abstraction

[[!redirects centipede mathematics]]

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## Introduction

Abstraction or generalization is a basic tactic in mathematics research. By formulating an argument or placing a concept in 'proper generality', one not only strengthens results and widens applicability, but one also perceives much more clearly what is truly at stake within the sphere of concepts where the mathematics takes place.

Without too much of an exaggeration, one could say that the development of modern algebra generally, and categorical algebra in particular, has been driven by the need to extract the essence of mathematical structures and situate them within axiomatic frameworks of general type. This is certain true of such concepts as [[topos]], [[abelian category]], [[scheme]], [[cohomology theory]], and countless others.

Finding the 'right' generality (as opposed to merely maximal generality) of concepts is a kind of dialectical process, in which generality in the strictly logical sense must be balanced against other considerations such as simplicity, ease of application, suggestiveness, aesthetics, etc. The result of such persistent conceptual polishing is often a theory that becomes concentrated in its well-chosen definitions, from which key summary results flow naturally.

Occasionally throughout the history of mathematics (and physics, etc.), the drive to abstraction has met with suspicion. To some extent this may be a generational phenomenon, where shifting conceptual landscapes and pressure to relearn the foundations of a subject may meet with resistance from mathematicians long accustomed to older insights. There can also be justice in the charge that generality for the sake of generality is a sterile pursuit, and a typical counter-reaction is that further layers of abstraction should "prove themselves", e.g., by leading to solutions of venerable problems.

## Styles of abstraction

### Centipede mathematics

**Centipede mathematics**  is where you take an extant mathematical concept and see how many parts you can strip away without completely destroying its ability to function properly.  Steven Krantz has attributed this term to Antoni Zygmund.
From _[Mathematical Apocrypha Redux](http://books.google.com/books?id=LMC5UeaStKwC&pg=PA6&lpg=PA6&dq=%22centipede+mathematics%22&source=bl&ots=aivICIZzMU&sig=2rMkg3u6tMhjQyfI-C46BZThiyk&hl=en&ei=kF46Ss_vC4yMsgPb0JD-Bg&sa=X&oi=book_result&ct=result&resnum=9)_: 'You take a centipede and pull off ninety-nine of its legs and see what it can do.'  (We do not recommend doing this to an actual centipede, unless it is a [land-mine destroying robot](http://www.washingtonpost.com/wp-dyn/content/article/2007/05/05/AR2007050501009.html?hpid=topnews).)

For example, you can start with the concept of [[abelian group]] and try removing some things:

* remove commutativity to get [[groups]];
* then remove inverses to get [[monoids]];
* or remove associativity instead of inverses to get [[loop(algebra)|loops]];
* etc (see [[group]] for more).

What still works?  Abelian groups form an [[abelian category]], while groups only form a [[semi-abelian category]], so [[cohomology]] gets more complicated (see [[nonabelian cohomology]]) but still makes sense.  Monoid [[representation]]s (say on [[vector space]]s) make as much sense as group representations, but loop representations are not apparently meaningful.  (But like nonabelian cohomology, maybe somebody will make sense of them some day.)  Describing homomorphisms and presentations of monoids takes a little more care than for groups, while the naïve definitions continue to work for loops.  Sometimes things actually work better; [[group object]]s make sense only in a [[cartesian monoidal category]], while [[monoid object]]s make sense in any [[monoidal category]].  By seeing what are the salient features for making sense of things like cohomology and representation theory, we also see that we can generalise these, not by simply removing clauses from the definition of group, but by reworking the concept into such things as [[category|categories]] or [[quantum group]]s.

Centipede mathematics in the context of [[foundations]] is often called __[[reverse mathematics]]__.  Here we remove axioms from [[set theory]] (or some other form of foundations) and consider what theorems can still be proved, which can be proved if reformulated into (obviously) classically equivalent forms, and which are lost entirely (ideally provably so).  On the Lab, some of us like to do this for [[constructive mathematics]], but it is also important for [[internalization]].  (Mainstream reverse mathematics has more to do with [[predicative mathematics]] than with either of these and works with subsystems of [[higher-order arithmetic]].)

Like '[[general abstract nonsense]]', the term 'centipede mathematics' can vary from tongue-in-cheek to derogatory.  Certainly there is little value in proving isolated theorems like 'Hemidemisemicontinuous functions are Leibniz-integrable.' when one never really needs to integrate such general functions in such a restrictive way (both terms here are made up, to avoid insulting the creators of any real-life useless theorems).  But sometimes thinking about what is really necessary points the way to really fruitful generalisations; like every judgement in mathematics, good taste is required.

## Illustrative quotes

From [[Lawvere]], in [An interview with F. William Lawvere](#picado):

> What is the primary tool for such summing up of the essence of ongoing
mathematics?  Algebra!  Nodal points in the progress of this kind of research occur when, as in the case with the finite number of axioms for the metacategory of categories, all that we know so far can be expressed in a single sort of algebra.  I am proud to have participated with Eilenberg, Mac Lane, Freyd, and many others, in bringing about the contemporary awareness of Algebra as Category Theory.

From [Miles Reid](#reid), p. 123:

> Many students of the time could apparently not think of any higher ambition than Étudier les EGAs. The study of category theory for its own sake (surely one of the most sterile of all intellectual pursuits) also dates from this time; Grothendieck himself can't necessarily be blamed for this, since his own use of categories was very successful in solving problems.

From [[Tom Leinster]], referring to the quotation from Reid, [Leinster2010](#leinster):

> I've become rather fond of that quotation, though not for the reason that Reid intended. 'Sterile' doesn't only mean infertile or unproductive. It's also what you want surgical instruments to be: clean, uncontaminated, disease-free. No one wants to be operated on with a dirty scalpel.

> I'm going to clarify that point, because I can hear a sarcastic voice saying 'oh, so you want to do category theory uncontaminated by real mathematics'. Category theory is at its best, I think, when it reflects what mathematicians actually do and shows how to do it better, when it functions as the [mathematics of mathematics](http://cheng.staff.shef.ac.uk/misc/4000.pdf).

From [[Categories for the Working Mathematician]], p. 108:

> One may speculate as to why the discovery of adjoint functors was so delayed. (...) Bourbaki just missed... Bourbaki's idea of universal construction was devised to be so general as to include more [than the problem of finding a left adjoint] ... [his] formulation lacks the symmetry of the adjunction problem ... and so missed a basic discovery; this discovery was left to a younger man, perhaps one less beholden to tradition or to fashion. Put differently, good general theory does not search for the maximum generality, but for the right generality.

## Related entries

* [[abstract general, concrete general and concrete particular]]

* [[concrete structure]]

## References

* Jorge Picado, _An interview with F. William Lawvere_, ([pdf](http://www.mat.uc.pt/~picado/lawvere/interview.pdf))
 {#picado}

* Miles Reid, _Undergraduate Algebraic Geometry_, available from [Reid's web page](https://homepages.warwick.ac.uk/staff/Miles.Reid/) as ([pdf](https://homepages.warwick.ac.uk/staff/Miles.Reid/MA4A5/UAG.pdf))
 {#reid}

* Tom Leinster, _A Perspective on Higher Category Theory_, $n$-Category Café post (March 8, 2010), ([link](https://golem.ph.utexas.edu/category/2010/03/a_perspective_on_higher_catego.html))
 {#leinster}

* [[Saunders Mac Lane]], _Categories for the Working Mathematician_ ($2^{nd}$ edition), Springer Verlag New York, 1998. ([pdf](http://www.maths.ed.ac.uk/~aar/papers/maclanecat.pdf))


AccCat

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## Idea

By $AccCat$ (often just $Acc$ or $\mathbf{Acc}$ etc.) one typically means the ([[very large category|very large]]) [[2-category]] whose

* [[objects]] are [[accessible categories]],

* [[1-morphisms]] are [[accessible functors]]

* [[2-morphisms]] are [[natural transformations]].

Hence [[forgetful functor|forgetting]] the accessiblity conditions yields a (non-full) [[2-functor]] from $AccCat$ to the 2-category [[Cat]] of all categories, [[functors]] and [[natural transformations]].

## Properties

### 2-Limits

\begin{proposition}
\label{AccCatHasAllPIELimits}
  The [[2-category]] [[AccCat]] has all (lax) [[2-limits]] and these are [[preserved limit|preserved]] by the inclusion $AccCat \to$ [[Cat]].
\end{proposition}
This appears as [Makkai & Paré (1989), Thm. 5.1.6, Cor. 5.1.8](#MakkaiParé1989), [Adámek & Rosický (1994) around Thm. 2.77](#AdámekRosický1994).

\begin{proposition}
Given a [[cosmos for enrichment]] $\mathcal{V}$ which is ([[symmetric monoidal category|symmetric monoidal]] [[closed monoidal category|closed]] and) [[locally presentable category|locally presentable]], then the [[2-category]] $\mathcal{V}$-[[AccCat]] of $\mathcal{V}$-[[enriched category|enriched]] [[accessible categories]] has all [[PIE 2-limits]] and [[split idempotent|splittings]] of [[idempotent]] [[equivalence in a 2-category|equivalences]], equivalently it has all [[flexible 2-limits]] as well as [[2-pullbacks]] along [[isofibrations]].

The analogous statements holds for $\mathcal{V}$-enriched and *[[conical limit|conically]]* accessible categories, in which case the [[forgetful functor]] $\mathcal{V}\text{-}ConAccCat \to \mathcal{V}\text{-}Cat$ [[preserved limit|preserves]] these [[2-limits]].
\end{proposition}
This is [Lack & Tendas (2023), Thm. 5.5, Thm. 5.9](#LackTendas23).

\begin{proposition}
**(directed unions)**
\linebreak
The [[2-category]] [[AccCat]] has [[directed colimits|directed]] [[2-colimits]] of systems of [[fully faithful functors]].  If there is a proper class of [[strongly compact cardinals]], then it has directed colimits of systems of [[faithful functors]].
\end{proposition}

[Paré & Rosický (2013)](#ParéRosický13)

## Related entries

* [[accessible category]] (see the respective section [there](accessible+category#TheTwoCategoryOfAccessibleCategories))

* [[Cat]]

## References

* {#MakkaiParé1989} [[Michael Makkai]], [[Robert Paré]],  _Accessible categories: The foundations of categorical model theory_,  Contemporary Mathematics 104. American Mathematical Society (1989) [[ISBN:978-0-8218-7692-3](https://bookstore.ams.org/conm-104)]

* {#AdámekRosický1994} [[Jiří Adámek]], [[Jiří Rosický]], *[[Locally presentable and accessible categories]]*, London Mathematical Society Lecture Note Series **189**, Cambridge University Press (1994) [[doi:10.1017/CBO9780511600579](https://doi.org/10.1017/CBO9780511600579)]

* {#ParéRosický13} [[Robert Paré]], [[Jiří Rosický]], *Colimits of accessible categories*, Math. Proc. Cambr. Phil. Soc. **155** (2013) 47-50 [[doi:10.1017/S0305004113000030](https://doi.org/10.1017/S0305004113000030), [arXiv:1110.0767](http://arxiv.org/abs/1110.0767)]

* {#LackTendas23} [[Stephen Lack]], [[Giacomo Tendas]], *Virtual concepts in the theory of accessible categories*, Journal of Pure and Applied Algebra **227** 2 (2023) 107196 [[arXiv:10.1016/j.jpaa.2022.107196](https://doi.org/10.1016/j.jpaa.2022.107196), [arXiv:2205.11056](https://arxiv.org/abs/2205.11056)]


acceleration

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## Definition

Given a [[Cartesian space]] $V$ and a [[smooth curve]] $r:\mathbb{R} \to V$, the **acceleration** of $r$ is given by the [[second derivative]] of $r$: $a \coloneqq \partial_t^2 r$.

## See also

* [[curve]]
* [[velocity]]
* [[jerk]]

## References

See the references at *[[differential geometry of curves and surfaces]]*.

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

[[!redirects accelerations]]


accessible (infinity,1)-category

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## Idea

The notion of **accessible $(\infty,1)$-category** is the generalization of the notion of [[accessible category]] from [[category theory]] to [[(∞,1)-category]] theory.

