cause annoyance, but how about calling them "$+k$-transformations", owing to their incrementing dimensions by $k$; or if we don't like the $+$ prefix, one might call them $k$-vexilors, because they tend to generate flags of period $k$.

_Toby_:  Interesting; can you explain more about how they generate flags?  (Maybe that\'s something to put in a new section here, or you could just give a reference.)

[[JCMcKeown]]: Just from reading above "... and more generally $j$-morphisms in $C$ to $(j+k)$-morphisms in $D$"... ahah! Now I see what you're getting at.  I've got my head fixed on _endo_-functors; where if you wanted to (I don't mean it's a *good* idea.  Who knows?) you can consider iterations of the underlying function that is the $+k$-transformation.

[[Mike Shulman]]: FWIW, Sjoerd Crans has called these things **k-transfors**, and speaks of something being *transforial* as a general term including both "functorial" and "natural."

_Toby_:  I\'m inclined to say that we should go with that!

[[Mike Shulman]]: I'm not sure how serious you are... but I've always thought it was a proposal that deserved to be taken more seriously than it seems to have been.  The reference is "Localizations of Transfors," _K-Theory_ 2004 (I can't find a free version online).

_Toby_:  I\'m perfectly serious.  The term *should* be indexed primarily by $k$, with $n$ only if one really insists.  I didn\'t want to make up my own word, but if Crans has published one, then why not use it?  I should be able to check that reference the next time that I visit the UCR library (about once a week).

[[Mike Shulman]]: No argument here (about indexing by $k$).  Also $(n,k)$-transformation sounds to me like something to do with [[(n,r)-categories]], but there of course the comma denotes something completely different.

[[Todd Trimble]]: I like $k$-transfor.

[[Mike Shulman]]: [Found it](http://home.tiscali.nl/secrans/papers/lotr.html)

_Toby_:  Excellent!  Since [[Finn Lawler|Finn]] and [[JCMcKeown]] have not been active lately, I\'ll move it over with that paper as a guide (or you can).

I would like to *also* mention '$(n,k)$-transformation' (or maybe '$n$-$k$-transfor'?) as a possible term, however, since some people might want to specify $n$ just as some people like to say '$n$-functor'.

_Toby_:  One could also say '$n$-natural $k$-transformation', which fits (what Crans claims on page 2 to be standard) '$2$-natural transformation' for a strict $(2,1)$-transformation.  But I still like '$k$-transfor' when $n$ is suppressed (which should be the default).

_Mike_: What about "$n$-categorical $k$-transfor" if it is necessary to specify $n$?

_Toby_:  That works too.  (Well, I don\'t like 'categorical', but that\'s a separate issue.)
=--

## Related concepts

* [[homotopy]]

* functors

  * [[functor]]

  * [[2-functor]]

  * [[(∞,1)-functor]]

* transformations

  * [[natural transformation]]

  * [[pseudonatural transformation]] / [[lax natural transformation]]

  * [[(∞,1)-natural transformation]]

  * [[(∞,n)-natural transformation]]

* modifications

  * [[modification]]

## References for the globular approach
[[Camell Kachour]]: Kamel Kachour, Définition algébrique des cellules non-strictes, Cahiers de Topologie et de Géométrie Différentielle Catégorique (2008), volume 1, pages 1–68.

[[Camell Kachour]]: Steps toward the Weak ω-category of the Weak ω-categories in the globular setting, Published Categories and General Algebraic Structures with Applications (2015).

[[!redirects (n,j)-transformation]]
[[!redirects j-transformation]]
[[!redirects k-transformation]]
[[!redirects n-transformation]]
[[!redirects (n,j)-transformations]]
[[!redirects (n,k)-transformation]]
[[!redirects (n,k)-transformations]]
[[!redirects j-transformations]]
[[!redirects k-transformations]]
[[!redirects n-transformations]]
[[!redirects k-transfor]]
[[!redirects k-transfors]]
[[!redirects transfors]]

[[!redirects n-natural transformation]]
[[!redirects n-natural transformations]]


Transformation Groups

This entry compiles material related to the textbook

* [[Tammo tom Dieck]],

  \linebreak

  **Transformation Groups**

  \linebreak

  de Gruyter 1987

  [doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372)

on [[transformation groups]]/[[topological G-spaces]], [[equivariant homotopy theory]] and [[equivariant cohomology]].

