action monad

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## Idea

The _action monad_ or _writer monad_ is a construction generalizing many seemingly different concepts across mathematics and [[computer science]].
It may intuitively be understood in the following ways, where throughout we fix a [[group]] or [[monoid]] $M$.

* It is the [[monad]] associated to the [[free-forgetful adjunction]] between [[G-set|M-sets]] (sets equipped with an $M$-[[action]]) and [[sets]].

* It is the monad whose [[algebra over a monad|algebras]] are $M$-sets and whose morphisms are [[equivariant maps]].

* It is the [[monad in computer science|monad modeling computational effects]] that can be added or concatenated (using the multiplication map of $M$), such as writing into a log file or standard output to screen. In this case, $M$ is usually the [[free monoid]] on a fixed [[alphabet]] (i.e. the [[data type]] "[[string (in computer science)|String]]").

If one, more generally, replaces [[sets]] and a [[monoid]] with a [[monoidal category]] and an [[internal monoid]], a similar construction can be given. This generalizes, for example, the [[action]] that [[rings]] have on their [[modules]], and that [[Lie groups]] have on [[manifolds]].

## Definition

### In Sets
 {#DefinitionInSets}

Let $M$ be a [[monoid]] with [[unit]] $e:1\to M$ and [[multiplication]] $\cdot:M\times M \to M$. The (left) **$M$-action monad** is a [[monad]] on [[Set]] where

* The [[endofunctor]] maps a set $X$ to the set $M\times X$;

* The unit is given by
\begin{tikzcd}
X \ar{r}{\cong} & 1\times X \ar{r}{e\times\mathrm{id}} & M\times X
\end{tikzcd}
for each object $X$;

* The multiplication is given by
\begin{tikzcd}
M\times (M\times X) \ar{r}{\cong} & (M\times M)\times X \ar{r}{\cdot\times\mathrm{id}} & M\times X
\end{tikzcd}
for each object $X$.

The monad axioms follow from the monoid axioms for $M$.

There exists an analogous monad for _right_ actions, whose endofunctor maps $X$ to $X\times M$.

### In any monoidal category
 {#DefinitionInMonoidalCategory}

More generally, let $(C,\otimes,1)$ be a [[monoidal category]].
Let $M$ be a [[monoid object]] in $C$ with [[unit]] $e:1\to M$ and [[multiplication]] $\cdot:M\otimes M \to M$. The (left) **$M$-action monad** is a [[monad]] on $C$ where

* The [[endofunctor]] maps an object $X$ of $C$ to the object $M\otimes X$;

* The unit is given by
\begin{tikzcd}
X \ar{r}{\cong} & 1\otimes X \ar{r}{e\times\mathrm{id}} & M\otimes X
\end{tikzcd}
for each object $X$;

* The multiplication is given by
\begin{tikzcd}
M\otimes (M\otimes X) \ar{r}{\cong} & (M\otimes M)\otimes X \ar{r}{\cdot\otimes\mathrm{id}} & M\otimes X
\end{tikzcd}
for each object $X$.

Again, the monad axioms follow from the monoid axioms for $M$. (And again there is an analogous notion for right actions).

## Algebras

The [[algebra over a monad|algebras]] over the action monad for a monoid (or group) $M$ can be seen as the [[G-space|$M$-spaces]], i.e. sets or spaces equipped with an [[action]] of $M$ (hence the name).

Plugging in the definition, an algebra over this monad is then a set $A$ together with a map $e:M\times A\to A$, such that the following diagrams commute.
The algebra diagrams
 \begin{tikzcd}
 A \ar{dr}[swap]{\mathrm{id}} \ar{r}{\eta} & M\times A \ar{d}{e} &&   M \times M \times A \ar{d}{\mu} \ar{r}{\mathrm{id}_M\times e} & M\times A \ar{d}{e} \\
 & A && M \times A\ar{r}{e} & A .
 \end{tikzcd}

say equivalently that for all $a\in A$ and $m,n\in M$,
 $$
 1\cdot a= a, \qquad (m n)\cdot a = m\cdot (n\cdot a) .
 $$

 In other words, a $T_M$-algebra is exactly a set equipped with an $M$-[[action]] in the traditional group-theoretical sense.

This gives a way to talk about monoid and group [[actions]] [[internally]] to any [[monoidal category]], giving the notion of a [[module object]].

### Examples
 {#Examples}

* In the category of [[manifolds]] with their [[cartesian product]], an [[internal group]] is a [[Lie group]]. The algebras of the corresponding action monad are the manifolds equipped with the action of the Lie group, such as [[homogeneous spaces]].

* The same can be said about [[topological spaces]] and continuous actions of [[topological groups]].

* In the category [[Ab]] of [[abelian groups]] with their [[tensor product]], an [[internal monoid]] is a [[ring]]. The algebras of the resulting action monad are the [[modules]] over that ring.

* In the category of [[endofunctors]] of a category $C$, an [[internal monoid]] is the same as a [[monad]] $T$. An algebra of the resulting action monad is an instance of a [[module over a monad|module over a monad]] $T$.

## Properties

* For the case of [[group actions]] on [[sets]], a [[G-set|$G$-set]] is [[free action|free as an action]] if and only if it is [[algebra over a monad#free_algebras|free as an algebra]] over the action monad.

* The action monad $X\mapsto M\times X$ on [[Set]] is canonically [[strong monad|strong]], with the strength given by the [[associator]] (see [[strong monad#examples]]). It is [[commutative monad|commutative]] if and only if $M$ is commutative as a monoid.

* More generally, in an arbitrary [[symmetric monoidal category]], the action monad induced by tensoring with an [[internal monoid]] is commutative if and only the monoid object is commutative. (See for example [Brandenburg 2014, Exp. 6.3.12](#Brandenburg2014).)

* Any action monad canonically [[distributive law|distributes]] over a [[strong monad]] $T$, with the distributive law $M \otimes T X \to T(M\otimes X)$ induced by the strength.

## Examples

### As logging-effect in computer science
 {#AsLoggingEffectInComputerScience}

As a [[monad in computer science]], the action monad is known under the name of **writer monad**, since it encodes the computational effect of programs "writing logging messages" like into a [[stream]].

The following explanation is taken from [Perrone, Example 5.1.14](#notesperrone).

Let $M$ be a [[monoid]], and let's write it additively. Denote by $T_M$ its _right_ writer monad.