It is a means to handle $(\infty,1)$-categories that are not [[essentially small (∞,1)-category|essentially small]] in terms of small data.

An _accessible_ $(\infty,1)$-category is one which may be [[large category|large]], but can entirely be _accessed_ as an $(\infty,1)$-category of "conglomerates of objects" in a small $(\infty,1)$-category -- precisely: that it is a category of $\kappa$-small [[ind-object]]s in some small $(\infty,1)$-category $C$.

A $\kappa$-accessible $(\infty,1)$-category which in addition has all [[(∞,1)-colimits]] is called a  _[[locally presentable (∞,1)-category|locally ∞-presentable]]_ or a _$\kappa$-[[compactly generated (∞,1)-category]]_.

## Definition

Let $\kappa$ be a [[regular cardinal]].

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

A [[(∞,1)-category]] $\mathcal{C}$ is **$\kappa$-accessible** if it satisfies the following equivalent conditions:

1. There is a [[small (∞,1)-category]] $\mathcal{C}^0$ and an [[equivalence of (∞,1)-categories]]

   $$
     \mathcal{C} \simeq Ind_\kappa(C^0)
   $$

   of $\mathcal{C}$ with the  [[(∞,1)-category of ind-objects]], relative $\kappa$, in $\mathcal{C}^0$.

1. The $(\infty,1)$-category $\mathcal{C}$

   1. is [[locally small (∞,1)-category|locally small]]

   1. has all $\kappa$-[[filtered colimits]]

   1. the full [[sub-(∞,1)-category]] $\mathcal{C}^\kappa \hookrightarrow \mathcal{C}$ of $\kappa$-[[compact objects]] is an [[essentially small (∞,1)-category]];

   1. $\mathcal{C}^\kappa \hookrightarrow \mathcal{C}$ generates $\mathcal{C}$ under $\kappa$-[[filtered (∞,1)-colimits]].

1. The $(\infty,1)$-category $\mathcal{C}$

   1. is [[locally small (∞,1)-category|locally small]]

   1. has all $\kappa$-[[filtered colimits]]

   1. there is _some_ [[essentially small (∞,1)-category|essentially small]]$\,$ [[sub-(∞,1)-category]] $\mathcal{C}' \hookrightarrow \mathcal{C}$ of $\kappa$-[[compact objects]] which generates $\mathcal{C}$ under $\kappa$-[[filtered (∞,1)-colimits]].

The notion of accessibility is mostly interesting for _large_ (∞,1)-categories. For

* If $\mathcal{C}$ is small, then there exists a $\kappa$ such that $\mathcal{C}$ is $\kappa$-accessible if and only if $\mathcal{C}$ is an [[idempotent-complete (∞,1)-category]].

Generally, $\mathcal{C}$ is called an **accessible $(\infty,1)$-category** if it is $\kappa$-accessible for some regular cardinal $\kappa$.

=--

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

These conditions are indeed equivalent.

=--

For the first few this is [[Higher Topos Theory|HTT, prop. 5.4.2.2]]. The last one is in [[HTT|HTT, section 5.4.3]].

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

An [[(∞,1)-functor]] between accessible $(\infty,1)$-categories that preserves $\kappa$-filtered colimits is called an **[[accessible (∞,1)-functor]]** .

=--

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

Write $(\infty,1)AccCat \subset (\infty,1)Cat$ for the 2-[[sub-(∞,1)-category]] of [[(∞,1)Cat]] on

* those objects that are accessible $(\infty,1)$-categories;

* those morphisms for which there is a $\kappa$ such that the [[(∞,1)-functor]] is $\kappa$-continuous and preserves $\kappa$-[[compact object]]s.

=--

So morphisms are the [[accessible (∞,1)-functor]]s that also preserves [[compact object]]s. (?)

This is [[Higher Topos Theory|HTT, def. 5.4.2.16]].

## Properties

### Stability under various operations
 {#StabilityUnderOperations}

+-- {: .num_theorem}
###### Theorem

If $C$ is an accessible $(\infty,1)$-category then so are

* for $K$ a small simplicial set the [[(∞,1)-category of (∞,1)-functors]] $Func(K,C)$;

* for $p : K \to C$ a small [[diagram]], the [[over quasi-category]] $C_{/p}$ and under-quasi-category $C_{p/}$.

=--

This is [[Higher Topos Theory|HTT]] section 5.4.4, 5.4.5 and 5.4.6.

+-- {: .num_theorem}
###### Theorem

The [[(∞,1)-pullback]] of accessible $(\infty,1)$-categories in [[(∞,1)Cat]] is again accessible.

=--

This is [[Higher Topos Theory|HTT, section 5.4.6]].

Generally:

+-- {: .num_theorem}
###### Theorem

The $(\infty,1)$-category $(\infty,1)AccCat$ has all small [[(∞,1)-limit]]s and the inclusion

$$
  (\infty,1)AccCAT \hookrightarrow (\infty,1)CAT
$$

preserves these.

=--

This is [[Higher Topos Theory|HTT, proposition 5.4.7.3]].

## Related concepts

* [[compactly generated (∞,1)-category]]

[[!include locally presentable categories - table]]

## References

Theory of [[accessible categories|accessible 1-categories]]:

* {#AdamekRosicky} [[Jiří Adámek]], [[Jiří Rosický]], _[[Locally presentable and accessible categories]]_, Cambridge University Press, (1994)

Theory of accessible $(\infty,1)$-categories:

* [[Jacob Lurie]], Section 5.4 of _[[Higher Topos Theory]]_

See also:

* {#Rezk2021} [[Charles Rezk]], *Generalizing accessible ∞-categories*, 2021 ([pdf](https://faculty.math.illinois.edu/~rezk/accessible-cat-thoughts.pdf)).

[[!redirects accessible (infinity,1)-categories]]
[[!redirects accessible (∞,1)-category]]
[[!redirects accessible (∞,1)-categories]]


accessible (infinity,1)-functor

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## Idea

The generalization of the notion of [[accessible functor]]  from [[category theory]] to  [[(∞,1)-category]] theory.

## Definition

\begin{definition}
An [[(∞,1)-functor]]
$F \;\colon\; C \to D$ is **accessible** if $C$ is an [[accessible (∞,1)-category]] and there is a [[regular cardinal]] $\kappa$ such that $F$ preserves $\kappa$-small [[filtered (infinity,1)-category|filtered]]$\,$[[(infinity,1)-colimits|$(\infty,1)$-colimits]].
\end{definition}
This appears as [[Higher Topos Theory|HTT, def. 5.4.2.5]].

## Properties

\begin{proposition}\label{AdjointFunctorsAreAccessible}
**([[adjoint (infinity,1)-functors|adjoint $(\infty,1)$-functors]] are accessible)**
\linebreak
If an [[(infinity,1)-functor|$(\infty,1)$]]-functor between [[accessible (∞,1)-categories]] has a left or right [[adjoint (∞,1)-functor]], then it is itself accessible.
\end{proposition}
([[Higher Topos Theory|HTT, prop. 5.4.7.7]])

## References

* [[Jacob Lurie]], Section 5.4.2 of:  _[[Higher Topos Theory]]_ (2009)

[[!redirects accessible (infinity,1)-functor]]
[[!redirects accessible (∞,1)-functor]]
[[!redirects accessible (infinity,1)-functors]]
[[!redirects accessible (∞,1)-functors]]
[[!redirects accessible (∞,1)-functor]]

[[!redirects accessible infinity1-functor]]


accessible category

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## Idea

An accessible [[category]] is a possibly [[large category]] which is however essentially determined by a [[small category]], in a certain way.

## Definition

Let $\kappa$ be a [[regular cardinal]]. Recall an object $X: \mathcal{C}$ is $\kappa$-compact iff $\mathcal{C}(X,-)$ commutes with $\kappa$-filtered colimits.

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

A [[locally small category]] $\mathcal{C}$ is **$\kappa$-accessible** if:

1.  the category has $\kappa$-[[directed colimits]] (or, equivalently, $\kappa$-filtered colimits), and

1.  there is a [[set]] of $\kappa$-[[compact objects]] that generate the category under $\kappa$-directed colimits.

Then $\mathcal{C}$ is an **accessible category** if there exists a $\kappa$ such that it is $\kappa$-accessible.

=--

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

Unlike for [[locally presentable categories]], it does not follow that if $\mathcal{C}$ is $\kappa$-accessible and $\kappa\lt \lambda$ then $\mathcal{C}$ is also $\lambda$-accessible.  It is true, however, that for any accessible category, there are arbitrarily large cardinals $\lambda$ such that $C$ is $\lambda$-accessible.

=--

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

Equivalent characterizations include that $C$ is accessible iff:

* it is the category of [[models]] (in [[Set]]) of some small [[sketch]].

* it is of the form $\mathrm{Ind}_\kappa(S)$ for $S$ small, i.e. the $\kappa$-[[ind-object|ind-completion]] of a small category, for some $\kappa$.

* it is of the form $\kappa\,\mathrm{Flat}(S)$ for $S$ small and some $\kappa$, i.e. the category of $\kappa$-[[flat functor|flat]] functors from some small category to $Set$.

* it is the category of models (in $\mathbf{Set}$) of a suitable type of logical theory.

=--

The relevant notion of [[functor]] between accessible categories is

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

A [[functor]] $F\colon C\to D$ between accessible categories is
an **[[accessible functor]]**  if there exists a $\kappa$ such that $C$ and $D$ are both $\kappa$-accessible and $F$ preserves $\kappa$-[[filtered colimits]].

=--

## Properties
 {#Properties}

### Raising the index of accessibility

If $\mathcal{C}$ is $\lambda$-accessible and $\lambda\unlhd\mu$ (see [[sharply smaller cardinal]]), then $\mathcal{C}$ is $\mu$-accessible.  Thus, any accessible category is $\mu$-accessible for arbitrarily large cardinals $\mu$.

### Stability under various constructions

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

If $\mathcal{C}$ is an accessible category and $K$ is a [[small category]], then the [[functor category]] $Func(K^{op}, \mathcal{C})$ is again accessible.

=--

([Lurie, prop. 5.4.4.3](#Lurie))

+-- {: .num_prop #StabilityUnderInverseImage}
###### Proposition
**(preservation of accessibility under inverse images)**

Let $F : \mathcal{C} \to \mathcal{D}$ be a [[functor]] between [[locally presentable categories]] which preserves $\kappa$-[[filtered category|filtered]] [[colimits]], and let $\mathcal{D}_0 \subset \mathcal{D}$ be an accessible subcategory. Then the inverse image $f^{-1}(\mathcal{D}_0) \subset C$ is a $\kappa$-accessible subcategory.

=--

This appears as [[Higher Topos Theory|HTT, corollary A.2.6.5]].

+-- {: .num_prop}
###### Proposition
**(accessibility of fibrations and weak equivalences in a combinatorial model category)**

Let $\mathcal{C}$ be a [[combinatorial model category]], $Arr(\mathcal{C})$ its [[arrow category]], $W \subset Arr(\mathcal{C})$ the [[full subcategory]] on the weak equivalences and $F \subset Arr(\mathcal{C})$ the full subcategory on the fibrations. Then $F$, $W$ and $F \cap W$ are accessible subcategories of $Arr(\mathcal{C})$.

=--

This appears as [[Higher Topos Theory|HTT, corollary A.2.6.6]].

### Adjoint functor theorem

+-- {: .num_prop}
###### Proposition
**(adjoint functors)**

Every accessible functor satisfies the [[solution set condition]], and every left or right adjoint between accessible categories is accessible.  Therefore, the [[adjoint functor theorem]] takes an especially pleasing form for accessible categories that are complete and cocomplete (i.e. are [[locally presentable categories|locally presentable]]): a functor between such categories is a left (resp. right) adjoint iff it is accessible and preserves all small colimits (resp. limits).

=--

### Idempotence completeness

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

A [[small category]] is accessible precisely when it is [[idempotent complete category|idempotent complete]].

=--

[Makkai & Paré (1989)](#MakkaiParé1989) say that this means accessibility is an "almost pure smallness condition."