Beware the existence of a book of similar title, by the same author and largely on the same subjects:

* [[Tammo tom Dieck]], _[[Transformation Groups and Representation Theory]]_, Lecture Notes in Mathematics 766, Springer 1979 ([doi:10.1007/BFb0085965](https://link.springer.com/book/10.1007/BFb0085965))

See also:

* {#Bredon72} [[Glen Bredon]], *[[Introduction to compact transformation groups]]*, Academic Press  1972 ([ISBN 9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1)
, [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

#Contents#
* table of contents
{:toc}

## Chapter I Foundations

1. Basic notions

2. General remarks. Examples

3. Elementary properties

4. Functorial properties

5. Differentiable manifolds. Tubes and slices

   * [[equivariant differential topology]]

   * [[slice theorem]]

6. Families of subgroups

7. Equivariant maps

   * [[equivariant function]]

8. Bundles

   * [[equivariant bundle]]

9. Vector bundles

   * [[equivariant vector bundle]]

10. Orbit categories, fundamental groups, and coverings

   * [[orbit category]]

     * [[equivariant set of connected components]]

     * [[equivariant fundamental groupoid]]

     * equivariant [[universal covering space]]

11. Elementary algebra of transformation groups

## Chapter II Algebraic Topology

1. Equivariant CW-complexes

   * [[G-CW complex]]

2. Maps between complexes

3. Obstruction theory

4. The classification theorem of Hopf

   * [[equivariant Hopf degree theorem]]

5. Maps between complex representation spheres

6. Stable homotopy. Homology. Cohomology

   * [[G-spectrum]]

   * [[equivariant stable homotopy theory]]

   * [[equivariant cohomology]]

7. Homology with families

8. The Burnside ring and stable homotopy

   * [[equivariant stable Cohomotopy]]

9. Bredon homology and Mackey functors

   * [[Bredon cohomology]], [[Mackey functor]]

10. Homotopy representations

## Chapter III Localization

1. Equivariant bundle cohomology

2. Cohomology of some classifying spaces

3. Localization

4. Applications of localization

5. Borel-Smith functions

6. Further results for cyclic groups. Applications

## Chapter IV The Burnside Ring

1. Additive invariants

2. The Burnside ring

   * [[Burnside ring]]

3. The space of subgroups

4. Prime ideals

5. Congruences

6. Finiteness theorems

7. Idempotent elements

8. Induction categories

9. Induction theory

10. The Burnside ring and localization

\linebreak

category: reference


This page compiles material related to the textbook:

* [[Tammo tom Dieck]],

  **Transformation Groups and Representation Theory**

  Lecture Notes in Mathematics 766

  Springer 1979

  ([doi:10.1007/BFb0085965](https://link.springer.com/book/10.1007/BFb0085965))

on [[transformation groups]]/[[topological G-spaces]], [[representation theory]], [[equivariant homotopy theory]], [[equivariant cohomology]].

Beware the existence of a book of similar title, by the same author and largely on the same subjects:

* [[Tammo tom Dieck]], _[[Transformation Groups]]_, de Gruyter 1987  ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

See also

* {#Bredon72} [[Glen Bredon]], *[[Introduction to compact transformation groups]]*, Academic Press  1972 ([ISBN 9780080873596](https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1)
, [pdf](http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf))

\linebreak

[front matter](https://web.math.rochester.edu/people/faculty/doug/otherpapers/tomDieck-Front.pdf)

[Chapter 1: The Burnside ring of finite G-sets](https://web.math.rochester.edu/people/faculty/doug/otherpapers/tomDieck-Burnside.pdf)

* [[G-set]]

* [[Burnside ring]]

[Chapter 3: Lambda rings](https://web.math.rochester.edu/people/faculty/doug/otherpapers/tomDieck-lambda.pdf)

* [[lambda ring]]

* [[Adams operation]]

Chapter 4: Permutation representations

* [[permutation representation]]

[Chapter 5: The Burnside ring of a compact Lie group](https://web.math.rochester.edu/people/faculty/doug/otherpapers/tomDieck-Lie.pdf)

[Chapter 7: Equivariant homology and cohomolgy](https://web.math.rochester.edu/people/faculty/doug/otherpapers/tomDieck-homology.pdf)

* [[equivariant cohomology]]

* [[Bredon equivariant cohomology]]

* [[equivariant Adams conjecture]]

[Chapter 8: Equivariant homotopy theory](https://web.math.rochester.edu/people/faculty/doug/otherpapers/tomDieck-homotopy.pdf)

* [[G-space]]

* [[equivariant homotopy theory]]

* [[equivariant Hopf degree theorem]]

* [[equivariant stable cohomotopy]]

* [[Burnside ring is equivariant stable cohomotopy of the point]]

[Chapter 10: Geometric modules over the Burnside ring](https://web.math.rochester.edu/people/faculty/doug/otherpapers/tomDieck-geometric.pdf)

[Chapter 11: Homotopy equivalent stable G-vector bundles](https://web.math.rochester.edu/people/faculty/doug/otherpapers/tomDieck-equiv-bundle.pdf)

* [[J-homomorphism]]

[References](https://web.math.rochester.edu/people/faculty/doug/otherpapers/tomDieck-Back.pdf)

category: reference


transformation of adjoints

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

#Contents#
* table of contents
{:toc}

## Idea

There are several notions of [[homomorphisms]] between [[pairs]] of [[adjoint functors]], notably the notion of pairs of *conjugate* [[natural transformations]] (Def. \ref{ConjugateTransformationOfAdjoints} below).