A [[Kleisli morphism]] of $T_M$ is a [[morphism]] $k \colon X \to Y \times M$. We can interpret it as a process which, when given an input $x\in X$, does not just produce an output $y\in Y$, but also an element of $M$.

For example, it could be energy released by a chemical reaction, or waste, or a cost of the transaction. In [[computer science]], this is the behaviour of a function that computes a certain value, but that also writes into a log file (or to the standard output) that something has happened (the monoid operation being the concatenation of strings). For example, when you compile a TeX document, a log file is produced alongside your output file. Hence the name "writer monad".

 Let's now look at the [[Kleisli composition]]. If we have processes $k \colon X \to Y\times M$ and $h \colon Y\to Z\times M$, then $k\circ_{kl} Y \colon X\to Z\times M$ is given by

\begin{tikzcd}[row sep=0, column sep=large, nodes={scale=1.25}]
  X
  \ar{r}{k}
  &
  Y \times M
  \ar{r}{h\times \mathrm{id}_M}
  &
  Z \times M \times M
  \ar{r}{\mathrm{id}_X\times +}
  &
  Z \times M
  \\
  x \ar[mapsto]{r}
  &
  (y,m)
  \ar[mapsto]{r}
  &
  (z,n,m)
  \ar[mapsto]{r}
  &
  (z, n+m)
  \mathrlap{\,.}
\end{tikzcd}

What it does is the following:

1. It executes the process $k$ with an input $x\in X$, giving as output an element of $y\in Y$ as well as a cost (or extra output) $m\in M$.

1. It executes the process $k$ taking as input the $y\in Y$ produced by $k$, giving an element $z\in Z$ as well as an extra cost $n\in M$. (All of this while keeping track of the first cost $m$.)

1. The two costs $m$ and $n$ are summed (or the extra outputs are concatenated).

So, for example, the cost of executing two processes one after another is the sum of the costs. The same is true about the release of energy in a chemical reaction, and about waste.

Just as well, executing two programs one after another will produce a concatenation of text in a log file (or two log files).

### Quantum measurement
 {#QuantumMeasurement}

For $B$ a [[finite set]], and fixing any [[ground field]] $k$ (such as the [[complex numbers]]), consider the [[commutative monoid]] in $k$-[[vector spaces]] (i.e. the ordinary [[commutative algebra]])

$$
  \mathbb{1}^B
    \;=\;
  \oplus_B k
  \;\in\;
  CMon\big(Vect_k\big)
  \,=\,
  CAlg_k
$$

which may be thought of as [[generators and relations|generated]] from a $B$-[[indexed set]] of mutually orthogonal [[projection operators]].

Then the induxed $\mathbb{1}^B$-writer monad on $k$-[[VectorSpaces]] (in the sense [above](#DefinitionInMonoidalCategory)) is [[isomorphism|isomorphic]] to the linear version of the $B$-[[reader monad]].
This is related to the notion of [[quantum measurement]], and as such is discussed at: *[[quantum reader monad]]*.

## Related concepts

* [[stream]]

* [[list monad]]

* [[action]], [[G-set]], [[equivariant map]]

* [[internalization]]

* [[monoidal category]], [[monoid object]], [[group object]], [[module object]]

* [[monad]], [[algebra over a monad]]

* [[monad in computer science]],

  * [[function monad]]

[[!include reader-writer (co)monads -- table]]

## References

### General

Elementary exposition:

* {#notesperrone} [[Paolo Perrone]], Section 5 of: _Notes on Category Theory with examples from basic mathematics_, . [[arXiv:1912.10642](http://arxiv.org/abs/1912.10642)]

Proof that the construction of actions monads from monoids is part of an [[adjunction]] between monoids and [[strong monads]]:

* {#Wolff73} [[Harvey Wolff]], *Monads and Monoids on Symmetric Monoidal Closed Categories*, Archiv der Mathematik **24** (1973) 113–120 [[doi:10.1007/BF01228184](https://doi.org/10.1007/BF01228184)]

Generalization of this discussion to [[Frobenius algebras]]/[[Frobenius monads]] (cf. *[[cowriter comonads]]*):

* [[Martti Karvonen]], *Frobenius algebras in functor categories*, Oxford (2014) [[pdf](https://www.cs.ox.ac.uk/people/aleks.kissinger/theses/bob/Karvonen.pdf), [[Karvonen-FrobeniusMonads.pdf:file]]]

and analogous discussion in [[dagger-categories]]:

* [[Chris Heunen]], [[Martti Karvonen]], *Monads on dagger categories*, Theory and Applications of Categories **31** 35 (2016) 1016-1043 [[arXiv:1602.04324](https://arxiv.org/abs/1602.04324), [tac:31-35](http://www.tac.mta.ca/tac/volumes/31/35/31-35abs.html)]

Discussion of the [[Eilenberg-Moore category]] of action monads:

* {#Seal12} [[Gavin J. Seal]], Section 3 of: *Tensors, monads and actions*, Theory and Applications of Categories **28** 15 (2013) 403-434.  [[arXiv:1205.0101](http://arxiv.org/abs/1205.0101), [tac:28-15](http://www.tac.mta.ca/tac/volumes/28/15/28-15abs.html)]

Brief discussion in the context of [[algebraic geometry]]:

* {#Brandenburg2014} [[Martin Brandenburg]], Exp. 6.3.4 in: _Tensor categorical foundations of algebraic geometry_ [[arXiv:1410.1716](https://arxiv.org/abs/1410.1716), [urn:nbn:de:hbz:6-22359532742](https://nbn-resolving.de/urn:nbn:de:hbz:6-22359532742)]

### In computer science

Discussion of the [[writer monad]] as an effect-[[monad in computer science]]:

* [[Tarmo Uustalu]], p.23 of: *Monads and Interaction Lecture 1* [[pdf](https://cs.ioc.ee/~tarmo/mgs21/mgs1.pdf), [[Uustalu-Monads1.pdf:file]]], lecture notes for [MGS 2021](https://staffwww.dcs.shef.ac.uk/people/G.Struth/mgs21.html) (2021):

* {#Milewski19} [[Bartosz Milewski]] (compiled by Igal Tabachnik), §4.1 and §21.2.4 in: *Category Theory for Programmers*, Blurb (2019) [[pdf](https://github.com/hmemcpy/milewski-ctfp-pdf/releases/download/v1.3.0/category-theory-for-programmers.pdf), [github](https://github.com/hmemcpy/milewski-ctfp-pdf), [webpage](https://bartoszmilewski.com/2014/10/28/category-theory-for-programmers-the-preface/), [ISBN:9780464243878](https://www.blurb.com/b/9621951-category-theory-for-programmers-new-edition-hardco)]

[[!redirects action monads]]

[[!redirects monoid action monad]]
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[[!redirects writer monad]]
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action object > history

see _[[module object]]_


action of a category on a set

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## Definition

Given a (small) [[category]] $C$ and given a [[set]] $S$ there are (at least) the following two equivalent ways to define an [[action]] of $C$ on $S$.