### Categories of models over a theory

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

For a [[category]] $\mathcal{M}$ the following are equivalent:

* $\mathcal{M}$ is finitely [[accessible category|accessible]].

* $\mathcal{M}\simeq \mathbb{T}\text{-}Mod(Set)$ for some [[theory of presheaf type|theory $\mathbb{T}$ of presheaf type]].

=--

Moreover, one has the following result due to [[Christian Lair]]:

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

For a [[category]] $\mathcal{M}$ the following are equivalent:

* $\mathcal{M}$ is [[accessible category|accessible]].

* $\mathcal{M}$ is [[sketch|sketchable]].
=--

See also at _[categorical model theory](model+theory#CategoricalModelTheory)_.

### Well-poweredness and well-copoweredness

* Every accessible category $\mathcal{C}$ is [[well-powered category|well-powered]], since it has a small [[dense subcategory]] $\mathcal{A}$, for which the [[restricted Yoneda embedding]] $\mathcal{C} \to [\mathcal{A}^{op},Set]$ is fully faithful and preserves monomorphisms, hence embeds the subobject posets of $\mathcal{C}$ as sub-posets of those of $[\mathcal{A}^{op},Set]$.

* Every accessible category with [[pushouts]] is well-*copowered*.  This is shown in [Adamek-Rosicky, Proposition 1.57 and Theorem 2.49](#AdámekRosický1994).  Whether this is true for all accessible categories depends on what [[large cardinal]] properties hold: by Corollary 6.8 of Adamek-Rosicky, if [[Vopenka's principle]] holds then all accessible categories are well-copowered, while by Example A.19 of Adamek-Rosicky, if all accessible categories are well-copowered then there exist arbitrarily large [[measurable cardinals]].

### The 2-category of accessible categories
 {#TheTwoCategoryOfAccessibleCategories}

Write [[AccCat]] for the [[2-category]] whose

* [[objects]] are [[accessible categories]],

* [[1-morphisms]] are [[accessible functors]]

* [[2-morphisms]] are [[natural transformations]].

\begin{proposition}
\label{AccCatHasAllPIELimits}
  The [[2-category]] [[AccCat]] has all (lax) [[2-limits]] and these are [[preserved limit|preserved]] by the inclusion $AccCat \to$ [[Cat]].
\end{proposition}
This appears as [Makkai & Paré (1989), Thm. 5.1.6, Cor. 5.1.8](#MakkaiParé1989), [Adámek & Rosický (1994) around Thm. 2.77](#AdámekRosický1994).

\begin{proposition}
Given a [[cosmos for enrichment]] $\mathcal{V}$ which is ([[symmetric monoidal category|symmetric monoidal]] [[closed monoidal category|closed]] and) [[locally presentable category|locally presentable]], then the [[2-category]] $\mathcal{V}$-[[AccCat]] of $\mathcal{V}$-[[enriched category|enriched]] [[accessible categories]] has all [[PIE 2-limits]] and [[split idempotent|splittings]] of [[idempotent]] [[equivalence in a 2-category|equivalences]] (equivalently it has all [[flexible 2-limits]]), as well as [[2-pullbacks]] along [[isofibrations]].

The analogous statements holds for $\mathcal{V}$-enriched and *[[conical limit|conically]]* accessible categories, in which case the [[forgetful functor]] $\mathcal{V}\text{-}ConAccCat \to \mathcal{V}\text{-}Cat$ [[preserved limit|preserves]] these [[2-limits]].
\end{proposition}
This is [Lack & Tendas (2023), Thm. 5.5, Thm. 5.9](#LackTendas23).

\begin{proposition}
**(directed unions)**
\linebreak
The [[2-category]] [[AccCat]] has [[directed colimits|directed]] [[2-colimits]] of systems of [[fully faithful functors]].  If there is a proper class of [[strongly compact cardinals]], then it has directed colimits of systems of [[faithful functors]].
\end{proposition}

[Paré & Rosický (2013)](#ParéRosický13)

## Examples

- Every small [[discrete category]] is $\kappa$-accessible for every [[regular cardinal]] $\kappa$, since every discrete filtered diagram is trivial.

### Functor categories

See at _[Functor category -- Accessibility](functor+category#LocalPresentability)_.

## Related concepts

* [[class-accessible category]]

[[!include locally presentable categories - table]]

## References

### General

The term _accessible category_ is due to

* {#MakkaiParé1989} [[Michael Makkai]], [[Robert Paré]],  _Accessible categories: The foundations of categorical model theory_,  Contemporary Mathematics **104**, American Mathematical Society, (1989) [[ISBN:978-0-8218-7692-3](https://bookstore.ams.org/conm-104)]

Further monographs (with focus on [[locally presentable categories]]):

* {#AdámekRosický1994} [[Jiří Adámek]], [[Jiří Rosický]], *[[Locally presentable and accessible categories]]*, London Mathematical Society Lecture Note Series **189**, Cambridge University Press (1994) [[doi:10.1017/CBO9780511600579](https://doi.org/10.1017/CBO9780511600579)]

See also

* {#ParéRosický13} [[Robert Paré]], [[Jiří Rosický]], *Colimits of accessible categories*, Math. Proc. Cambr. Phil. Soc. **155** (2013) 47-50 [[doi:10.1017/S0305004113000030](https://doi.org/10.1017/S0305004113000030), [arXiv:1110.0767](http://arxiv.org/abs/1110.0767)]

* [[Jiří Adámek]], [[Francis Borceux]], [[Stephen Lack]], [[Jiří Rosický]], _A classification of accessible categories,_ Journal of Pure and Applied Algebra 175:7-30, 2002. [abstract](http://maths.mq.edu.au/~slack/papers/acc.html)

which further stratifies the accessible categories in terms of [[sound doctrines]].

The concept is studied in a 2-categorical setting in

* [[John Bourke]], _Accessible aspects of 2-category theory_ , arXiv:2003.06375 (2020). ([abstract](https://arxiv.org/abs/2003.06375))

A discussion of [[accessible (∞,1)-categories]] is in [section 5.4, p. 341](http://arxiv.org/PS_cache/math/pdf/0608/0608040v4.pdf#page=341)
of

* {#Lurie} [[Jacob Lurie]], _[[Higher Topos Theory]]_

Some recent developments in the theory of accessible categories can be found in a series of papers on [categorical model theory](model+theory#CategoricalModelTheory) and [[abstract elementary classes]] (many of which also contain some results about arbitrary accessible categories), such as:

* [[Tibor Beke]], [[Jiří Rosický]], *Abstract elementary classes and accessible categories*, 2011, [arxiv](https://arxiv.org/abs/1005.2910)

* Michael Lieberman, [[Jiří Rosický]], Sebastien Vasey, *Internal sizes in μ-abstract elementary classes*, [arxiv](https://arxiv.org/abs/1708.06782)

* Michael Lieberman, [[Jiří Rosický]], Sebastien Vasey , *Set-theoretic aspects of accessible categories*, [arxiv](https://arxiv.org/abs/1902.06777)

See also:

* {#Rezk2021} [[Charles Rezk]], *Generalizing accessible ∞-categories*, 2021 ([pdf](https://faculty.math.illinois.edu/~rezk/accessible-cat-thoughts.pdf))

### In enriched category theory
 {#ReferencesInEnrichedCategoryTheory}

Discussion of accessible [[enriched categories]] (see also [references on enriched locally presentable categories](locally+presentable+category#ReferencesInEnrichedCategoryTheory)):

* [[Francis Borceux]], [[Carmen Quinteriro]], *Enriched accessible categories*, Bull. Austral. Math. Soc. __54__ (1996) 489-501 [[doi:10.1017/S0004972700021900](https://doi.org/10.1017/S0004972700021900)]

* [[Francis Borceux]], [[Carmen Quinteriro]], [[Jiří Rosický]], *A theory of enriched sketches*, Theory and Applications of Categories, **4** 3 (1998) 47-72 [[tac:4-03](http://www.tac.mta.ca/tac/volumes/1998/n3/4-03abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/1998/n3/n3.pdf)]

* {#LackTendas23} [[Stephen Lack]], [[Giacomo Tendas]], *Virtual concepts in the theory of accessible categories*, Journal of Pure and Applied Algebra **227** 2 (2023) 107196 [[arXiv:10.1016/j.jpaa.2022.107196](https://doi.org/10.1016/j.jpaa.2022.107196), [arXiv:2205.11056](https://arxiv.org/abs/2205.11056)]

  > New characterizations of (enriched) accessible categories and introducing the notions of virtual orthogonality and virtual reflectivity.

[[!redirects accessible categories]]

[[!redirects enriched accessible category]]
[[!redirects enriched accessible categories]]


accessible functor

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#Contents#
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## Definition

A [[functor]]
$$
  F\colon C\to D
$$
is a **$\kappa$-accessible functor** (for $\kappa$ a [[regular cardinal]]) if $C$ and $D$ are both $\kappa$-[[accessible categories]] and $F$ preserves $\kappa$-[[filtered colimit]]s.  $F$ is an **accessible functor** if it is $\kappa$-accessible for some regular cardinal $\kappa$.

## Properties

It is immediate from the definition that accessible functors are closed under composition.

### Raising the index of accessibility

If $\lambda\le\kappa$, then every $\kappa$-filtered colimit is also $\lambda$-filtered, and thus if $F$ preserves $\lambda$-filtered colimits then it also preserves $\kappa$-filtered ones.  Therefore, if $F$ is $\lambda$-accessible and $C$ and $D$ are $\kappa$-accessible, then $F$ is $\kappa$-accessible.  Two conditions under which this happens are:

1. $C$ and $D$ are [[locally presentable categories]].

2. $\lambda$ is [[sharply smaller cardinal|sharply smaller]] than $\kappa$, i.e. $\lambda\lhd\kappa$.

In particular, for any accessible functor $F$ there are arbitrarily large cardinals $\kappa$ such that $F$ is $\kappa$-accessible, and if the domain and codomain of $F$ are locally presentable then $F$ is $\kappa$-accessible for all sufficiently large $\kappa$.

### Preserving presentable objects

For any accessible functor $F$, there are arbitrarily large cardinals $\kappa$ such that $F$ is $\kappa$-accessible and preserves $\kappa$-[[presentable objects]].  Indeed, this can be achieved simultaneously for any set of accessible functors.  See [Adamek-Rosicky, Theorem 2.19](#AdamekRosicky).

### Essential images of accessible functors

\begin{theorem}
Assuming the existence of a [[proper class]] of [[strongly compact cardinals]],
the following are equivalent for the [[essential image]] $K$ of an accessible functor:

* $K$ is [[complete]];
* $K$ is [[cocomplete]];
* $K$ is [[locally presentable]].

\end{theorem}

\begin{theorem}
Assuming the existence of a [[proper class]] of [[strongly compact cardinals]],
the closure of the image of an accessible functor under passage to subobjects is an accessible subcategory.
\end{theorem}

The existence of a [[proper class]] of [[strongly compact cardinals]]
can be weakened, see the paper of [Brooke-Taylor and Rosický](#BrookeTaylorRosicky).

## Examples

\begin{example}\label{AdjointsAreAccecible}
Given [[locally presentable categories]] $C$ and $D$ and a [[functor]] $F\colon C\to D$, if $F$ has a [[left adjoint|left]] or [[right adjoint]], then it is an accessible functor.
\end{example}

\begin{example}
By Example \ref{AdjointsAreAccecible} it follows that [[polynomial endofunctors]] of [[Set|$Set$]] are accessible, as they are [[composition|composites]] of [[adjoint functors]].
\end{example}

## Related concepts

* [[accessible (∞,1)-functor]].