## Definition

There are several layers of generality at which one may consider a notion of [[homomorphism]] *between* adjoint functors.

Here is a basic but important notion:

\begin{definition}\label{ConjugateTransformationOfAdjoints}
  **(conjugate transformations of adjoints)**
  \linebreak
  Given a [[pair]] of pairs of adjoint functors between the same [[categories]]

  \[
    \label{PairOfPairsOfAdjointFunctors}
    \array{
    \mathcal{C}
      \underoverset
        {\underset{R_1}{\longleftarrow}}
        {\overset{L_1}{\longrightarrow}}
        {\;\;\; \bot \;\;\;}
    \mathcal{D}
    \\
    \mathcal{C}
      \underoverset
        {\underset{R_2}{\longleftarrow}}
        {\overset{L_2}{\longrightarrow}}
        {\;\;\; \bot \;\;\;}
    \mathcal{D}
    }
  \]
then a [[pair]] of [[natural transformations]] between the adjoints of the same chirality, of this form
\[
  \label{PairOfTransformations}
  \lambda \,\colon\, L_1 \to L_2
  \;\;\;\;\;\;
  \rho \,\colon\, R_2 \to R_1
  \,,
\]
is called *conjugate for* [[MacLane (1971), §IV.7 (5)](#MacLane71)] or a *pseudo-transformation of* [[Harpaz & Prasma (2015), Sec. 2.2](#HarpazPrasma15)] the given adjunctions if they make the following [[diagram]] of [[natural transformations]] between [[hom-sets]] [[commuting diagram|commute]]:
\[
  \label{ConjugacyConditionOnAdjoints}
  \array{
    \mathcal{C}\big(
      L_2(-)
      ,\,
      (-)
    \big)
    &\overset{\sim}{\longrightarrow}&
    \mathcal{D}\big(
      (-)
      ,\,
      R_2(-)
    \big)
    \\
    \mathllap{{}^{
      \mathcal{C}\big(\lambda_{(-)},\,id_{(-)}\big)
    }}
    \Big\downarrow
    &&
    \Big\downarrow
    \mathrlap{{}^{
      \mathcal{D}\big(id_{(-)},\,\rho_{(-)}\big)
    }}
    \\
    \mathcal{C}\big(
      L_1(-)
      ,\,
      (-)
    \big)
    &\overset{\sim}{\longrightarrow}&
    \mathcal{D}\big(
      (-)
      ,\,
      R_1(-)
    \big)
    \mathrlap{\,,}
  }
\]
where the horizontal maps are the given hom-isomorphisms (see [there](adjoint+functor#eq:HomIsomorphismForAdjointFunctors)).

Such pairs of conjugate transformation [[composition|compose]] via composition of their component [[natural transformations]] (cf. *[[functor category]]*) to yields a [[category]]
\[
  \label{CatAdjCD}
  Cat_{adj}(\mathcal{C},\mathcal{D})
\]
whose

* [[objects]] are the [[left adjoint functors]] $\mathcal{C} \to \mathcal{D}$

* [[morphisms]] are the conjugate transformations between these

* [[composition]] is the composition of component [[natural transformations]].

\end{definition}

\begin{proposition}
  \label{UniqueConjugateMate}
  Given a pair of pairs of adjunctions as in (eq:PairOfPairsOfAdjointFunctors) and given (just) $\lambda$ as in (eq:PairOfTransformations) then there exists a unique $\rho$ as in (eq:PairOfTransformations) such that the conjugacy condition (eq:ConjugacyConditionOnAdjoints) holds.

In other words, the evident [[forgetful functor]] from $Cat_{adj}(\mathcal{C},\,\mathcal{D})$ (eq:CatAdjCD) to the [[hom-category]] $\Cat(\mathcal{C},\,\mathcal{D})$ (i.e. the [[functor category]]) is a [[fully faithful functor]] exhibiting a [[full subcategory]]-inclusion:
\[
  \label{FullInclusionOfConjugateTransformations}
  \mathcal{C},\,\mathcal{D} \,\in\, Obj(Cat)
  \;\;\;\;\;\;
  \vdash
  \;\;\;\;\;\;
  Cat_{adj}(\mathcal{C},\,\mathcal{D})
  \hookrightarrow
  Cat(\mathcal{C},\,\mathcal{D})
  \,.
\]
\end{proposition}
[[Mac Lane (1971), p. 98](#MacLane71)]

In fact:
\begin{proposition}
\label{ConjugacyMeansMated}
**(conjugate pairs are mates)**
\linebreak
  The conjugacy condition (eq:ConjugacyConditionOnAdjoints) means equivalently that $\lambda$ and $\rho$ are *[[mates]]* in the sense of [[2-category theory]].
\end{proposition}
This is [MacLane (1971) IV.7 Thm. 2 (6), p. 98](#MacLane71) (not using the "[[mate]]"-terminology, though, which is due to [Kelly & Street 2006](mate#KellyStreet06)).