### Action as a functor

An **action of a category** $C$ on a [[set]] $S$ is nothing but a [[functor]] $\rho : C \to$ [[Set]].

The particular set $S$ that this functor defines an [[action]] on is the disjoint union of sets that the functor assigns to the objects of $C$:

$$
  S = \bigsqcup_{c \in Obj(C)} \rho(c)
  \,.
$$

Given an element $s \in S$ which sits in the subset $\rho(c) \subset S$ associated with the object $c$ of $C$, it is acted on by all [[morphism]]s $c \stackrel{f}{\to} d$ in $C$ whose [[source]] is $c$. By the definition of [[functor]] every such morphism defines a map of sets

$$
  \rho(f) : (\rho(c) \subset S) \to (\rho(d) \subset S)
$$

and the the action of $f$ on $s \in \rho(c)$ under $\rho$ is

$$
 \rho(f) :  (s \in \rho(c)) \mapsto (\rho(f)(s) \in \rho(d))
  \,.
$$

In the case that $C$ has just a single object $\bullet$ the category $C$ is just a [[monoid]] (might for instance be a [[group]]), there is just a single set $S = \rho(\bullet)$ and we recover the ordinary notion of a [[monoid]] or [[group]] acting on a set.

Indeed this generalizes the instance (the motivating example for the notion of action) where $\rho:G\rightarrow \mathbf{Aut}(S)$ is a group action on a set $S$, since the notion of coproduct is a generalization of the notion of automorphism group since naively a [[cardinal]] is an isomorphism class of sets and the notion of coproduct in turn generalizes that of cardinal (see [[cardinal|there]]).

### Action as an algebra for a monad

An equivalent perspective on the above situation is often useful. To motivate this, notice that the decomposition $S = \sqcup_{c \in Obj(c)} \rho(c)$ of the set $S$ into subsets corresponding to objects of the category $C$ can equivalently be encoded in a map of sets

$$
  \lambda : S \to Obj(C)
$$
which sends each element of $S$ to the object of $c$ it corresponds to under the action.

(In the case that our category $C$ is a [[groupoid]] or even a [[Lie groupoid]] this map may be familiar as the _anchor map_ or _moment map_ of the action.)

But also the category $C$ itself comes with maps to $Obj(C)$: the [[source]] map $s$ and [[target]] map $t$, which are suggestively drawn as a [[span]] in [[Set]] by writing:

$$
  \array{
     && Mor(C)
     \\
     & {}^{s}\swarrow
     && \searrow^{t}
     \\
     Obj(C)
     &&&&
     Obj(C)
  }
  \,.
$$

Recall from the above discussion that a morphism $f : c \to d$ in $C$ could act on an element $s \in S$ if the image of $s$ under the anchor map $\lambda$ coincides with the source of $f$, i.e. with the image of $f$ under the source map $s$. Formally this means that the pairs of elements of $S$ and morphisms of $C$ which can be paired by the action live in the [[pullback]] set $S {}_\lambda \times_s Mor(C)$ (the fiber product):

$$
  \array{
     &&
     S {}_\lambda \times_s Mor(C)
     \\
     & {}^{pr_1}\swarrow && \searrow^{pr_2}
     \\
     S
     &&
     && Mor(C)
     \\
     & \searrow^{\lambda}&
     & {}^{s}\swarrow
     && \searrow^{t}
     \\
     &&
     Obj(C)
     &&&&
     Obj(C)
  }
  \,.
$$

Above we have seen that the action of $C$ on $S$ sends every element in this fiber product, which is a pair

$$
  (s \in \rho(c) \subset S, (c \stackrel{f}{\to} d) \in Mor(C))
$$

to an element $\rho(f)(s) \in \rho(d)$. So this is a map of sets $\rho : S {}_\lambda \times_s C \to S$. But a special such map, in that it satisfies a couple of conditions. One condition is that $s \in \rho(c)$ is taken to $\rho(d)$ by $f : c \to d$. This can be encoded by saying that $\rho$ extends to a morphism of [[span]]s from the pullback span above back to $S$:

$$
  \array{
     && S {}_\lambda \times_s Mor(C)
     \\
     & \swarrow && \searrow^{t \circ pr_2}
      \\
     pt &&\downarrow^{\rho}&& Obj(C)
     \\
     & \nwarrow && \nearrow_{\lambda}
     \\
     &&
     S
  }
$$

But $\rho$ satisfies yet another compatibility condition: so far we have only used the source-target mathcing condition of the functor $\rho : C \to Set$. There is also its _functoriality_, i.e. its respect for composition.

But composition in the category $C$ is itself naturally expressed in terms of morphisms of spans:

the set of composable morphisms $Mor(C) {}_t \times_s Mor(C)$ is itself the tip of a [[span]] arising from composing the [[span]] of $C$ with itself by [[pullback]]:

$$
  \array{
     &&&& Mor(C) {}_t\times_s Mor(C)
     \\
     &&& \swarrow
     && \searrow
     \\
     && Mor(C)
     &&&&
     Mor(C)
     \\
     &
     {}^s\swarrow
     && \searrow^t
     &&
     {}^s\swarrow
     && \searrow^t
     \\
     Obj(C)
     &&&&
     Obj(C)
     &&&&
     Obj(C)
  }
$$

and the composition operation $\circ$ in $C$ is a morphism from this composed span to the original span

$$
  \array{
     && Mor(C) {}_t \times_s Mor(C)
     \\
     & {}^{s \circ pr_1}\swarrow && \searrow^{t \circ pr_2}
      \\
     Obj(C) &&\downarrow^{\circ}&& Obj(C)
     \\
     & {}^{s}\nwarrow && \nearrow_{t}
     \\
     &&
     Mor(C)
  }
  \,.
$$

In total this gives us two different ways to map the total span with tip $S {}_\lambda \times_s Mor(C) {}_t \times_s Mor(C)$ obtained by composing the anchor map span with two copies of the span of $C$ back to the anchor map span

$$
  \array{
     && S {}_\lambda \times_s Mor(C) {}_t \times_s Mor(C)
     \\
     & {}^{}\swarrow && \searrow^{t \circ pr_3}
      \\
     pr &&\downarrow && Obj(C)
     \\
     & {}^{s}\nwarrow && \nearrow_{\lambda}
     \\
     &&
     S
  }
  \,.
$$

The **action property** of $\rho$, which is nothing but the functoriality of $\rho$ in the above description, says precisely that these two morphisms coincide.