* [[accessible monad]]

## References

The theory of accessible 1-categories is described in

* [[Michael Makkai]], [[Robert Paré]], _Accessible categories: The foundations of categorical model theory_ Contemporary Mathematics 104. American Mathematical Society, Rhode Island, 1989.
{#MakkaiPare}

* [[Jiří Adámek]], [[Jiří Rosický]], _[[Locally presentable and accessible categories]]_, Cambridge University Press, (1994)
 {#AdamekRosicky}

Essential images of accessible functors are considered in

* [[Jiří Rosický]], _More on directed colimits of models_, Applied Categorical Structures volume 2 (1994) pages 71–76, doi:[10.1007/BF00878503](https://doi.org/10.1007/BF00878503)

An improvement of Rosický's result is in

* {#BrookeTaylorRosicky} Andrew Brooke-Taylor, [[Jiří Rosický]], _Accessible images revisited_ ([arXiv:1506.01986](https://arxiv.org/abs/1506.01986)).

The theory of accessible $(\infty,1)$-categories is the topic of section 5.4 of

* [[Jacob Lurie]], _[[Higher Topos Theory]]_

[[!redirects accessible functors]]


accessible model category

# Accessible model categories

* table of contents
{: toc}

## Definition

An **accessible model category** is a [[model structure]] on a [[locally presentable category]] whose two [[weak factorization systems]] can be realized by [[functorial factorizations]] that are [[accessible functors]], that is that they are [[accessible weak factorization systems]].

This implies that in fact its weak factorization systems can be enhanced to [[algebraic weak factorization systems]] that are also accessible (see [[accessible wfs]]).  However, such algebraic structure is not given as data in the notion of accessible model category: for that, see [[algebraic model category]].

## Properties

### Left and right lifting

Accessible model structures can be both left- and right-lifted along adjunctions as long as the relevant "acyclicity condition" holds.  That is, let $U:A\to B$ be a functor between locally presentable categories and suppose $B$ is an accessible model category.  Then:

1. If $U$ is a right adjoint, then there is a model structure on $A$ in which the weak equivalences and fibrations are created by $U$ (i.e. $W_A = U^{-1}(W_B)$ and $F_A = U^{-1}(F_B)$) if and only if every map having the left lifting property with respect to $F_A$ lies in $W_A$.

1. If $U$ is a left adjoint, then there is a model structure on $A$ in which the weak equivalences and cofibrations are created by $U$ (i.e. $W_A = U^{-1}(W_B)$ and $C_A = U^{-1}(C_B)$) if and only if every map having the right lifting property with respect to $C_A$ lies in $W_A$.

Both of the lifted model structures are then again accessible.  See [[transferred model structure]].  For proofs, see [HKRS](#HKRS15) and its correction in [GKR](#GKR18).

## Related concepts

* [[accessible weak factorization system]]
* [[algebraic model category]]

[[!include algebraic model structures - table]]

## References

* {#Rosicky17} [[Jiri Rosicky]], *Accessible model categories*,  Appl Categor Struct (2017) 25: 187. [doi](https://doi.org/10.1007/s10485-015-9419-6), [arxiv](https://arxiv.org/abs/1503.05010)

* {#HKRS15} [[Kathryn Hess]], Magdalena Kędziorek, [[Emily Riehl]], [[Brooke Shipley]], _A necessary and sufficient condition for induced model structures_ ([arXiv:1509.08154](http://arxiv.org/abs/1509.08154)).  This paper contains an error, corrected by:

* {#GKR18} [[Richard Garner]], Magdalena Kedziorek, [[Emily Riehl]], _Lifting accessible model structures_, [arXiv:1802.09889](https://arxiv.org/abs/1802.09889)

* [[John Bourke]], *Equipping weak equivalences with algebraic structure*, [arxiv:1712.02523](https://arxiv.org/abs/1712.02523)

[[!redirects accessible model categories]]
[[!redirects accessible model structure]]
[[!redirects accessible model structures]]


accessible monad

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## Definition

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

An _[[accessible monad]]_ is a [[monad]] on an [[accessible category]] whose underlying [[functor]] is an [[accessible functor]].

=--

## Properties

### Category of algebras over an accessible monad
 {#CategoryOfAlgebras}

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

The Eilenberg-Moore category of a $\kappa$-accessible monad, def. \ref{AccessibleMonad}, is a  $\kappa$-[[accessible category]]. If in addition the category on which the monad acts is a $\kappa$-[[locally presentable category]] then so is the EM-category.

=--

([Adamek-Rosicky, 2.78](#AdamekRosicky))

Moreover, let $C$ be a [[topos]]. Then

* if a [[monad]] $T : C \to C$ has a [[right adjoint]] then $T Alg(C)= C^T$ is itself a topos;

* if a [[comonad]] $T : C \to C$ is [[exact functor|left exact]], then $T CoAlg(C) = C_T$ is itself a topos.

See at _[[topos of algebras over a monad]]_ for details.

## References

* [[Jiří Adámek]], [[Jiří Rosický]], _[[Locally presentable and accessible categories]]_, Cambridge University Press, (1994)
 {#AdamekRosicky}


accessible topological space

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* table of contents
{:toc}

## Definition

A [[topological space]] $(X,\tau)$ is called an **accessible topological space** or a **$T_1$-topological space** if it is both a [[Kolmogorov topological space]] and a [[symmetric topological space]]. Or equivalently, if its [[specialization preorder]] is [[equality]]. Or equivalently still, if all [[singletons]] are [[closed subsets]] ("[[closed point|points are closed]]").

[[!include main separation axioms -- table]]

## Properties
 {#Properties}

[[!include main separation axioms -- as lifting properties]]

## Examples

[[Leibniz's identity of indiscernibles]] implies that every [[set]] $X$ with its [[power set]] $\mathcal{P}(X)$ is an accessible topological space:
$$\left(\forall P \in \mathcal{P}(X).(x \in P) \iff (y \in P)\right) \implies x = y$$
for all $x \in X$ and $y \in X$, since the relation
$$\forall P \in \mathcal{P}(X).(x \in P) \iff (y \in P)$$
is an [[equivalence relation]].

## Related concepts

* [[specialization order]]
* [[separation axiom]]
* [[identity of indiscernibles]]

| [[preorder]] | [[partial order]] | [[equivalence relation]] | [[equality]] |
| - | - | - | - |
| [[topological space]] | [[ Kolmogorov topological space]] | [[symmetric topological space]] | [[accessible topological space]] |

## References

* Wikipedia, [T1 space](https://en.wikipedia.org/wiki/T1_space)

[[!redirects T1]]
[[!redirects T_1]]

[[!redirects T1 space]]
[[!redirects T1 spaces]]
[[!redirects T1-space]]
[[!redirects T1-spaces]]
[[!redirects T_1 space]]
[[!redirects T_1 spaces]]
[[!redirects T_1-space]]
[[!redirects T_1-spaces]]

[[!redirects symmetric topological space]]
[[!redirects symmetric topological spaces]]
[[!redirects T1 topological space]]
[[!redirects T1 topological spaces]]
[[!redirects T_1 topological space]]
[[!redirects T_1 topological spaces]]
[[!redirects T1-topological space]]
[[!redirects T1-topological spaces]]
[[!redirects T_1-topological space]]
[[!redirects T_1-topological spaces]]


accessible weak factorization system

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# Accessible weak factorization systems

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{: toc}

## Definition

A [[weak factorization system]] on a [[locally presentable category]] is **accessible** if it admits a [[functorial factorization]] that is an [[accessible functor]].

## Equivalent forms

\begin{theorem}
For a weak factorization system $(L,R)$ on a locally presentable category $M$, the following are equivalent.

1. $(L,R)$ is accessible.
2. There is a small category $J \to M^\to$ over the [[arrow category]] such that $R$ consists of the morphisms with coherent right lifting functions relative to $J$.
3. $(L,R)$ can be generated by the [[algebraic small object argument]].
4. $(L,R)$ can be equipped with the structure of an accessible [[algebraic weak factorization system]].

\end{theorem}

\begin{proof}
Properties of the algebraic small object argument yield (2)$\Rightarrow$(3)$\Rightarrow$(4), and (4)$\Rightarrow$(1) since an algebraic wfs is in particular a functorial factorization.  The remaining implication is the following lemma, which is Remark 3.1.8 of [HKRS15](#HKRS15), which relies on Theorem 4.3 of [Rosicky17](#Rosicky17).
\end{proof}

\begin{lemma}
\label{EnhancementToAlgebaicWeakFactorization}
Suppose $E$ is an accessible functorial factorization on a [[locally presentable category]] $M$, realizing a [[weak factorization system]].  Then there is an accessible [[algebraic weak factorization system]] realizing the same weak factorization system.
\end{lemma}
\begin{proof}
\label{ProofOfEnhancementToAlgebaicWeakFactorization}
Let $Coalg(L)$ be the category of [[algebra over an endofunctor|coalgebras]] for the [[pointed endofunctor|copointed endofunctor]] of the [[arrow category]] $M^\to$ induced by $E$, i.e. morphisms equipped with a section of their $E$-factorization exhibiting them as a retract of the first factor.  Then $Coalg(L)$ is locally presentable (being complete and a [[PIE limit]] construction from $M$).  Thus, it has a small dense subcategory $X$.  We can then apply [[Garner's small object argument]] to generate an algebraically-free algebraic weak factorization system from $X$.  The algebraic right-maps in this awfs are the morphisms with coherent lifting functions against $X$, which by density is the same as having a coherent lifting function against all of $Coalg(L)$, which is the same as being an algebra for the pointed endofunctor of $M^\to$ corresponding to $E$.  Thus this is an awfs with the same right-maps, hence the same underlying weak factorization system.
\end{proof}

Regarding point (4), note that being accessible is a *property* of a wfs, while being algebraic is a *structure* on it.  A given accessible wfs can admit many different algebraic realizations, and not all of them (even the accessible ones) may be produced by the ordinary algebraic small object argument (although they can all be produced by the fancier version involving a double category of maps as input).

## Properties

### Left and right lifting

Any accessible weak factorization system can be right-lifted along a right adjoint, or also left-lifted along a left adjoint, between locally presentable categories.  That is, if $U:A\to B$ is a functor between locally presentable categories and $(L,R)$ is an accessible weak factorization system of $B$, then:

1. If $U$ is a right adjoint, then there is an accessible wfs $(L',R')$ on $A$ such that $R' = U^{-1}(R)$.

1. If $U$ is a left adjoint, then there is an accessible wfs $(L',R')$ on $A$ such that $L' = U^{-1}(L)$.

See [HKRS](#HKRS15) and its correction in [GKR](#GKR18) for details.  In particular, this is useful for the construction of [[transferred model structures]].

## Related concepts

* [[algebraic weak factorization system]]
* [[accessible model structure]]

[[!include algebraic model structures - table]]

## References

* {#Rosicky17} [[Jiri Rosicky]], *Accessible model categories*,  Appl Categor Struct (2017) 25: 187. [doi](https://doi.org/10.1007/s10485-015-9419-6), [arxiv](https://arxiv.org/abs/1503.05010)

* {#HKRS15} [[Kathryn Hess]], Magdalena Kędziorek, [[Emily Riehl]], [[Brooke Shipley]], _A necessary and sufficient condition for induced model structures_ ([arXiv:1509.08154](http://arxiv.org/abs/1509.08154)).  This paper contains an error, corrected by:

* {#GKR18} [[Richard Garner]], Magdalena Kedziorek, [[Emily Riehl]], _Lifting accessible model structures_, [arXiv:1802.09889](https://arxiv.org/abs/1802.09889)

[[!redirects accessible weak factorization systems]]
[[!redirects accessible weak factorisation system]]
[[!redirects accessible weak factorisation systems]]
[[!redirects accessible wfs]]
[[!redirects accessible WFS]]


Achim Jung

* [personal page](https://www.cs.bham.ac.uk/~axj/)

## Selected writings

On [[domain theory]]:

* [[Samson Abramsky]], [[Achim Jung]], *Domain Theory*, in: *Handbook of Logic in Computer Science* **3**, Oxford University Press (1995) [[ISBN:9780198537625](https://global.oup.com/academic/product/handbook-of-logic-in-computer-science-9780198537625?cc=de&lang=en&), [pdf](https://www.cs.bham.ac.uk/~axj/pub/papers/handy1.pdf)]

category: people


Achim Krause

* [personal page](https://www.uni-muenster.de/IVV5WS/WebHop/user/krauseac/)

## Selected writings

On [[prismatic cohomology]] applied to [[algebraic K-theory]]:

* {#AKN22} [[Benjamin Antieau]], [[Achim Krause]], [[Thomas Nikolaus]], _On the K-theory of $\mathbb{Z}/p^{n}$ -- Announcement_ [[arXiv:2204.03420](https://arxiv.org/abs/2204.03420)[

category: people


Ackermann groupoid

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## Idea

An *Ackermann groupoid* is a particular type of [[algebra|algebraic]] [[mathematical structure]] that provides [[semantics]] for a flavour of [[relevance logic]], a weak form of [[substructural logic]].