Moreover:
\begin{proposition}
\label{HorizontalCompositionOfConjugateTransformations}
Conjugacy of transformations is compatible with [[horizontal composition]] $(-)\cdot(-)$ of [[natural transformations]] as [[2-morphisms]] in [[Cat]] ("[[whiskering]]"), so that (eq:FullInclusionOfConjugateTransformations) extends to a [[horizontal composition]]-[[functor]]:
$$
  \mathcal{C}
  ,\,
  \mathcal{D}
  ,\,
  \mathcal{E}
  \;\in\;
  Cat
  \;\;\;\;\;\;\;\;\;\;
    \vdash
  \;\;\;\;\;\;\;\;\;\;
  \array{
  Cat_{adj}(\mathcal{D},\,\mathcal{E})
  \times
  Cat_{adj}(\mathcal{C},\,\mathcal{D})
  &\overset{}{\longrightarrow}&
  Cat_{adj}(\mathcal{C},\,\mathcal{E})
  \\
  \big(
    (\lambda, \rho)
    ,\,
    (\lambda', \rho')
  \big)
  &\mapsto&
  \big(
    \lambda' \cdot \lambda
    ,\,
    \rho \cdot \rho'
  \big)
  }
  \,.
$$
\end{proposition}
[[MacLane (1971), §IV.8 Thm. 2 & Exc. 1 (p. 102)](#MacLane71)]

Therefore, from Prop. \ref{UniqueConjugateMate} and Prop. \ref{HorizontalCompositionOfConjugateTransformations} we have:
\begin{definition}\label{CatAdj}
The ([[very large category|very large]]) [[wide subcategory|wide]] and [[locally full sub-2-category]] [[CatAdj|$Cat_{adj}$]] of [[Cat]]
\[
  \label{CatAdjInsideCat}
  Cat_{adj} \longrightarrow Cat
\]
whose

* [[objects]] are categories,

* [[1-morphisms]] are [[left adjoint]] [[functors]]

* [[2-morphisms]] are [[natural transformations]] which are conjugate in the sense of Def. \ref{ConjugateTransformationOfAdjoints}.

\end{definition}
[[MacLane (1971), §IV.8 below (1), p. 102](#MacLane71)]

## Properties

### Relation to bifibrations

\begin{proposition}
  Under the [[Grothendieck construction]], the [[Grothendieck fibrations]] which arise from [[pseudofunctors]] $\mathcal{B} \longrightarrow Cat$ that factor through [[CatAdj|$Cat_{adj}$]] (eq:CatAdjInsideCat) are equivalently the [[bifibrations]].
\end{proposition}
\begin{proof}
A Grothendieck construction on a pseudofunctor yielding a bifibration is fairly immediately equivalent to all base change functors having an adjoint on the respective side (e.g. [Jacobs (1998), Lem. 9.1.2](bifibration#Jacobs98)). By the fact that $Cat_{adj} \to Cat$ is a [[locally full sub-2-category]] (Prop. \ref{UniqueConjugateMate}) this already means that the given pseudofunctor factors through  $Cat_adj$, and essentially uniquely so.
\end{proof}
See also [Harpaz & Prasma (2015), Prop. 2.2.1](#HarpazPrasma15).

## Related concepts

* [[monad transformation]]

* [[bifibration]]

* [[model structures on Grothendieck fibrations]]

* [[double category of model categories]]

## References

* {#MacLane71} [[Saunders MacLane]], Sections IV.7-8 of: _[[Categories Work|Categories for the working mathematician]]_, Graduate Texts in Mathematics, Springer (1971) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

See also:

* {#HarpazPrasma15} [[Yonatan Harpaz]], [[Matan Prasma]], Section 2.2. of: _The Grothendieck construction for model categories_, Advances in Mathematics **281** (2015) 1306-1363 [[arXiv:1404.1852](https://arxiv.org/abs/1404.1852), [10.1016/j.aim.2015.03.031](https://doi.org/10.1016/j.aim.2015.03.031)]

  > (in the context of [[model structures on Grothendieck constructions]])

[[!redirects transformations of adjoints]]

[[!redirects conjugate transformation of adjoints]]
[[!redirects conjugate transformations of adjoints]]

[[!redirects transformation of adjoint functors]]
[[!redirects transformations of adjoint functors]]

[[!redirects conjugate transformation of adjoint functors]]
[[!redirects conjugate transformations of adjoint functors]]

[[!redirects natural transformation of adjoint functors]]
[[!redirects natural transformations of adjoint functors]]

[[!redirects conjugate natural transformation of adjoint functors]]
[[!redirects conjugate natural transformations of adjoint functors]]


transgression

#Contents#
* table of contents
{:toc}

## Idea

Transgression is a tool in [[algebraic topology]] to transfer [[cohomology]] classes from one [[space]] to another without needing a [[morphism]] between them.  Rather, one has a third space with morphisms to each of the two spaces.  Of course, not just _any_ third space will do.