Abstractly this says that

* a (small) [[category]] is a [[monad]] in [[span|spans]] in [[Set]];

* the [[action]] of a [[category]] on a [[set]] is an [[algebra]] for this [[monad]].

+--{.query}
Generalizing this slightly, it should be possible to associate an action of a category $C$ on a category $\coprod_{c\in C_0}\rho(c)$ to a functor $\rho:C\rightarrow \Cat$ with the expectation, that this then is just a module for $C$ as a monad.
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action of a monoidal category

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# Contents
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## Idea
 {#Idea}

In [[vertical categorification]] of how [[monoids]]/[[monoid objects]] $A$ may [[action|act]] on other objects $N$ ([[action objects]], [[module objects]]) inside an ambient [[monoidal category]] by [[maps]]

$$
  A \otimes N \longrightarrow N
  \,.
$$

satisfying the [action property](action#eq:ActionProperty), so a [[monoidal category]] $\mathcal{A}$ may act on other *[[categories]]* $\mathcal{N}$ by [[functors]]

$$
  \oslash
  \;\colon\;
  \mathcal{A} \times \mathcal{N}
  \longrightarrow
  \mathcal{N}
$$

subject to [[associators]], [[unitors]] and [[coherence conditions]] for [[action objects]] [[coherence|coherently]] [[internalization|internalized]] into the [[2-category]] [[Cat]], analogous to the laws in a [[monoidal category]].

At least if some linear structure is present and respected (such as when $\mathcal{A}$ qualifies as a [[2-ring]]) it is natural to speak of *[[module categories]]* over $\mathcal{A}$ (see also at *[[n-module|$n$-module]]*).

Similarly, compatible actions from both sides, such as for a [[bimodule]], give a notion of *[[bimodule category]].

Note that the term **actegory**, introduced by [McCrudden (2000)](#McCrudden00)[^1], is often used in the literature for this concept, and consequently **[[biactegory]]** for the two-sided case. However, since "actegory" is a single transposition away from "category", we prefer to use the explicit terminology on this page and elsewhere.

[^1]: From [Cockett & Pastro (2007)](#CockettPastro07): "The term *actegory* is used to describe the situation of a monoidal category “acting” on a category.  They first appeared (under a different name) in the work of Bénabou as a simple example of a bicategory. B. Pareigis developed the theory of actegories (again under a different name) and showed there usefulness in the representation theory of monoids and comonoids. The word “actegory” was first suggested at the Australian Category Seminar and first appeared in print in the thesis of P. McCrudden where they were used to study categories of representations of coalgebroids."

## Definition

For any [[category]] $A$, the category of [[endofunctors]] $End(A)$ is [[monoidal category|monoidal]] with respect to the (horizontal) composition (the composition of functors and the [[Godement product]] for natural transformations).

Given a [[monoidal category]] $(C,\otimes,I,l,r,a)$ a (left or right) __$C$-module category__ is a category $A$ together with a (left or right) coherent action of $C$ on $A$. Depending on author and context, the left coherent action of $C$ on $A$ is a morphism of monoidal categories $C\to End(A)$ in the lax, colax, pseudo or strict sense (most often in pseudo-sense) or, in another terminology, a monoidal, comonoidal, strong monoidal or strict [[monoidal functor]]. Right coherent actions correspond to the monoidal functors into the category $End(A)$ with the opposite tensor product.

$C$-module categories, colax $C$-equivariant functors and natural transformations of colax $C$-equivariant functors form a [[strict 2-category]] $_C Act^c$. A [[monad]] in $_C Act^c$ amounts to a pair of a monad in $Cat$ and a [[distributive law]] between the monad and an action of $C$.

The notion of $C$-action (hence a $C$-module category) is easily extendable to [[bicategories]] (see Baković's thesis).

+-- {: .num_defn #ModuleCategory}
###### Definition
A **(left) $\mathcal{C}$-module category** is

1. a category $\mathcal{A}$;
1. a functor $\oslash : \mathcal{C} \times \mathcal{A} \to \mathcal{A}$ called the *action*;
1. a natural isomorphism $\lambda_a : a \to I \oslash a$ called the *unitor*;
1. a natural isomorphism $\alpha_{c,d,a} : c \oslash (d \oslash a) \to (c \otimes d) \oslash a$ called the *actor*;

satisfying a pentagonal and two triangular laws (see [KJ01](#KJ01), diagg. (1.1)-(1.3)) that witness the coherence of $\lambda$ and $\alpha$ with the unitors and associators of $\mathcal{C}$.
=--

## Connection with enrichment
If a category $D$ is [[enriched category|enriched]] in $C$ with [[copowers]], then the copower structure forms a module category on the ordinary category underlying $D$.

Conversely, if module category is such that the functor $(-)\oslash d:C\to D$ has a [[right adjoint]] for all objects $d$ of $D$, then the right adjoints $D(d,-):D\to C$ provide an enrichment of $D$ in $C$ for which the action is a copower. See [KJ01](#KJ01).

More generally, supposing $C$ is small, the following data are equivalent: (1) a $C$-module category $D$; (2) an enrichment of $D$ in the category $\hat C$ of [[presheaves]] on $C$ (i.e. a [[locally graded category]]), with copowers by [[representables]]. Here we regard $\hat C$ as a monoidal category with the [[Day convolution]].

Starting from a $C$-module category $D$, consider the enrichment given by $D(d,d')=hom(-\oslash d,d'):C^{op}\to Set$. If these presheaves are representable, this is what it means to be enriched in $C$ for which the action is a copower. If they are not representable, it is still an enrichment, and the copowers by representables are the action.

For this reason, many concepts from enriched category theory make sense for module categories too.

## Related Concepts

* [[action object]], [[module object]]

* [[2-module]], [[n-module]]

* [[monoidal actegory]]

## References

* [[Bodo Pareigis]], _Non-additive ring and module theory I. General theory of monoids_, Publ. Math. Debrecen 24 (1977), 189--204. MR 56:8656; _Non-additive ring and module theory II. C-categories, C-functors, and C-morphisms_, Publ. Math. Debrecen 24 (351--361) 1977.