Note that "groupoid" here does not mean *[[groupoid]]*, but *[[magma]]*. The terminology comes from logic, rather than category theory.

## Definition

An *Ackermann groupoid* is a [[partially ordered set|partially ordered]] [[magma]] $(M,\circ, 1,\leq)$ that is left unital ($1\circ a = a$ for all $a\in M$), and has a binary operation, "implication", written $b\to c$ satisfying $a \leq b\to c$ if and only if $a\circ b \leq c$.

This might be called an _implicational Ackermann groupoid_, since it provides semantic models for an implicational fragment of logic, together with [[intensional conjunction]] (here $\to$ models implication, analogous to [[linear implication]] in [[linear logic]]). A _positive_ Ackermann groupoid upgrades the underlying [[poset]] to a [[distributive lattice]], permitting the interpretation of additional logical connectives, namely (classical) [[logical conjunction]] and [[logical disjunction]].

## Example

Every [[Church monoid]] is an Ackermann groupoid.

## Related concepts

* [[relevance monoidal category]]
* [[relevance logic]]
* [[Church monoid]]
* [[Heyting algebra]]

## References

Ackermann groupoids were introduced in

* Robert K. Meyer and Richard Routley, _Algebraic analysis of entailment I_, Logique et Analyse NOUVELLE SÉRIE, Vol. 15, No. 59/60 (1972) pp407-428, [JSTOR](https://www.jstor.org/stable/44083856)

and named for [[Wilhelm Ackermann]].

[[!redirects Ackermann groupoids]]
[[!redirects Ackermann magma]]
[[!redirects Ackermann magmas]]


acoustics

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\tableofcontents

\section{Idea}

A branch of [[physics]] that studies [[mechanical waves]].

\section{See also}

* [[sound]]
* [[vibration]]
* [[noise]]
* [[music]]
* [[wave]]
* [[wave equation]]

\section{References}

See also:

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


action

> this entry is about the notion of _action_ in algebra (of one algebraic object on another object). For the notion of _[[action functional]]_ in [[physics]] see there.

***

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# Actions
* table of contents
{: toc}

## Idea
 {#Idea}

There are various variants of the notion of something _acting_ on something else. They are all closely related.

The simplest concept of an action involves one [[set]], $X$, acting on another set $Y$ and such an action is given by a [[function]] from the [[product]] of $X$ with $Y$ to $Y$

$$
  act\colon X \times Y \to Y
  \,.
$$

For fixed $x \in X$ this produces an [[endofunction]] $act(x,-) \colon Y \to Y$ and hence some "transformation" or "action" on $Y$. In this way the whole of $X$ acts on $Y$.

Here $(x\mapsto act(x,-))$ is the _[[currying|curried]]_ function $\widehat{act}\colon X \to Y^Y$ of the original $act$, which maps $X$ to the [[function set|set]] of [[endofunctions]] on $Y$.  Quite generally one has these two perspectives on actions.

Usually the key aspect of an action of some $X$ is that $X$ itself carries an algebraic structure, such as being a [[group]] (or just a [[monoid]]) or being a [[ring]] or an [[associative algebra]], which is also possessed by $Y^Y$ and preserved by the curried action $\widehat{act}$.  Note that if $Y$ is any set then $Y^Y$ is a monoid, and when $X$ acts on it one calls it an [[MSet|X-set]].  For $Y^Y$ to have a ring/algebra structure, $Y$ must be some sort of [[abelian group]] or [[vector space]] with the action by [[linear functions]]; then one calls the action also a _[[module]]_ or _[[representation]]_.

In terms of the uncurried action $X\times Y\to Y$, the "preservation" condition says roughly speaking that acting consecutively with two elements in $X$ is the same as first multiplying them and then acting with the result:

\[
  \label{ActionProperty}
  act(x_2,act(x_1,y)) = act(x_2\cdot x_1, y)
  \,.
\]

To be precise, this is the condition for a *left action*; a *right action* is defined dually in terms of a map $Y\times X\to Y$.  If $X$ has no algebraic structure, or if its relevant structure is commutative, then there is no essential difference between the two; but in general they can be quite different.

This _action property_ can also often be identified with a [[functor]] property: it characterizes a [[functor]] from the [[delooping]] $\mathbf{B}X$ of the [[monoid]] $X$ to the [[category]] (such as [[Set]])  of which $Y$ is an object. The distinction between left and right actions is mirrored in the variance; acting on the left yields a covariant functor, whereas acting on the right is expressed via contravariance.

In this way essentially every kind of [[functor]], [[n-functor]] and [[enriched functor]] may be thought of as defining a generalized kind of action. This perspective on actions is particularly prevalent in [[enriched category theory]], where for instance [[coends]] may be thought of as producing [[tensor products]] of actions in this general functorial sense.

Under the [[Grothendieck construction]] (or one of its variants), this perspective turns into the perspective where an action of $X$ is some [[bundle]] $Y/X$ over $\mathbf{B}X$, whose [[fiber]] is $Y$:

$$
  \array{
    Y &\longrightarrow& Y/X
    \\
    && \downarrow
    \\
    && \mathbf{B}X
  }
  \,.
$$

Here the total space $Y/X$ of this bundle is typically the "weak" [[quotient]] (for instance: [[homotopy quotient]]) of the action, whence the notation. If one thinks of $\mathbf{B}X$ as the [[classifying space]] for the $X$-[[universal principal bundle]], then this bundle $Y/X \to \mathbf{B}X$ is the $Y$-[[fiber bundle]] which is [[associated bundle|associated]] via the action to this universal bundle. For more on this perspective on actions see at _[[∞-action]]_.

## Definitions

### Actions of a group
 {#ActionsOfAGroup}

An **action** of a [[group]] $G$ on an [[object]] $S$ in a [[category]] $\mathcal{C}$ is a [[representation]] of $G$ on $S$, that is a [[group homomorphism]] $\rho \colon G \to Aut(S)$, where $Aut(S)$ is the [[automorphism group]] of $S$ in $\mathcal{C}$.

Group actions, especially [[continuous function|continuous]] actions on [[topological spaces]], are also known as _transformation groups_ (starting around [Klein 1872, Sec. 1](#Klein1872), see also [Koszul 65](#Koszul65) [Bredon 72](#Bredon72), [tom Dieck 79](#tomDieck79), [tom Dieck 87](#tomDieck87)).  Alternatively, if the group $G$ that acts is understood, one calls ([Bredon 72, Ch. II](#Bredon72)) the space $X$ equipped with an action by $G$ a _[[topological G-space]]_ (or *[[G-set]]*, *[[G-manifold]]*, etc., as the case may be).

As indicated above, a more abstract but equivalent definition regards the group $G$ as a category (a [[groupoid]]), denoted $\mathbf{B} G$, with a single object $\ast$.  Then an _action_ of $G$ in the category $C$ is equivalently a [[functor]] of the form
$$
  \rho \colon \mathbf{B} G \to \mathcal{C}
$$
Here the object $S$ of the previous definition is the $\rho(\ast)$ of that single object.

Concretely, if $\mathcal{C}$ is a category like [[Set]], then an action is equivalently a [[function]]
$$
  \array{
    G \times S &\longrightarrow& S
    \\
    (g,s) &\mapsto& \rho(g)(s)
  }
$$
which satisfies the _action property_

\[
  \label{ActionPropertyOfGroupActions}
  \underset{
    g_1, g_2, s
  }{\forall}
  \;\;\;
  \rho(g_1 \cdot g_2)(s)
  \;=\;
  \rho(g_1)
  \big(
    \rho(g_2)(s)
  \big)
\]
and
$$
  \underset{s}{\forall}
  \;\;\;
  \rho(e)(s) \;=\; s
$$

### Actions of a monoid

More generally we can define an _action_ of a [[monoid]] $M$ in the category $C$ to be a functor
$$\rho: \mathbf{B} M \to C $$
where $\mathbf{B} M$ is (again) $M$ regarded as a one-object category.

The _category of actions_ of $M$ in $C$ is then defined to be the [[functor category]] $C^{\mathbf{B} M}$. When $C$ is $Set$ this is called [[MSet]].

Considering this in [[enriched category theory]] yields the [[internalization|internal]] notion of _[[action objects]]_.

### Actions of a category

One can[^2] also define an [[action of a category on a set|action of a category]] $D$ on the category $C$ as a functor from $D$ to $C$, but usually one just calls this a [[functor]].

[^2]: One example of this relatively rare usage is [[William Lawvere]]: _Qualitative Distinctions Between Some Toposes of Generalized Graphs_, Contemporary Mathematics 92 (1989) in which this sense of action is routinely used.

Another perspective on the same situation is: a (small) category is a [[monad]] in the category of [[span]]s in [[Set]]. An action of the category is an algebra for this monad. See [[action of a category on a set]].

On the other hand, an action of a [[monoidal category]] (not _in_ a monoidal category, as above) is called an [[module category]] (also "[[actegory]]"). This notion can be expanded of course to actions _in_ a [[monoidal bicategory]], where in the case of $Cat$ as monoidal bicategory it specializes to the notion of module category.

### Actions of a group object

Suppose we have  a category, $C$, with binary [[product]]s and a [[terminal object]] $*$. There is an alternative way of viewing group actions in [[Set]], so that we can talk about an action of a [[group object]], $G$, in $C$ on an object, $X$, of $C$.

By the adjointness relation between cartesian product, $A\times ?$, and function set, $?^A$, in [[Set]], a group homomorphism

$$\alpha: G\to Aut(X)$$
corresponds to a function

$$act: G\times X\to X$$
which will have various properties encoding that $\alpha$ was a homomorphism of groups:

$$act(g_1g_2,x) = act(g_1,act(g_2,x))$$

$$act(1,x) = x$$

and these can be encoded diagrammatically.

Because of this, we can **define** an action of a group object, $G$, in $C$ on an object, $X$, of $C$ to be a morphism
$$act: G\times X\to X$$
satisfying conditions that certain diagrams (left to the reader) encoding these two rules, commute.

The advantage of this is that it does not require the category $C$ to have internal automorphism group objects for all objects being considered.

As an example, only [[locally compact topological space|locally compact topological spaces]] have well-behaved topological automorphism groups, and thus actions of topological spaces on topological spaces must either be restricted to actions on locally compact spaces, or else be defined as above.

As another example, within the category of [[profinite group|profinite groups]] viewed as topological groups, not all objects have automorphism groups for which the natural topology is profinite. Thus profinite group actions on (the underlying topological space of) a profinite group must either be given in this form, or else be restricted to actions on profinite groups for which the automorphism group is naturally profinite.

### Actions of a monoid object

Suppose we have  a category, $C$, with binary [[product]]s and a [[terminal object]] $*$. There is an alternative way of viewing monoid actions in [[Set]], so that we can talk about an action of a [[monoid object]], $M$, in $C$ on an object, $X$, of $C$.

By the adjointness relation between cartesian product, $A\times ?$, and function set, $?^A$, in [[Set]], a monoid homomorphism

$$\alpha: M\to End(X)$$
corresponds to a function

$$act: M\times X\to X$$
which will have various properties encoding that $\alpha$ was a homomorphism of monoids:

$$act(m_1m_2,x) = act(m_1,act(m_2,x))$$

$$act(1,x) = x$$

and these can be encoded diagrammatically.

Because of this, we can **define** an action of a monoid object, $M$, in $C$ on an object, $X$, of $C$ to be a morphism
$$act:M\times X\to X$$
satisfying conditions that certain diagrams (left to the reader) encoding these two rules, commute.