The topological set up is as follows.  We have two [[topological space]]s, $X$ and $Y$, and a [[generalised cohomology theory]] $E^*(-)$.  We want to be able to transfer ( _transgress_ ) $E^*$-classes from $X$ to $Y$.  We find a third space, $Z$, which has morphisms to both $X$ and $Y$.  We usually write this in the following way:

$$
\begin{matrix}
Z & \to& X \\
\downarrow \\
Y
\end{matrix}
$$

(Note to self: replace with SVG, write $f \colon Z \to X$ and $g \colon Z \to Y$.)

An important example is that where we have some parameter space $\Sigma$ (for instance the circle $\Sigma = S^1$) and set $Z = \Sigma \times [\Sigma,X]$,  the [[product]] of $\Sigma$ with the [[internal hom|mapping space]] $[\Sigma,X]$ of maps from $\Sigma$ to $X$ (for instance the [[loop space]] of $X$). Then the morphism on the right is taken to be evaluation and the morphism on the left is projection onto the mapping space $[\Sigma,X]$:

$$
  \array{
    \Sigma \times [\Sigma, X] &\stackrel{ev}{\to}& X
    \\
    \downarrow^{\mathrlap{p_2}}
    \\
    [\Sigma,X]
  }
  \,.
$$

This setup induces **transgression to mapping spaces** in the following.

Now we apply the generalised cohomology theory to this diagram.  As it is a cohomology theory, the arrows reverse.  We have part of our route from $E^*(X)$ to $E^*(Y)$, namely from $E^*(X)$ to $E^*(Z)$.  However, the way from $E^*(Y)$ to $E^*(Z)$ is blocked: it is a one-way street and we want to go the _wrong_ way.

However, under certain special circumstances, cohomology theories admit push-forward maps.  That is, if $g \colon Z \to Y$ is particularly nice, there is a map $g_! \colon E^*(Z) \to E^*(Y)$.  The notation $g_!$ (pronounced "g shriek") has the explanation that it is a surprise that there is a map in this direction[^shriek].
Note that the push-forward map, $g_! \colon E^*(Z) \to E^*(Y)$, often results in a change of degree. This is described at [[fiber integration]] and [[Pontrjagin-Thom collapse map]].

[^shriek]: To the best of [[Andrew Stacey|my]] knowledge, this remark is attributable to [[Ralph Cohen]].

Thus the key to transgression is to understand the conditions where the map $g \colon Z \to Y$ admits a push-forward map.  The simplest case is where $Z$ is a product space, $Y \times W$, and $W$ is [[orientable]] for the generalised cohomology theory $E^*(-)$.  This means that $W$ has a fundamental class in $E^*(W)$ and evaluation on this class gives a morphism $E^*(Y \times W) \to E^*(Y)$.

## Definition

### Transgression of differential forms

See at _[[transgression of differential forms]]_,

### By pullback and fiber integration

(...)

### Transgression of forms through fiber bundles

A related construction called _transgression_ is the following

For $P \to X$ a [[fiber bundle]] with typical fiber $i : F \hookrightarrow P $ and $[\omega] \in H_{dR}^n(F)$ a class in [[de Rham cohomology]] of the fiber, we say this is _transgressive_ if

* there exists a form $cs \in \Omega^{n}(P)$ on the total space of the bundle;

* such that restricted along $F \hookrightarrow P$ to the fiber it is $\sim \omega$;

* and such that $d cs$ is the pull-back of a form $\kappa \in \Omega^{n+1}(X)$ on the base along the bundle projection .

See for instance section 9 of ([Borel](#Borel)).

This construction really exhibits transgression as a special case of the [[connecting homomorphism]] in cohomology

$$
  H^n(F) \to H^{n+1}(X)
$$

that is induced from the [[short exact sequence]] of cocycles

$$
  ker(i^*) \hookrightarrow \Omega^\bullet(P) \stackrel{i^*}{\to} \Omega^\bullet(F)
$$

by restricting to those elements for which it factors through

$$
  \Omega^\bullet(X) \hookrightarrow ker(i^*)
  \,.
$$

For more on this see [[Chern-Simons element]].

## Examples

### Transgression in de Rham Cohomology ###

With a [[cohomology theory]] that has a good geometric model, such as [[de Rham cohomology]], there is often a similar geometric model for the push-forward map.  In the case of de Rham cohomology, it goes by the name of _[[fiber integration|integration along the fibres]]_.