* {#KJ01} [[Max Kelly]], [[George Janelidze]], _A note on actions of a monoidal category_, Theory and Applications of Categories, Vol. 9, 2001, No. 4, pp 61--91 [link](http://www.tac.mta.ca/tac/volumes/9/n4/9-04abs.html)

* {#McCrudden00} P. McCrudden, *Categories of representations of coalgebroids*, Advances in Mathematics __154__ 2 (2000) 299--332 [[doi:10.1006/aima.2000.1926](https://doi.org/10.1006/aima.2000.1926)]

* P. Schauenburg, _Actions of monoidal categories and generalized Hopf smash products_, J. Algebra __270__ (2003) 521--563 (remark: actegories with action in the strong monoidal sense)

* [[Zoran Škoda]], _Distributive laws for actions of monoidal categories_, [arXiv:0406310](https://arxiv.org/abs/math/0406310), _Equivariant monads and equivariant lifts versus a 2-category of distributive laws_, [arXiv:0707.1609](https://arxiv.org/abs/0707.1609)

* [[J. R. B. Cockett]], Craig Pastro, _The logic of message-passing_ [arXiv:math/0703713](https://arxiv.org/abs/math/0703713).

A recent survey of many basic definitions and operations on actegories is

* [[Matteo Capucci]], Bruno Gavranović, _Actegories for the working amthematician_, [arXiv:2203.16351](https://arxiv.org/abs/2203.16351)

A generalisation from [[monoidal categories]] to [[bicategories]] is studied in, where actions are called **representations**:

* [[Robert Gordon]], and [[John Power]]. _Enrichment through variation_, Journal of Pure and Applied Algebra 120.2 (1997): 167-185.

[[!redirects actegory]]
[[!redirects actegories]]


acyclic assembly lemma

##Idea

The **Acyclic Assembly Lemma** gives conditions for the [[total complex]] of a [[double complex]] of [[modules]] over a ring to be acyclic.

A full statement and proof can be found in [[An Introduction to Homological Algebra]] by [[Charles Weibel|Chuck Weibel]] on pages 59 and 60 (Lemma 2.7.3).

##Reference:

* [[Charles Weibel]], _[[An introduction to homological algebra]]_,  Cambridge Studies in Adv. Math. 38, CUP 1994


acyclic fibration

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## Definition

In a [[model category]], an **acyclic fibration** or **trivial fibration** is a [[morphism]] which is both a [[fibration]] and a [[weak equivalence]].

Dually, an **acyclic cofibration** or **trivial cofibration** is a morphism which is both a [[cofibration]] and a weak equivalence.

## Examples

* [[acyclic Kan fibration]]

[[!redirects acyclic fibration]]
[[!redirects acyclic fibrations]]
[[!redirects acyclic cofibration]]
[[!redirects acyclic cofibrations]]
[[!redirects trivial fibration]]
[[!redirects trivial fibrations]]
[[!redirects trivial cofibration]]
[[!redirects trivial cofibrations]]


acyclic graph

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## Idea

Recall that an [[undirected graph]] $G$ consists of a set $V$ of _vertices_ and a set $E$ of unordered pairs of vertices. A **path** in $G$ is a finite sequence of vertices $x_0, \ldots, x_n$ such that each pair $x_i x_{i+1}$ belongs to $E$. If $x_n = x_0$, then the path is called a **cycle**. A graph is **connected** if it is nonempty and if for every distinct pair of vertices $x, y$, there is a path for which $x_0 = x$ and $x_n = y$. A graph is **acyclic** if the only cycles are paths

$$x_0, \ldots, x_{n-1}, x_n, x_{n-1}, \ldots, x_0$$

where steps are retraced; an [[acyclic graph]] is also called a _[[forest]]_. A *[[tree]]* is a connected forest.

Each forest is a [[disjoint union|disjoint sum]] (a [[coproduct]] in the category of graphs) of trees. Removal of an edge of a tree results in a forest.

## References

See also:

* Wikipedia, *[Polytree](https://en.wikipedia.org/wiki/Polytree)*

[[!redirects acyclic graphs]]

[[!redirects forest]]
[[!redirects forests]]

[[!redirects rooted forest]]
[[!redirects rooted forests]]


acyclic group

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## Idea

A [[discrete group]] $\Gamma$ is called an _acyclic group_ if its [[classifying space]] $B \Gamma$ is an "acyclic space" in that its [[ordinary cohomology]] $H^\bullet(B \Gamma, \mathbb{Z})$ vanishes in all positive degree, so that

$$
  H^\bullet(B \Gamma, \mathbb{Z}) \simeq \mathbb{Z}
  \,.
$$

Hence to cohomology these spaces look like [[contractible topological spaces]].
The existence of acyclic groups is one reason why [[rational homotopy theory]] restricts attention to [[simply-connected topological spaces]].

## References

The first two non-trivial examples of an acyclic group were apparently given in

* G. Baumslag, K.W. Gruenberg, _Some reflections on cohomological dimension and freeness, J. Algebra 6 (1967), 394–409.

* D. B. A. Epstein, _A group with zero homology, Proc. Camb. Phil. Soc. 64
(1968), 599–601

See also

* A. J. Berrick, _The acyclic group dichotomy_ ([arXiv:1006.4009](https://arxiv.org/abs/1006.4009))

* Wikipedia, _[Acyclic group](https://en.wikipedia.org/wiki/Acyclic_space#Acyclic_groups)_

[[!redirects acyclic groups]]


acyclic Kan fibration

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## Definition

A [[Kan fibration]] that at the same time is a [[weak homotopy equivalence]] is called an **acyclic Kan fibration**. (Also: a _trivial Kan fibration_.)

## Properties

Acyclic Kan fibrations are the [[acyclic fibrations]] in the [[classical model structure on simplicial sets]] $sSet_{Qu}$, hence those morphisms which have the [[right lifting property]] against [[monomorphisms]] (degreewise [[injections]]) of [[simplicial sets]].