The advantage of this is that it does not require the category $C$ to have internal endomorphism monoid objects for all objects being considered.

### Actions of a set

The action of a set on a set was defined above; it consists of a function $act: X\times Y\to Y$.  This can equivalently be represented by a [[quiver]] with $Y$ as its vertices, with its edges labeled by elements of $X$, and such that each vertex has exactly one arrow leaving it with each label.  (This is a sort of "Grothendieck construction".)  It is also the same as a simple (non halting) [[deterministic automaton]], with $Y$ the set of states and $X$ the set of inputs.

That an action is a type of edge labeled quiver can be seen by explicitly giving the product [[projection]] functions, $p_1$ and $p_2$, of $X\times Y$.

$$X\overset{\quad p_1 \quad}{ \leftarrow}X\times Y\underoverset{\quad act \quad}{p_2}{\rightrightarrows}Y$$

The shape of this diagram corresponds to that of an edge labeled quiver:

$$Labels\overset{\quad label \quad}{ \leftarrow}Edges\underoverset{\quad target \quad}{source}{\rightrightarrows}Vertices$$

While the set $X$ has no algebraic structure to be preserved, the action $act$ generates a unique **[[free category]] action** $act^{*}:X^{*}\times Y\to Y$ where $X^{*}$ is the [[free monoid]] on $X$ containing [[path|paths]] of $X$ elements. The monoidal structure of $X^*$ is preserved: two actions in succession is equal to the action of the concatenation of their paths.

$$act^{*}(x^{*}_2,act(x^{*}_1,y)) = act^{*}(x^{*}_2\cdot x^{*}_1, y) $$

An action of a set $X$ in itself is also called a __[[binary operation]]__, and the set $X$ is called a __[[magma]]__.

## Examples

* A [[representation]] is a "linear action".

* In [[symplectic geometry]] one considers [[Hamiltonian actions]].

* In [[topology]]: [[topological G-space]]

  * [[circle action]]

  * [[group action on an n-sphere]]

(...)

## Related concepts
 {#RelatedConcepts}

* [[action object]]

* **action**, [[∞-action]],

  * [[conjugation action]], [[adjoint action]]

  * [[diagonal action]]

  * [[transitive action]], [[free action]], [[regular action]]

  * [[proper action]], [[properly discontinuous action]]

* [[coaction]]

* [[representation]], [[∞-representation]]

* [[module]], [[∞-module]]

* [[action monad]]

* [[associated bundle]], [[associated ∞-bundle]]

* [[quotient]], [[quotient stack]], [[quotient type]]

* [[representation theory]], [[invariant theory]]

* [[equivariant homotopy theory]]

  * [[Borel model structure]]

* [[biaction]]

## References

### Group actions

On [[group actions]], mostly in [[TopologicalSpaces]], hence in the form of [[topological G-spaces]]:

Historical origins:

* {#Klein1872} [[Felix Klein]], _[[Vergleichende Betrachtungen über neuere geometrische Forschungen]]_ (1872) Mathematische Annalen volume 43, pages 63–100 1893  ([doi:10.1007/BF01446615](https://doi.org/10.1007/BF01446615))

  English translation by M. W. Haskell:

  _A comparative review of recent researches in geometry_, Bull. New York Math. Soc. 2, (1892-1893), 215-249. ([euclid:1183407629](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-2/issue-10/A-comparative-review-of-recent-researches-in-geometry/bams/1183407629.full), LaTeX version retyped by Nitin C. Rughoonauth: [arXiv:0807.3161](https://arxiv.org/abs/0807.3161))

Introduction of group actions into ([[quantum physics|quantum]]) [[physics]] (cf. *[[Gruppenpest]]*):

* [[Hermann Weyl]], §III.1 in: *Gruppentheorie und Quantenmechanik*, S. Hirzel, Leipzig (1931), translated by H. P. Robertson: *The Theory of Groups and Quantum Mechanics*, Dover (1950) [[ISBN:0486602699](https://store.doverpublications.com/0486602699.html), [ark:/13960/t1kh1w36w](https://archive.org/details/ost-chemistry-quantumtheoryofa029235mbp/page/n15/mode/2up)]

Textbook accounts:

* {#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))

* {#tomDieck79} [[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))

* {#tomDieck87} [[Tammo tom Dieck]], *[[Transformation Groups]]*, de Gruyter 1987  ([doi:10.1515/9783110858372]( https://doi.org/10.1515/9783110858372))

Lecture notes:

* {#Koszul65} [[Jean-Louis Koszul]], _Lectures on Groups of Transformations_, Tata Institute 1965 ([pdf](http://www.math.tifr.res.in/~publ/ln/tifr32.pdf), [[KoszulGroupsOfTransformations.pdf:file]])

* [[Patrick Morandi]], _Group actions_ ([[MorandiGroupActions.pdf:file]])

[[!redirects action]]
[[!redirects actions]]
[[!redirects group action]]
[[!redirects group actions]]
[[!redirects monoid action]]
[[!redirects monoid actions]]

[[!redirects left action]]
[[!redirects left actions]]

[[!redirects right action]]
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[[!redirects transformation groups]]


action (physics) - table

| [[action functional]] | [[kinetic action]] | [[interaction]] | [[path integral]] [[measure]] |
|--|--|--|--|
| $\exp(-S(\phi)) \cdot \mu = $ | $\exp(-(\phi, Q \phi)) \cdot$  | $\exp(I(\phi)) \cdot$ | $\mu$  |
| [[BV-complex|BV]] [[differential]] | [[elliptic complex]] +  |  [[antibracket]] with interaction + |  [[BV-Laplacian]] |
| $d_q =$ | $Q$ + | $\{I,-\}$ + | $\hbar \Delta$   |


action functional

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Variational calculus
+-- {: .hide}
[[!include variational calculus - contents]]
=--
#### Physics
+-- {: .hide}
[[!include physicscontents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Idea

In [[physics]] the [[dynamics]] of a [[physical system]] may be encoded by a [[nonlinear functional]] -- called the **action functional** -- on its [[configuration space]]:

* in [[classical mechanics]] and [[classical field theory]] -- by the **action principle** or **[[principle of least action]]** -- the extrema of the action functional -- obtained by [[variational calculus]] and given by [[Euler-Lagrange equation]]s -- encode the physically observable configurations  ;

* in [[quantum mechanics]] and [[quantum field theory]] the evolution of the [[quantum state]]s is encoded by the integral -- the [[path integral]] -- of the exponentiated action functional over the space of field configurations.

For emphasis the description of dynamics by action functionals is called the **Lagrangean** approach. Another formulation of dynamics in physics that does not involve an action functional explicitly is [[Hamiltonian mechanics]] on [[phase space]]. At least in certain classes of cases the relation and equivalence of both approaches is understood. Generally the formulation of [[quantum field theory]] in terms of action functionals suffers from a lack of precise understanding of what the [[path integral]] over the action functional really means.

\begin{imagefromfile}
  "file_name": "action_functional_on_a_napkin.jpg",
  "float": "left",
  "margin": {
    "top": 0,
    "right": 10,
    "bottom": 10,
    "left": 0,
    "unit": "px"
  },
  "alt": "Action functional on a napkin",
  "caption": "Taken from A Zee, Fearful Symmetry"
\end{imagefromfile}

## Definition

Let $\mathbf{H}$ be the ambient [[(∞,1)-topos]] with a [[natural numbers object]] and equipped with an additive [[continuum]] [[line object]] $\mathbb{A}^1$ (see there). Let $C \in \mathbf{H}$ be the [[configuration space]] of a physical system. Then an **action functional** is a morphism

$$
  \exp(\tfrac{i}{\hbar} S(-)) : C \to \mathbb{A}^1 / \mathbb{Z}
  \,
$$

(here $\hbar$ refers to [[Planck's constant]]).

If $\mathbf{H}$ is a [[cohesive (∞,1)-topos]] then there is an intrinsic differential of the action functional to a morphism

$$
  \mathbf{d} \exp(\tfrac{i}{\hbar} S(-)) : C \to \mathbf{\flat}_{dR}\mathbb{A}^1/\mathbb{Z}
  \,.
$$

The [[equation]]

$$
  \mathbf{d} \exp(\tfrac{i}{\hbar} S(-)) = 0
$$

is the [[Euler-Lagrange equation]] of the system. It characterizes the [[critical locus]] of $S$ is the [[covariant phase space]] inside the configuration space: the space of classically realized trajectories/histories of the system. If $\mathbf{H}$ models [[derived geometry]] then this critical locus is presented by a [[BRST-BV complex]].

### Local action functionals (traditional theory)
 {#LocalActionFunctional}

An action functional is called **local** if it arises from integration of a [[Lagrangian]].

In traditional theory this is interpreted as follows: an action functional $S : C \to \mathbb{A}^1$ is called local if

* the [[configuration space]] $C$ is the space $C = \Gamma_X(E)$ of [[section]]s of a [[fiber bundle]] $E \to X$ over some parameter space ([[spacetime]] $X$);

* there is a [[Lagrangian]] density $J_\infty(E) \to \Omega^{\dim X}(X)$ on the [[jet bundle]] of $E$;

* on a section/field configuration $\phi : X \to E$ the action $S$ takes the value

  $$
    S(\phi) = \int_X L(j_\infty(\phi))
    \,,
  $$

  where $j_\infty(\phi) = (\phi, \partial_i \phi, \cdots)$ is the jet-prolongation of $\phi$ (the collection of all its higher partial derivatives).

Consider action functional for on a configuration space of
[[smooth function]]s from the line to a [[smooth manifold]] $X$.

We can consider

1. $ S(q) = \int_a^b L(q,\dot{q}) \,\mathrm{d}t $, where $q$ is a path through configuration space, on the time interval $[a,b]$, with derivative $\dot{q} = \mathrm{d}q/\mathrm{d}t$.  When minimising the action, we fix the values of $q(a)$ and $q(b)$.
2. $ L(q,\dot{q}) = \int_{S} \mathcal{L}(q,\dot{q}) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z $, where now $q$ is a configuration of fields on $S$, which is a region of space.  We fix boundary conditions on the boundary of $S$ (typically that $q$ and $\dot{q}$ go to zero if $S$ is all of space).
3. $ S(q) = \int_{R} \mathcal{L}(q,\dot{q}) \,\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t $, where now $q$ is a configuration of fields on $R$, which a region of spacetime, with time derivative $\dot{q} = \partial{q}/\partial{t}$.  We fix boundary conditions on the boundary of $R$.

The formulation of (3) above is still not manifestly coordinate independent.  However, $\mathrm{d}x\,\mathrm{d}y\,\mathrm{d}z\,\mathrm{d}t$ is simply the [[volume form]] on spacetime and $\dot{q}$ is merely one choice of coordinate on [[configuration space|state space]] and could just as easily be replaced by a derivative with respect to any timelike coordinate on spacetime (or drop coordinates altogether).

### Extended local action functionals in (higher) gauge theory
 {#ExtendedLocalInGaugeTheory}

For [[gauge theories]] and [[higher gauge theories]] the configuration spaces of the physical system are in general not plain [[manifolds]] or similar, but are [[orbifolds]] or more generally [[smooth groupoids]], [[smooth ∞-groupoids]]. (An exposition of and introduction to much of the following is at _[[geometry of physics]]_.)

For instance for $G$ a [[Lie group]] and $\mathbf{B}G_{conn}$ the smooth [[moduli stack]] of $G$-[[principal connections]] (see at [[connection on a bundle]]), then the [[smooth groupoid]] of $G$-[[gauge field]] configurations is the [[internal hom]]/[[mapping stack]] $[\Sigma, \mathbf{B}G_{conn}] \in $[[Smooth∞Grpd]] (or some concretification thereof, see at _[geometric of physics -- differential moduli](geometry%20of%20physics#DifferentialModuli)_: this is the [[smooth groupoid]] whose [[objects]] are $G$-[[gauge field]]-configurations on $\Sigma$ ([[connection on a bundle|connections]] on $G$-[[principal bundles]] over $\Sigma$), and whose [[morphisms]] are [[gauge transformations]] between these. The [[infinitesimal space|infinitesimal]] approximation to this [[smooth ∞-groupoid]], its _[[∞-Lie algebroid]]_ is the (off-shell) [[BRST complex]] of the theory. The tangent to the $n$-fold [[higher gauge transformations]] becomes the $n$-fold _ghosts_ in the BRST complex.