Using the notation above, with the simple case of $Z = Y \times W$, we have the formula

$$
\alpha \in H^k_{dR}(X) \mapsto \int_W f^* \alpha \in H^{k - \dim W}_{dR}(Y)
$$

### Transgression of de Rham cohomology to loop spaces

A particular application of this is in respect to the cohomology of [[loop spaces]], and in particular to [[iterated integrals]].  The aim here is to transfer information from a space $X$ to its [[free loop space object|free loop space]], $L X$.  The intermediate space in this situation is $S^1 \times L X$ with evaluation as the map to $X$ and projection to $L X$.  Note that although $L X$ is rather large, the [[fibre]] is $S^1$ and it is that which needs to be controlled.

If we worked in the realm of pure [[algebraic topology]] (i.e. didn't care about geometry) this transfer would be almost trivial.  Indeed, if we worked with _based_ loops, it would be a tautology since then the push-forward $H^*(S^1 \wedge \Omega X) \to H^{*-1}(\Omega X)$ is simply the suspension isomorphism.  (We use the [[smash product]] as we are working with _based_ spaces here.)

However, we wish to have a geometric interpretation of this, and so work in the realm of [[differential topology]]: i.e. with smooth [[manifold]]s (albeit possibly infinite dimensional).  We therefore have the diagram:

$$
\array{
S^1 \times L M & \to & M \\
\downarrow \\
L M
}
$$

The general lore of transgression says that this induces a map in cohomology $H^k_{dR}(M) \to H^{k-1}_{dR}(L M)$ via

$$
\alpha \mapsto \int_{S^1} ev^*\alpha.
$$

As with any cohomological construction, there is always the question as to whether or not it is purely cohomological or whether or not it can be lifted to whatever-it-was that defined the cohomology theory.  In this case, that is [[differential form]]s.  In this case we are in the happy circumstance that the lift exists.  Expressed purely in terms of differential forms, the formula doesn't change.  However, we can think of differential forms as "that which acts on [[tangent vector]]s" and ask for a formulation that involves tangent vectors.  This can make the integration step clearer.

To reformulate transgression thus we need to understand tangent vectors on $L M$.  The simplest way to view a tangent vector on $L M$ is to identify $T L M$ with $L T M$.  That is, we view a tangent vector $x \in T_\gamma L M$ as a loop in $T M$ with the property that $x(t) \in T_{\gamma(t)} M$.  In one view ($T L M$), $x$ tells $\gamma$ which direction to move in; in the other view ($L T M$), each $x(t)$ tells each $\gamma(t)$ which direction to move in.

That done, we have an obvious evaluation map $S^1 \times T L M \to T M$, $x \mapsto x(t)$ and so, given a differential form on $M$, we can evaluate it on tangent vectors coming from $L M$.

However, that's not all we need.  There's a very special [[vector field]] on $L M$ which we want to throw into the mix.  This is the vector field which assigns to $\gamma \in L M$ the tangent vector $\gamma' \in L T M$.  This is the rotation vector field: if we rotate the circle on $S^1 \times L M$ then to keep the evaluation map consistent, we have to rotate the loops in $L M$ as well.

So starting with a $k$-form, say $\alpha$, on $M$, we fix a loop $\gamma \in L M$, $t \in S^1$, and take $k-1$ tangent vectors at $\gamma$, say $x_1, \dots, x_{k-1}$.  We evaluate these at $t$ to get $x_1(t), \dots, x_{k-1}(t) \in T_{\gamma(t)} M$ and also throw in $\gamma'(t)$.  That gives $k$ vectors at $T_{\gamma(t)} M$ on which we can evaluate $\alpha$:

$$
\alpha(x_1(t),x_2(t),\dots,x_{k-1}(t),\gamma'(t))
$$

This is a number.  Hence, by varying $t \in S^1$, it defines a function $S^1 \to \mathbb{R}$.  Integrating this function around $S^1$ yields a number:

$$
\int_{S^1} \alpha(x_1(t),x_2(t),\dots,x_{k-1}(t),\gamma'(t)) dt
$$

Thus we have a map $\bigotimes^{k-1} T(L M) \to \mathbb{R}$.  It can be shown that it is alternating and $(k-1)$-linear, and that it varies smoothly over $L M$, hence defines an element of $\Omega^{k-1}(L M)$.