In particular, this implies that:
\begin{proposition}\label{AcyclicKanFibrationsAreSurjective}
Acyclic Kan fibrations are  (degreewise) [[surjective]], in that they have the [[right lifting property]] against any [[empty bundle]] $\varnothing \xhookrightarrow{\;} S$.
\end{proposition}
\begin{remark}
Prop. \ref{AcyclicKanFibrationsAreSurjective} is in contrast to plain [[Kan fibrations]], see [this remark](horn#ZeroSimplexHasNoHorns) about the empty horn, and see the discussion on surjective Kan fibrations [there](Kan+fibration#SurjectiveKanFibrations).
\end{remark}

In fact:
\begin{prop}
Acyclic Kan fibrations are precisely the morphisms of [[simplicial sets]] that have the [[right lifting property]] against all [[boundary of a simplex|simplex boundary inclusions]].
\end{prop}
See [this Prop.](classical+model+structure+on+simplicial+sets#AcyclicKanFibrationsAsRLPAgainstBoundaryInclusions) at *[[classical model structure on simplicial sets]]*.
This is part of the statement that $sSet_{Qu}$  is a [[cofibrantly generated model category|cofibrantly generated]] (see [this Prop.](classical+model+structure+on+simplicial+sets#AsACofibrantlyGeneratedModelCategory)).

## Related entries

* [[simplicial homotopy theory]]

[[!redirects acyclic Kan fibrations]]

[[!redirects trivial Kan fibration]]
[[!redirects trivial Kan fibrations]]


acyclic object

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## Definition

In [[homological algebra]]:

For $F : \mathcal{A} \to \mathcal{B}$ a [[left exact functor|left exact]] [[additive functor]] between [[abelian categories]], an [[object]] $A \in \mathcal{A}$ is $F$-**acyclic** if the right [[derived functor]] of $F$ has no cohomology on $A$ in positive degree

$$
  (p \gt 0) \Rightarrow R^p F A = 0
  \,.
$$

## Properties

A [[resolution]] by $F$-acyclic objects serves to compute the [[derived functor]] of $F$. See at _[derived functor in homological algebra -- Via acyclic resolutions](derived+functor+in+homological+algebra#ViaAcyclicResolutions)_

## References

See most references on [[homological algebra]].

Also:

* [[Michael Barr]], *Acyclic models*, Canadian J. Math. **48** 2 (1996) 258–273 [[doi:10.4153/CJM-1996-013-x](https://doi.org/10.4153/CJM-1996-013-x), [pdf](https://www.math.mcgill.ca/barr/papers/acmod.pdf), [[Barr-AcyclicModels1996.pdf:file]]]

* [[Michael Barr]], *Acyclic models*, CRM Monograph Series **17** (2002) [[ams:crmm-17](https://bookstore.ams.org/crmm-17), [pdf](https://www.math.mcgill.ca/barr/papers/acycmod.pdf), [[Barr-AcyclicModels2002.pdf:file]]]

[[!redirects acyclic objects]]

[[!redirects acyclic model]]
[[!redirects acyclic models]]


acyclic type

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## Definition

In [[univalent type theory]] with [[pushout types]], a type $A$ is an **acyclic type** if its [[suspension type]] $\Sigma A$ is a [[contractible type]]. Equivalently, a type $A$ is acyclic if its reduced integral homology vanishes.

## Related concepts

* [[suspension type]]

* [[epimorphism]]

* [[acyclic object]]

* [[acyclic group]]

## References

* [[Tom de Jong]], *Acyclic types and epimorphisms in HoTT*, [[Homotopy Type Theory Electronic Seminar Talks]], 17 November 2022 ([slides](https://www.uwo.ca/math/faculty/kapulkin/seminars/hottestfiles/Jong-2022-11-17-HoTTEST.pdf), [video](https://www.youtube.com/watch?v=vohAYPAAWxs))

* [[Ulrik Buchholtz]], [[Tom de Jong]], [[Egbert Rijke]], *Epimorphisms and Acyclic Types in Univalent Mathematics*, ([arXiv:2401.14106](https://arxiv.org/abs/2401.14106))

[[!redirects acyclic type]]
[[!redirects acyclic types]]


Adam

Contributor _Adam_ is somebody who once was a grad student at UC Berkeley.  Not a member of the Logic group. That's all we know.

Possibly [Adam Megacz](http://www.megacz.com).

category: people


Adam Brandenburger

* [personal page](https://www.adambrandenburger.com/)

* [institute page](https://www.stern.nyu.edu/faculty/bio/adam-brandenburger)

## Selected writings

[[sheaf theory|Sheaf theoretic]] discussion of [[quantum contextuality]] ([[hidden variable theories]], [[Kochen-Specker theorem]]):

* {#AbramskyBrandenburger11} [[Samson Abramsky]], [[Adam Brandenburger]], _The Sheaf-Theoretic Structure Of Non-Locality and Contextuality_, New Journal of Physics **13** (2011) 113036 [[arXiv:1102.0264](https://arxiv.org/abs/1102.0264), [doi:10.1088/1367-2630/13/11/113036](https://doi.org/10.1088/1367-2630/13/11/113036)]

category: people


Adam Chlipala

* [webpage](http://adam.chlipala.net/)

## Selected writings

On [[software verification]] via [[dependent type theory]] with [[Coq]]:

* [[Adam Chlipala]], *Implementing Certified Programming Language Tools in Dependent Type Theory*, Berkeley (2007) [[UCB/EECS-2007-113](https://www2.eecs.berkeley.edu/Pubs/TechRpts/2007/EECS-2007-113.html), [pdf](https://www2.eecs.berkeley.edu/Pubs/TechRpts/2007/EECS-2007-113.pdf),  [web](http://adam.chlipala.net/papers/ChlipalaPhD/)]

* [[Adam Chlipala]], _Certified programming with dependent types_, MIT Press 2013 ([ISBN:9780262026659 ](https://mitpress.mit.edu/books/certified-programming-dependent-types), [pdf](http://adam.chlipala.net/cpdt/cpdt.pdf),  [book webpage](http://adam.chlipala.net/cpdt/))

On [[internal categories in homotopy type theory]]:

* [[Jason Gross]], [[Adam Chlipala]], [[David Spivak]], *Experience Implementing a Performant Category-Theory Library in Coq*,  In:  *Interactive Theorem Proving. ITP 2014* Lecture Notes in Computer Science, **8558** Springer (2014)  $[$[arXiv:1401.7694](http://arxiv.org/abs/1401.7694), [doi:10.1007/978-3-319-08970-6_18](https://doi.org/10.1007/978-3-319-08970-6_18)$]$

On [[cryptography]]:

* Andres Erbsen, Jade Philipoom, [[Jason Gross]], Robert Sloan, [[Adam Chlipala]], *Simple High-Level Code for Cryptographic Arithmetic - With Proofs, Without Compromises*, [2019 IEEE Symposium on Security and Privacy (SP)](https://ieeexplore.ieee.org/xpl/conhome/8826229/proceeding) $[$[doi:10.1109/SP.2019.00005](https://doi.org/10.1109/SP.2019.00005)$]$

category: people


Adam Falkowski

* [webpage](http://www.fuw.edu.pl/~afalkows/Work/work.html)

* particle physics blog: _[Resonaances](http://resonaances.blogspot.fr/)_

## Related $n$Lab pages

* [[Horava-Witten theory]]

* [[flavour anomaly]]

* [[hydrogen line]]

category: people


Adam G. Riess

* [Wikipedia entry](https://en.wikipedia.org/wiki/Adam_Riess)

* [Institute page](https://physics-astronomy.jhu.edu/directory/adam-riess/)

## Selected writings

On observation of [[supernovae]] interpreted as indicating a [[positive number|positive]] [[cosmological constant]] ([[dark energy]], the origin of the modern [[standard model of cosmology]]):

* {#Riess1998} [[Adam G. Riess]] et al., *Observational Evidence from Supernovae for an Accelerating Universe and a Cosmological Constant*,  The Astronomical Journal, **116** 3 (1998) [[doi:10.1086/300499](https://iopscience.iop.org/article/10.1086/300499)]

* {#RiessEtAl16} [[Adam G. Riess]] et al., _A 2.4% Determination of the Local Value of the Hubble Constant_,  The Astrophysical Journal **826** 1 (2016) [[arXiv:1604.01424](https://arxiv.org/abs/1604.01424), [doi:10.3847/0004-637X/826/1/56](https://iopscience.iop.org/article/10.3847/0004-637X/826/1/56)]

category: people

[[!redirects Adam Riess]]


Adam Keenan

* [webpage](http://www.math.utah.edu/~keenan/)

## Selected writings

On [[differential topology]]:

* [[Mladen Bestvina]] (notes by [[Adam Keenan]]), _Differentiable Topology and Geometry_, 2002 ([pdf](http://www.math.utah.edu/~keenan/manifoldsnotes.pdf), [[BestvinaKeenanDifferentialTopology.pdf:file]])

On the [[ADE classification]] of [[finite subgroups of SU(2)]] and [[free action|free]] [[finite group]] [[actions]] on [[n-spheres]]:

* {#Keenan03} [[Adam Keenan]], _Which finite groups act freely on spheres?_, 2003 ([pdf](http://www.math.utah.edu/~keenan/actions.pdf))

category: people


Adam Krupicka


Adam Lewandowski

## Selected writings

On [[glueballs]] via [[AdS-QCD]]:

* [[Emanuel Katz]], [[Adam Lewandowski]], [[Matthew Schwartz]], _Tensors Mesons in AdS/QCD_, Phys. Rev. D74:086004, 2006 ([arXiv:hep-ph/0510388](https://arxiv.org/abs/hep-ph/0510388))

category: people


Adam Marsh

* [webpage](https://cal.berkeley.edu/adamjmarsh)

## Selected writings

On [[differential geometry]] and [[topology]]/[[homotopy theory]] in [[mathematical physics]] (such as [[fiber bundles in physics]]):

* [[Adam Marsh]], _Mathematics for Physics: An Illustrated Handbook_, World Scientific 2018 ([doi:10.1142/10816](https://doi.org/10.1142/10816), [book webpage](https://www.mathphysicsbook.com/))

  chapter 10: _Gauge Theories and Fiber Bundles: Definitions, Pictures, and Results_ ([arXiv:1607.03089](https://arxiv.org/abs/1607.03089))

category: people


Adam Petcher

* [personal page](https://adam.petcher.net/)

## Selected writings

On [[verified software]] in [[cryptography]]:

* [[Adam Petcher]], *A Foundational Proof Framework for Cryptography*, 2014 ([pdf](https://dash.harvard.edu/bitstream/handle/1/17463136/PETCHER-DISSERTATION-2015.pdf), [harvard:17463136](https://dash.harvard.edu/handle/1/17463136))

* [[Adam Petcher]], Greg Morrisett, *The Foundational Cryptography Framework*,  In: R. Focardi, A. Myers (eds.) *Principles of Security and Trust* POST 2015. Lecture Notes in Computer Science, vol 9036. Springer, Berlin, Heidelberg ([doi:10.1007/978-3-662-46666-7_4](https://doi.org/10.1007/978-3-662-46666-7_4))

category: people


Adams category

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## Idea

The _Adams category_, named after ([Adams 74](#Adams74)), was historically one of the first constructions of the [[stable homotopy category]], following ideas in ([Boardman 65](#Boardman65)), see ([Lewis-May-Steinberger 86, pages 1-2](#LewisMaySteinberger86)) for recollection of the historical development and critical comments on the definition.

The Adams category is defined to be the category of [[CW-spectra]] together with [[left homotopy|left]] [[homotopy classes]] (via [[cylinder spectra]]) of "eventually defined" functions between them.

This was originally advertised as being a definition not involving tools from [[category theory]]. Arguably this is also its main deficiency when it comes to working with it (this is the "polemic" of [Lewis-May-Steinberger 86, preamble](#LewisMaySteinberger86)). For modern alternatives see at _[[stable homotopy category]]_.

## References

The definition is due to

* {#Adams74} [[Frank Adams]], Part III, section 2 of _[[Stable homotopy and generalised homology]]_, 1974

following

* {#Boardman65} [[Michael Boardman]], _Stable homotopy theory_, mimeographed notes, University of Warwick, 1965 onward

The term "Adams category" for this starts to be used for instance in

* [[Harold Hastings]], _On function spectra_, Proceedings of the AMS, volume 44, Number 1, May 1974 ([pdf](http://www.ams.org/journals/proc/1974-044-01/S0002-9939-1974-0343258-4/S0002-9939-1974-0343258-4.pdf))

An account following ([Adams 74](#Adams74)) is also in

* {#Switzer75} [[Robert Switzer]], section 8 of _Algebraic Topology - Homotopy and Homology_, Die  Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 212, Springer-Verlag, New York, N. Y., 1975.