More generally $G$ here can by any [[smooth ∞-group]], such as the [[circle n-group]] $\mathbf{B}^{n-1}U(1)$ or the [[String 2-group]] or the [[Fivebrane 6-group]], and so on, in which case $[\Sigma, \mathbf{B}G_{conn}]$ is the [[smooth ∞-groupoid]] of [[higher gauge theory|higher gauge field]], [[gauge transformations]] between these, [[higher gauge transformations]] between those, and so on.

Notice that this means in particular that in [[higher geometry]] a [[gauge theory]] is a [[sigma-model]] [[quantum field theory]]: one whose [[target space]] is not just a plain [[manifold]] but is a [[moduli stack]] of gauge field configurations.

A [[gauge invariance|gauge invariant]] action functional is then a morphism of [[smooth ∞-groupoids]]

$$
  \exp( i S(-)) \colon [\Sigma, \mathbf{B}G_{conn}] \to U(1)
  \,.
$$

This is of particular interest, again, if it is _local_. In fact, in this context now we can also ask that it is "extended" in the sense of [[extended topological quantum field theory]]: that we have an action functional not only in top dimension, being a function, but also in codimension 1, being a [[prequantum bundle]], and in higher codimension, being a [[prequantum n-bundle]].

This is notably the case for all (higher) gauge theories of [[schreiber:infinity-Chern-Simons theory]] type, such as ordinary [[Chern-Simons theory]] and such as ordinary [[Dijkgraaf-Witten theory]], as well as its higher generalizations. In these cases the action functional $\exp(i S(-)) \colon [\Sigma, \mathbf{B}G_{conn}]$ arises itself from [[transgression]] of an [[extended Lagrangian]] that is defined on the universal [[moduli stack]] of gauge field configurations $\mathbf{B}G_{conn}$ itself, namely from a [[universal characteristic class]] in higher nonabelian [[differential cohomology]] of the form

$$
  \mathbf{L} \colon \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn}
  \,.
$$

Here $\mathbf{B}^n U(1)_{conn}$ is the universal [[smooth infinity-groupoid|smooth]] [[moduli infinity-stack]] for [[circle n-bundles with connection]]. Such a morphism of moduli stacks locally takes a [[connection on a bundle|connection]] [[differential form]] $A$ to a [[Chern-Simons form]] $CS(A)$, but globally it sends the underlying [[principal bundle]] to a [[circle n-group|circle (n-1)-group]] [[principal ∞-bundle]] and accordingly acts globally on the connection.
This is hence a fully local [[Lagrangian]]: an _[[extended Lagrangian]]_. Alternatively, one may think of this whole morphism as modulating a [[prequantum circle n-bundle]] on the universal moduli stack $\mathbf{B}G_{conn}$ of [[gauge fields]] itself.

For instance for ordinary [[Chern-Simons theory]] here $n = 3$ $G$ is a [[semisimple Lie group]] and $\mathbf{L}$ is a smooth and differential refinement of the [[first Pontryagin class]]/[[second Chern class]], or of an integral multiple of that (the "level" of the theory). In this case $\mathbf{L}$ may also be thought of as modulating the universal [[Chern-Simons circle 3-bundle]]. If instead $G$ is a [[discrete group]] then $\mathbf{L}$ is a [[cocycle]] in the $U(1)$-[[group cohomology]] and this is the [[extended Lagrangian]] of [[Dijkgraaf-Witten theory]].

This [[extended Lagrangian]] becomes an extended action functional after [[transgression]]: the operation of [[fiber integration in ordinary differential cohomology]] refines to a morphism of moduli stacks of the form

$$
  \exp(2 \pi i \int_{\Sigma_k} (-))
  \colon
  [\Sigma_k, \mathbf{B}^n U(1)_{\mathrm{conn}} ]
  \to
  \mathbf{B}^{n-k}U(1)_{conn}
 \,,
$$

where $\Sigma$ is an [[orientation|oriented]] [[closed manifold]] of [[dimension]] $k$. This morphism locally simply takes a [[differential n-form]] to its ordinary [[integration of differential forms]] over $\Sigma_k$, but globally it takes the correct [[higher holonomy]] of [[circle n-bundles with connection]].

Combining this with an extended [[schreiber:∞-Chern-Simons theory]] [[Lagrangian]] $\mathbf{L} \colon \mathbf{B}G_{conn} \to \mathbf{B}^n U(1)_{conn}$ as above yields for each dimension $k$ a [[prequantum circle n-bundle]] on the space of gauge field configurations over $\Sigma_k$, by forming the [[transgression]] composite

$$
  \exp(i S(-))
  \coloneqq
  \exp(2 \pi i \int_{\Sigma_k} [\Sigma_k, \mathbf{L}])
   \;\; \colon \;\;
  [\Sigma_k, \mathbf{B}G_{conn}]
  \stackrel{[\Sigma_k, \mathbf{L}]}{\to}
  [\Sigma_k, \mathbf{B}^n U(1)_{conn}]
  \stackrel{\exp(2 \pi i \int_{\Sigma_k}(-))}{\to}
  \mathbf{B}^{n-k}U(1)_{conn}
  \,.
$$

This morphism locally takes the local [[differential form]] incarnation $A$ of a [[connection on an ∞-bundle]]  to the exponentiation of the [[integration of differential forms]] $\int_\Sigma CS(A)$ of some higher [[Chern-Simons form]], but globally it computes the correct [[higher holonomy]] of the higher [[circle n-bundle with connection]] over the universal moduli stack of fields, as modulated by the extended [[Lagrangian]] $\mathbf{L}$.

## Examples

For [[spacetime]] [[field theory]]:

* [[Einstein-Hilbert action]]

  * [[Einstein-Maxwell theory]]

  * [[Einstein-Yang-Mills theory]]

  * [[Einstein-Yang-Mills-Dirac theory]]

  * [[Einstein-Maxwell-Yang-Mills-Dirac-Higgs theory]]

For [[branes]]:

* [[Nambu-Goto action]]

* [[Dirac-Born-Infeld action]]

* [[Perry-Schwarz action]]

* A large class of examples of action functionals arises in [[schreiber:∞-Chern-Simons theory]]. See there for details.

## Related concepts

[[!include action (physics) - table]]

* [[principle of extremal action]], [[Euler-Lagrange equations]]

* [[path integral]] [[quantization]]

* [[effective action functional]], [[background field formalism]]

* [[parent action functional]]

[[!include extended prequantum field theory - table]]

## References

Historical and expository accounts of the "[[principle of extremal action]]" or "action principle", for short:

* Agamenon R. E. Oliveira, *History of Two Fundamental Principles of Physics: Least Action and Conservation of Energy*, Advances in Historical Studies **3** 2 (2014) [[doi:10.4236/ahs.2014.32008](http://dx.doi.org/10.4236/ahs.2014.32008)]

* Walter Dittrich, *The Development of the Action Principle -- A Didactic History from Euler-Lagrange to Schwinger*, SpringerBriefs in Physics, Springer (2021) [[doi:10.1007/978-3-030-69105-9](https://doi.org/10.1007/978-3-030-69105-9)]

* [[Douglas Cline]], *Variational Principles in Classical Mechanics*, University of Rochester (2021) [[pdf](http://classicalmechanics.lib.rochester.edu/pdf/Variational_Principles_in_Classical_Mechanics_3e_final_fbcover.pdf), online version]

Textbook account in the context of [[gauge theories]]:

* [[Marc Henneaux]], [[Claudio Teitelboim]], §1.1.1 in: _[[Quantization of Gauge Systems]]_, Princeton University Press (1992) [[ISBN:9780691037691](https://press.princeton.edu/books/paperback/9780691037691/quantization-of-gauge-systems), [jstor:j.ctv10crg0r](https://www.jstor.org/stable/j.ctv10crg0r)]

Lecture notes with more details are in the section _[Lagrangians and Action functionals](geometry+of+physics#LagrangiansAndActionFunctionals)_ of

* _[[geometry of physics]]_ .

Discussion of extended higher local action functional for (higher) gauge theories of generalized [[schreiber:∞-Chern-Simons theory]] type are discussed in

* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Hisham Sati]], _[[schreiber:Extended higher cup-product Chern-Simons theories]]_

The extended local action functionals for ordinary 3d [[Chern-Simons theory]]/[[Dijkgraaf-Witten theory]] and for 7d [[String 2-group]] Chern-Simons theory are constructed in

* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Jim Stasheff]], _[[schreiber:Cech cocycles for differential characteristic classes]]_

and discussed further in

* [[Domenico Fiorenza]], [[Urs Schreiber]], [[Hisham Sati]], _[[schreiber:7d Chern-Simons theory and the 5-brane]]_

A comprehensive discussion in a general context of higher differential geometry is in

* [[Urs Schreiber]], _[[schreiber:differential cohomology in a cohesive topos]]_

[[!redirects action functional]]
[[!redirects action functionals]]
[[!redirects action functinal]]

[[!redirects local action functional]]
[[!redirects local action functionals]]


action group > history

< [[action group]]
[[!redirects action group -- history]]


action groupoid

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###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Representation theory
+-- {: .hide}
[[!include representation theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given an [[action]] $\rho$ of a [[group]] $G$ on a [[set]] $S$, the action groupoid $S//G$ is a bit like the quotient set $S/G$ (the set of $G$-orbits).  But, instead of taking elements of $S$ in the same $G$-orbit as being [[equality|equal]] in $S/G$, in the action groupoid they are just [[isomorphism|isomorphic]].  We may think of the action groupoid as a [[homological resolution|resolution]] of the usual quotient.  When the action of $G$ on $S$ fails to be free, the action groupoid is generally better-behaved than the quotient set.

The action groupoid also goes by other names, including '[[weak quotient]]'.  It is a special case of a '[[pseudo colimit]]', as explained below. It is also called a "[[semidirect product]]" and then written $S \rtimes G$. The advantage of this is that it accords with the generalisation to the action of a group $G$ on a groupoid $S$, which is relevant to orbit space considerations, since if $G$ acts on a space $X$ it also acts on the fundamental groupoid of $X$; this is fully developed in  "Topology and Groupoids", Chapter 11.

## Definition

### In category theory
Given an [[action]]
$\rho : S \times G \to S$ of a group $G$ on the set $S$, the _action groupoid_ $S//G$ (or, more precisely, $S//_\rho G$) is the [[groupoid]] for which:

* an object is an element of $S$

* a morphism from $s \in S$ to $s' \in S$ is a group element $g \in G$ with $g s = s'$.  So, a general morphism is a pair $(g,s) : s \to g s$.

* The composite of $(g,s) : s \to g s = s'$ and $(g',s'):
s' \to g's'$ is $(g' g, s) : s \to g' g s$.

Equivalently, we may define the _action groupoid_ $S//G$ to be the groupoid
$$
  \array{
     && S \times G
     \\
     & {}^{s := p_1}\swarrow
     &&
     \searrow^{t = \rho}
     \\
     S
     &&&&
     S
  }
$$
with composition
$$(S \times G) \times_{t,s} (S \times G) \simeq S \times G \times G \to S \times G$$
given by the product in $G$.