The reason why $\int_{S^1} ev^*\alpha$ takes the form of the above formula is the following:

$$ev^*: \Omega^k(M) \to \Omega^k(S^1\times LM)= \oplus_{i+j=k} \Omega^i(S^1) \otimes\Omega^j(LM)  = \Omega^0(S^1) \otimes \Omega^k(LM) \oplus \Omega^1(S^1)\otimes \Omega^{k-1}(LM). $$

Thus we may write

$$ev^*\alpha= \alpha_0\otimes \alpha_k  + \alpha_{1}\otimes  \alpha_{k-1}, $$

where $\alpha_k\in \Omega^k(LM)$, $\alpha^{k-1}\in \Omega^{k-1}(LM)$,  and $\alpha_1\in \Omega^1(S^1)$, $\alpha^0\in \Omega^0(S^1)$. Then contracting with $S^1$ via $\int_{S^1}$, only the second term survives. Thus

\[ \label{int}  (\int_{S^1} ev^*\alpha)_{\gamma}  (x_1, ..., x_{k-1})
=\int_{S^1} \alpha_{1} \otimes \alpha_{k-1} (x_1, ..., x_{k-1} ) .\]

On the other hand, let $(v_i, x_i) \in T_t S^1 \times T_{\gamma}LM $ be tangent vectors of $T(S^1\times LM)$ for $i=1, \dots, k$.    Then

$$ev^*\alpha_{(t, \gamma)}((v_1, x_1), \dots, (v_k, x_k)) = \alpha_{k-1}|_{\gamma} \otimes \alpha_1 |_t((v_1, x_1), \dots, (v_k, x_k)) + \alpha_k|_{\gamma}\otimes \alpha_0|_t((v_1, x_1), \dots, (v_k, x_k)).$$

At the same time, $T_{\gamma} ev( v, x) = v \gamma' + x$. This can be seen by taking a variation (a small path) $(t+v\epsilon, \gamma^\epsilon)$ representing $(v, x)$ (thus $\gamma^0=\gamma$). Then

$$T_{\gamma} ev( v, x)= \frac{d}{d\epsilon}|_{\epsilon=0}(\gamma^\epsilon(t+v\epsilon))=\frac{d}{d\epsilon}|_{\epsilon=0}\gamma^0(t+v\epsilon)+ \frac{d}{d\epsilon}|_{\epsilon=0}\gamma^\epsilon(t) = v\gamma'(t)+x(t),$$

which is a tangent vector at $\gamma(t)$. Thus

$$ev^*\alpha|_{(t, \gamma)}((v_1, x_1), \dots, (v_k, x_k))= \alpha_{\gamma(t)}(T_\gamma ev (v_1, x_1), \dots, T_\gamma ev (v_k, x_k)) $$

$$
= \alpha_{\gamma(t)}(x_1(t), \dots, x_k(t)) + \alpha_{\gamma(t)}(v_1 \gamma'(t), x_2(t), \dots, x_{k}(t)) + c.p.
$$

Thus $\alpha_k=\alpha$, $\alpha_0=1$, $\alpha_{k-1}=\iota(\gamma')\alpha$, and $\alpha_1=dt$. Combining with equation \eqref{int}, we obtain the desired formula (might be up to a sign).

There is much more to tell in this particular story.  It is possible to replacing $S^1$ by a [[simplex]] (but keeping $L M$ the same) and so build up more complicated objects in $\Omega^*(L M)$.  This is the starting point of Chen's theory of [[iterated integrals]] and further developments of the theory can be found in the work of Jones, Getzler, and Petrack.

### Transgression in group cohomology

See at *[[transgression in group cohomology]]*.

### Transgression to mapping spaces in terms of internal homs
 {#InternalHom}

> just a rough remark, for the moment

At least for some cases of transgression to [[mapping space]]s, the concept has an [[nPOV]] description in terms of [[internal hom]]s.

This makes crucial use of the [[nPOV]] notion of [[cohomology]], as described there.

Let $\mathbf{H} = $ [[Top]] $\simeq$ [[∞Grpd]]. For $X \in \mathbf{H}$ and for $\Sigma$ a $k$-dimensional closed oriented [[manifold]], and for $K$ an [[abelian group]] that is injective as a $\mathbb{Z}$-module, we may identify transgression to the mapping space $[\Sigma,X]$ as the composite

$$
  trans_\Sigma : \mathbf{H}(X,\mathbf{B}^n K)
  \stackrel{[\Sigma,-]}{\to}
  \mathbf{H}([\Sigma,X], [\Sigma, \mathbf{B}^n K])
  \stackrel{(\tau_{n-k})_*}{\to}
  \mathbf{H}([\Sigma,X], \tau_{n-k} [\Sigma, \mathbf{B}^n K])
  \simeq
  \mathbf{H}([\Sigma,X], \mathbf{B}^{n-k} K)
  \,.
$$

Here the first step is application of the [[internal hom]], then the second is postcomposition with [[truncated|truncation]], and then the last step uses the [[universal coefficient theorem]]. Note that the transgression map $trans_\Sigma$ depends on the choice of an equivalence $\tau_{n-k} [\Sigma, \mathbf{B}^n K]\simeq \mathbf{B}^{n-k} K$. Up to homotopy, this choice is precisely the datum of the [[orientation]] on $\Sigma$.

> more details go here....

This plays a role in the [[quantization]] process that yields [[FQFT]]s. For an application see [[Dijkgraaf-Witten theory]].