Comments on the historical development are in

* {#LewisMaySteinberger86} [[L. Gaunce Lewis]], [[Peter May]], M. Steinberger, preamble of  _Equivariant stable homotopy theory_, Springer Lecture Notes in Mathematics, 1986 ([pdf](http://www.math.uchicago.edu/~may/BOOKS/equi.pdf))

in the spirit of

> There is much to love in [[Stable homotopy and generalised homology|Adams'  book]], but not in the foundational part on CW spectra. ([[Peter May]], [MO comment](http://math.stackexchange.com/a/53783/58526))

More recent textbook accounts include

* {#Kochman96} [[Stanley Kochman]], section 3.3 of _[[Bordism, Stable Homotopy and Adams Spectral Sequences]]_, AMS 1996

Lecture notes include

* [[Cary Malkiewich]], section 2.1 of _The stable homotopy category_, 2014 ([pdf](http://math.stanford.edu/~carym/stable.pdf))

[[!redirects Adams' category]]


Adams conjecture

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## Idea

The _Adams conjecture_ is a statement about triviality of [[spherical fibrations]] associated to certain formal differences of [[vector bundles]] ([[K-theory]] classes) via the [[J-homomorphism]]. The conjecture was stated in ([Adams 63, conjecture 1.2](#Adams63)), for vector bundles of rank up to two over a [[finite CW-complex]], which was proven in ([Adams 63, theorem 1.4](#Adams63)).  A general proof was then given in ([Quillen 71](Quillen71)).

The Adams conjecture/Adams-Quillen theorem serves a central role in the identification of the [[image of the J-homomorphism]].

## Statements

Let $X$ be (the [[homotopy type]] of) a [[topological space]]. For $V \;\colon\; X \longrightarrow B O$ classifying a real [[vector bundle]] on $X$, the corresponding [[spherical fibration]] is classified by the composite

$$
  J(V)
   \;\colon\;
  X \stackrel{V}{\longrightarrow} B O \stackrel{J}{\longrightarrow} B GL_1(\mathbb{S})
$$

with the delooped [[J-homomorphism]].
This descends to a map from [[topological K-theory]] to [[spherical fibrations]].

Now for $L$ a [[line bundle]] on some $X$ and for non-vanishing $k \in \mathbb{Z}$, [[John Adams]] observed that the [[spherical fibration]] associated with the difference $L^{\otimes k} - L \in K O(X)$ has the property that some $k$-fold multiple of it has trivial spherical fibration, hence that there is $N \in \mathbb{N}$ for which

$$
  J\left(
    \oplus^{k^N} (L^{\otimes k} - L)
  \right)
  = 0
  \,.
$$

Noticing that $L \mapsto L^{\otimes^k} = \Psi^k(L)$ is the $k$th [[Adams operation]] on [[K-theory]] applied to the [[line bundle]] $L$, [[John Adams]] then conjectured that the above is true for all [[vector bundles]] $V$ in the form

$$
  J\left(
    \oplus^{k^N} (\Psi^k(V) - V)
  \right)
  = 0
  \,.
$$

## Related entries

* [[J-homomorphism]]

* [[Schur index]]

## References

### General

The conjecture originates in:

* {#Adams63} [[John Adams]], _On the groups $J(X)$ I: Topology_, 2 (1963) pp. 181–195

Textbook accounts:

* [[Dai Tamaki]], [[Akira Kono]], Section 4.7 in: _Generalized Cohomology_, Translations of Mathematical Monographs, American Mathematical Society, 2006 ([ISBN: 978-0-8218-3514-2](https://bookstore.ams.org/mmono-230))

Review:

* [[Akhil Mathew]], _The Adams conjecture I_ ([web](http://amathew.wordpress.com/2013/01/23/the-adams-conjecture-i/))

* [[Michael Hopkins]] (notes by [[Akhil Mathew]]), Lecture 30 in: _Spectra and stable homotopy theory_, 2012 ([pdf](http://math.uchicago.edu/~amathew/256y.pdf), [[HopkinsMathewStableHomotopyTheory.pdf:file]])

The [[proof]] of the Adams conjecture is originally due to

* {#Quillen71} [[Daniel Quillen]], _The Adams conjecture_, Topology, vol 10, 1971  ([pdf](http://math1.unice.fr/~cazanave/Gdt/ImJ/Quillen.pdf))

The proof using [[algebraic geometry]] is due to

* [[Dennis Sullivan]], _Genetics of homotopy theory and the Adams conjecture_, The Annals of Mathematics, Second Series, Vol. 100, No. 1 (Jul., 1974), pp. 1-79 ([JSTOR](http://www.jstor.org/stable/1970841), [pdf](http://math1.unice.fr/~cazanave/Gdt/ImJ/Sullivan.pdf))

Yet another proof via [[Becker-Gottlieb transfer]] is due to

* J. Becker, D. Gottlieb, _The transfer map and fiber bundles_ Topology , 14 (1975) {#BeckerGottlieb75}

### In equivariant cohomology
 {#ReferencesInEquivariantCohomology}

The generalization to [[equivariant cohomology]] ([[equivariant K-theory]]) is discussed in

* [[Tammo tom Dieck]], theorem 11.3.8 in _[[Transformation Groups and Representation Theory]]_ Lecture Notes in Mathematics 766 Springer 1979

* Z. Fiedorowicz, H. Hauschild, [[Peter May]], theorem 0.4 of _Equivariant algebraic K-theory_, _Equivariant algebraic K-theory_, Algebraic K-Theory. Springer, Berlin, Heidelberg, 1982. 23-80 ([pdf](http://math.uchicago.edu/~may/PAPERS/40.pdf))

* Henning Hauschild, [[Stefan Waner]], theorem 0.1 of _The equivariant Dold theorem mod $k$ and the Adams conjecture_, Illinois J. Math. Volume 27, Issue 1 (1983), 52-66. ([euclid:1256065410](https://projecteuclid.org/euclid.ijm/1256065410))

* Kuzuhisa Shimakawa, _Note on the equivariant $K$-theory spectrum_, Publ. RIMS, Kyoto Univ. **29** (1993), 449-453 ([pdf](http://www.ems-ph.org/journals/show_pdf.php?issn=0034-5318&vol=29&iss=3&rank=5), [doi](https://doi.org/10.2977/prims/1195167052))

* Christopher French, theorem 2.4 in _The equivariant $J$–homomorphism for finite groups at certain primes_, Algebr. Geom. Topol. Volume 9, Number 4 (2009), 1885-1949 ([euclid:1513797069](https://projecteuclid.org/euclid.agt/1513797069))

[[!redirects Adams conjectures]]

[[!redirects Adams' conjecture]]
[[!redirects Adams' conjectures]]

[[!redirects equivariant Adams conjecture]]
[[!redirects equivariant Adams conjectures]]