We can denote the morphisms in $S//G$ by

$$S//G:=\{s\stackrel{g}{\to} \rho(s,g) | s\in S, g\in G\}.$$

### In (∞,1)-category theory

+-- {: .num_defn #ActionInfinityGroupoid}
###### Definition
Let $C$ be an ($\infty$,1)-category, let $G\in Grpd(C)$ be a groupoid object in $C$, let $X\in C$ be an object. Then the [[simplicial object]]

$$
  \array{
      \cdots
    &
      \underoverset{\to}{\to}{\to}
    &
      X\times_{G_0}G\times_p G
    &
      \rightrightarrows
    &
      X\times_{G_0}G
    &
      \to
    &
      X
  }
$$

such that the degree-wise projections give a simplicial map

$$
  \array{
        \cdots
      &
        \underoverset{\to}{\to}{\to}
      &
        X\times_{G_0}G\times_p G
      &
        \rightrightarrows
      &
        X\times_{G_0}G
      &
        \to
      &
        X
    \\
      &&
        \downarrow
      &&
        \downarrow
      &&
        \downarrow^a
    \\
        \cdots
      &
        \underoverset{\to}{\to}{\to}
      &
        G\times_p G
      &
        \rightrightarrows
      &
        G
      &
        \xrightarrow{p}
      &
        G_0
  }
$$

is called an _action of_ $G$ _on_ $X$. The colimit $colim\; X\times_{G_0}^{\times_\bullet}$ is called _action $\infty$-groupoid of_ $G$ _on_ $X$.

=--

## Interpretations

On top of the above explicit definitions, there are several useful ways to think of action groupoids.

Recall that the action $\rho$ is equivalently thought of as a functor

$$
  \rho : \mathbf{B}G \to Sets
$$

from the [[group]] $G$ regarded as a one-object groupoid, denoted $\mathbf{B}G$.

This functor sends the single object of $\mathbf{B}G$ to the set $S$.

### As a pseudo colimit

$S//G$ is the [[2-limit|2-colimit]] of $\rho$, i.e., the [[category of elements]] of $\rho$.

$$
  S//G \simeq colim_{\mathbf{B}G} \rho
  \,.
$$

The universal cocone consists of cells of the form

$$
  \array{
     S &&\stackrel{\rho(g)}{\to}&& S
     \\
     & \searrow &\stackrel{\simeq}{\Leftarrow}& \swarrow
     \\
     && S//G
  }
  \,,
$$

where the 2-morphism is uniquely specified and in components given by $s \mapsto (s \stackrel{g}{\to} \rho(s,g))$.

### As associated universal bundle

Let $Set_*$ be the category of pointed sets and $Sets_* \to Sets$ be the canonical forgetful functor.
We can think of this as the "universal $Set$-bundle".

Then $S//G$ is the pullback

$$
  \array{
    S//G
    &\to&
    Sets_*
    \\
    \downarrow
    &&
    \downarrow
    \\
    \mathbf{B}G
    &\stackrel{\rho}{\to}&
    Sets
  }
  \,.
$$

One place where we discussed this is the comment [It was David Roberts who apparently first noticed...](http://golem.ph.utexas.edu/category/2007/08/on_hess_and_lack_on_bundles_of_1.html#c019094).

Notice also that an action of $G$ on the set $S$ gives rise to a morphism $p: S \rtimes G \to G$ which has the property of unique path lifting, or in other words is a discrete opfibration. It is also called a covering morphism of groupoids, and models nicely covering maps of spaces.

Higgins used this idea to lift presentations of a group $G$ to presentations of the  covering morphism of $G$ derived from the action of $G$ on cosets, and so to apply graph theory to obtain old and new subgroup theorems in group theory.

### As a stack

In the case where the action is [[internalization|internal]] to sets with structure, such as internal to [[Diff]] one wants to realize the action groupoid as a [[Lie groupoid]]. That Lie groupoid in turn may be taken to present a [[differentiable stack]] which then usually goes by the same name $S//G$.

## Properties

### Relation to representation theory

The action groupoids $X//G$ of a group $G$ come equipped with a canonical map to $\mathbf{B}G \simeq \ast //G$. Regarded via this map as objects in the [[slice (infinity,1)-category|slice]] of groupoids over $\mathbf{B}G$, action groupoids are in fact [[equivalence|equivalent]] to the actions that they arise from.

For more on this see at _[[infinity-action]]_ and also at _[[geometry of physics -- representations and associated bundles]]_.

## Action $\infty$-groupoid {#ActionooGrpd}

All of this goes through almost verbatim for actions in the context of [[(∞,1)-category theory]].

Let $G$ be an [[∞-group]] in that $\mathbf{B}G$ is an [[∞-groupoid]] with a single object. An action of $G$ on an [[(∞,1)-category]] is an [[(∞,1)-functor]]

$$
  \rho : \mathbf{B}G \to (\infty,1)Cat
$$

to [[(∞,1)Cat]]. This takes the single object of $\mathbf{B}G$ to some $(\infty,1)$-category  $V$.

Again we want to **define** the _action groupoid_ $V//G$ as the [[limit in a quasi-category|(∞,1)-categorical colimit]] over the action:

$$
  V//G := \lim_\to \rho
  \,.
$$

By the result [described here](http://ncatlab.org/nlab/show/limit+in+a+quasi-category#WithValInooGrpd) this is, as before, equivalent to the pullback of the "universal $(\infty,1)Cat$-bundle" $Z \to (\infty,1)Cat$, namely to the [[coCartesian fibration]]

$$
  \array{
    V//G &\to& Z
    \\
    \downarrow && \downarrow
    \\
    \mathbf{B}G &\stackrel{\rho}{\to}& (\infty,1)Cat
  }
$$

classified by $\rho$ under the [[(∞,1)-Grothendieck construction]]. As before, we can continue a [[fiber sequence]] to the left by adjoining the $(\infty,1)$-categorical pullback along the point inclusion $* \to \mathbf{B}G$

$$
  \array{
    V&\to& V//G &\to& Z
    \\
    \downarrow && \downarrow && \downarrow
    \\
    {*} &\to& \mathbf{B}G &\stackrel{\rho}{\to}& (\infty,1)Cat
  }
  \,.
$$

The resulting total $(\infty,1)$-pullback rectangle is the fiber of $Z \to (\infty,1)Cat$ over the $(\infty,1)$-category $V$, which is $V$ itself, as indicated.

## Some comments

 * If the action of a Lie group $G$ on the manifold $X$ is free and proper, what you get is a manifold $X/G$.

 *  If the action of a Lie group $G$ on the manifold $X$ is not necesssarily free and proper, what you get is a Lie groupoid, denoted (among other symbols) by $X//G$.

## Related concepts

* [[global quotient orbifold]], [[quotient stack]], [[homotopy quotient]]

* the [[groupoid cardinality]] of action groupoids is given by the [[class formula]]

## References

* P.J. Higgins, 1971, "Categories and Groupoids", van
Nostrand, {New   York}. Reprints in Theory and Applications of Categories, 7 (2005) pp 1--195.

* [[Ronnie Brown]], *Topology and groupoids*, Booksurge (2006) [[webpage](http://groupoids.org.uk/topgpds.html), [pdf](https://groupoids.org.uk/pdffiles/topgrpds-e.pdf)]

* John Armstrong's article, [Groupoids (and more group actions)](http://unapologetic.wordpress.com/2007/06/09/groupoids-and-more-group-actions/)

* John Baez, [TWF 249](http://math.ucr.edu/home/baez/week249.html)

[[!redirects action groupoids]]
[[!redirects action group]]
[[!redirects action groups]]
[[!redirects groupoid quotient]]


action Lie algebroid

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###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

For $V$ a  [[space]], $G$ a [[group]] and $\rho : G\times V \to V$ a  [[action]] of $G$ on $V$, we have the corresponding [[action groupoid]]. If everything is sufficiently smooth, this is a [[Lie groupoid]] denoted $V//_\rho G$.

The _action Lie algebroid_ of $\rho$ is the [[Lie algebroid]] that corresponds to this Lie groupoid (under [[Lie integration]]).

The [[Chevalley-Eilenberg algebra]] of an action Lie algebroid is in [[physics]] known as a [[BRST complex]].

## Details

Let $G$ be a [[Lie group]], $V$ a [[smooth manifold]] and $\rho : G \times V \to V$ a smooth [[action]]. Write $V//G$ for the corresponding [[action groupoid]], itself a Lie groupoid.  The [[Lie algebroid]] $Lie(V//G)$ corresponding to this is the [[action Lie algebroid]].

The [[Chevalley-Eilenberg algebra]] of the action Lie algebroid is

$$
  CE(Lie(V//G)) = (\wedge^\bullet_{C^\infty(V)} (C^\infty(V) \otimes\mathfrak{g}^*), d_{\rho})
  \,,
$$

where the differential acts on functions $f \in C^\infty(V)$ by

$$
  d_\rho : f \mapsto \rho(-)(-)^* f \in C^\infty(V)\otimes \mathfrak{g}^*
  \,.
$$

Explicitly, for $t \in \mathfrak{g}$ this sends $f$ to the function
$(d_\rho f)(t)$ which is the derivative along $t \in T_e G$ of the function $G \times V \stackrel{\rho}{\to}V \stackrel{f}{\to} \mathbb{R}$.

Even more explicitly, if we choose local coordinates $\{v^k\} : \mathbb{R}^{dim V} \to V$ on a patch, and choose a basis $\{t^a\}$ of $\mathfrak{g}^*$ then we have that restricted to this patch the differential is on generators given by

$$
  d_\rho : f \mapsto \rho^k{}_a t^a \wedge \partial_k f
$$

$$
  d_\rho : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c
  \,.
$$

Specifically for $V$ a finite dimensional [[vector space]], $\rho : G$ a _linear_ action, $\{v^k\}$ a choice of basis of that vector space and $f$ a _[[linear function]]_ $f= f_k v^k$ , we have that $(f_k := \partial_k f) \in \mathbb{R}^{dim V}$ are the components vector of the dual vector given by $V$ in this basis, and the above gives the [[matrix multiplication]] form of the action

$$
  d_\rho : v^k \mapsto t^a \rho_a{}^k{}_l  v^l
  \,.
$$

Notice for completeness that the equation $(d_\rho)^2 = 0$ is equivalent to the [[Jacobi identity]] of the Lie bracket and the action property of $\rho$:

$$
  d_\rho d_\rho v^k =
   (t^a \wedge t^b \rho_a{}^k{}_r \rho_b{}^r{}_l
    -
    \frac{1}{2}C^a{}_{b c}t^b \wedge t^c \rho_a{}^k{}_l )
     v^l
  \,.
$$

These local formulas shall be useful below for recognizing from our general abstract definition of covariant derivative the formulas traditionally given in the literature. For that notice that in the above local coordinates further restricting attention to linear actions, the [[Weil algebra]] of the action Lie algebroid is given by

$$
  W(Lie(V//G)) = (\wedge^\bullet_{C^\infty(\mathbb{R}^{dim V})} ( \Gamma(T^* \mathbb{R}^{dim V}) \oplus \mathfrak{g}^* \oplus \mathfrak{g}^*[1]), d_{W_\rho})
$$

where the differential is given on generators by

$$
  d_{W_\rho} : v^k \mapsto \rho_a{}^k{}_l t^a \wedge v^l + d_{dR} v^k
$$

$$
  d_{W_\rho} : t^a \mapsto - \frac{1}{2} C^a{}_{b c} t^b \wedge t^c + r^a
$$

and where the uniquely induced differential on the shifted generators -- the one encoding [[Bianchi identities]] -- is

$$
  d_{W_\rho}  : d_{dR} v^k
      \mapsto
        \rho_a{}^k{}_k r^a \wedge v^l
      - \rho_a{}^k{}_l t^a \wedge  d_{dR} v^l
$$

and

$$
  d_{W} : r^a \mapsto C^a{}_{b c} t^b \wedge r^c
  \,.
$$

Notice that we may identify the [[delooping]] Lie groupoid $\mathbf{B}G$ of $G$ with the action groupoid of the trivial action on the point, $\mathbf{B}G \simeq *//G$.  On Lie algebroids this morphism is dually the inclusion

$$
  CE(Lie(V//G)) \leftarrow CE(\mathfrak{g})
$$

that is the identity on $\mathfrak{g}^*$.

## Applications

* A [[covariant derivative]] is the 1-form [[curvature]] of [[∞-Lie algebra valued forms|Lie algebroid valued differential forms]] with values in an action Lie algebroid.

* [[equivariant de Rham cohomology]]

## Related concepts

* [[action groupoid]], [[homotopy quotient]], [[Borel construction]]

* [[BRST complex]]

[[!redirects action Lie algebroids]]