> more details, and more polishing later...

## Related entries

* [[transgression of bundle gerbes]]

## References

The classical notion of transgression of forms through fiber bundles is described in section 9 of

* {#Borel} [[Armand Borel]], _Topology of Lie groups and characteristic classes_  Bull. Amer. Math. Soc. Volume 61, Number 5 (1955), 397-432. ([EUCLID](http://projecteuclid.org/euclid.bams/1183520007))

Transgression as [[pullback in cohomology]] along the [[evaluation map]] followed by [[fiber integration]] over the base space is considered in:

for [[differential forms]]:

* [[Jean-Luc Brylinski]], §3.5 in *Loop Spaces, Characteristic Classes, and Geometric Quantization*, Birkhäuser (1993)  [[doi:10.1007/978-0-8176-4731-5](https://doi.org/10.1007/978-0-8176-4731-5)]

for [[Deligne cohomology]] (and with an eye towards defining the [[higher holonomy]] of [[bundle gerbes with connection]]):

* [[Ernesto Lupercio]], [[Bernardo Uribe]], p. 2 of: *Holonomy for Gerbes over Orbifolds*, J. Geom.Phys. **56** (2006) 1534-1560 [[arXiv:math/0307114](https://arxiv.org/abs/math/0307114), [doi:10.1016/j.geomphys.2005.08.006](https://doi.org/10.1016/j.geomphys.2005.08.006)]

  > (over [[orbifolds]])

* [[Christian Bär]], [[Christian Becker]], Def. 9.1 in: *Differential Characters and Geometric Chains*, in: *Differential Characters*, Lecture Notes in Mathematics **2112**, Springer (2014)  [[arXiv:1303.6457](https://arxiv.org/abs/1303.6457), [doi:10.1007/978-3-319-07034-6_1](https://doi.org/10.1007/978-3-319-07034-6_1)]

Transgression of [[group cohomology]] for [[discrete groups]] to cohomology of [[inertia groupoids]] (see at *[[transgression in group cohomology]]*):

* {#Willerton08} [[Simon Willerton]], Section 1 of: *The twisted Drinfeld double of a finite group via gerbes and finite groupoids*, Algebr. Geom. Topol. 8 (2008) 1419-1457 ([arXiv:math/0503266](https://arxiv.org/abs/math/0503266))

* [[Jean-Louis Tu]], [[Ping Xu]], Section 3 of: *The ring structure for equivariant twisted K-theory*, J. Reine Angew. Math. 635 (2009), 97–148 ([arXiv:math/0604160](https://arxiv.org/abs/math/0604160), [doi:10.1515/CRELLE.2009.077](https://doi.org/10.1515/CRELLE.2009.077))

* {#AdemRuanZhang07} [[Alejandro Adem]], [[Yongbin Ruan]], [[Bin Zhang]], Section 4 of: _A Stringy Product on Twisted Orbifold K-theory_, Morfismos (10th Anniversary Issue), **11** 2 (2007) 33-64 [[arXiv:math/0605534](https://arxiv.org/abs/math/0605534), [Morfismos pdf](www.morfismos.cinvestav.mx/Portals/morfismos/SiteDocs/Articulos/Volumen11/No2/Zhang/arz.pdf)]

The [[image]] of the transgression operation from [[bundle gerbes]] ([[bundle gerbe with connection|with connection]]) to [[complex line bundles]] ([[connection on a vector bundle|with connection]]) on the [[free loop space]] of their base space is characterized (as consisting of the "[[fusion bundle|fusion bundles]]") in:

* [[Konrad Waldorf]], *Transgression to Loop Spaces and its Inverse, I: Diffeological Bundles and Fusion Maps*, Cah. Topol. Geom. Differ. Categ., 2012, Vol. LIII, 162-210 [[arXiv:0911.3212](https://arxiv.org/abs/0911.3212), [cahierstgdc:LIII](http://cahierstgdc.com/index.php/volumes/volume-liii-2012)]

* [[Konrad Waldorf]], *Transgression to Loop Spaces and its Inverse, II: Gerbes and Fusion Bundles with Connection*, Asian Journal of Mathematics **20** 1 (2016) 59-116 [[arXiv:1004.0031](https://arxiv.org/abs/1004.0031), [doi:10.4310/AJM.2016.v20.n1.a4](https://dx.doi.org/10.4310/AJM.2016.v20.n1.a4)]

* [[Konrad Waldorf]], *Transgression to Loop Spaces and its Inverse, III: Gerbes and Thin Fusion Bundles*, Advances in Mathematics **231** (2012) 3445-3472 [[arXiv:1109.0480](https://arxiv.org/abs/1109.0480), [doi:10.1016/j.aim.2012.08.016](https://doi.org/10.1016/j.aim.2012.08.016)]

[[!redirects transgressions]]