essential image

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# The essential image of a functor
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## Idea

The notion of _essential image_ is supposed to be an adaptation of the notion of [[image]] from a [[category theory|1-categorical]] to the
[[2-category|2-categorical]] context [[Cat]], i.e. to the image of [[functors]]. But some care has to be exercised.

For another definition of image of a functor, see [(2,1)-image](http://ncatlab.org/nlab/show/image#InfImageExamples).

## Definitions

(A concrete realization of) the __essential image__ of a [[functor]] $F: A\to B$ between [[category|categories]] or $n$-[[n-category|categories]] is the smallest [[replete subcategory]] of the target $n$-category $B$ containing the __image__ of $F$.  (The *image* is, in turn, the smallest [[subcategory]] which contains all the $n$-cells which are strictly the images of $n$-cells in $A$.)

Note that if $F$ is not [[pseudomonic functor|pseudomonic]], then its essential image, defined in this way, need not be equivalent to its ordinary image.

## Considerations of the principle of equivalence

Note that the property of "belonging to the image" (said of an object or morphism) breaks the [[principle of equivalence]] of [[category theory]]; of two [[equivalence|equivalent]] objects, one may belong while the other does not.  Passing to the essential image removes this, so that the property of "belonging to the essential image" respects the [[principle of equivalence]].

Of course, the property of "being equal to the essential image" (said of a subcategory) violates the [[principle of equivalence]], as is the property of "being replete".  But $D$ is a replete subcategory of $C$ if and only if the property of belonging to $D$ (said of an object or morphism) does not violate the principle of equivalence.

## Remarks

* the *image* of a functor $F:A\to B$  may contain some morphisms or cells which are not the images of any cell in $A$, namely the [[composite|compositions]] of $B$-composable chains of such image cells, whose preimage cells do not form any $A$-composable chain;
* the *essential image* in addition contains precisely all equivalent $k$-cells to the $k$-cells of the image for all $0 \leq k \leq n$.

[[!redirects essential images]]


essential sublocale

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## Idea

Essential [[sublocales]] are a generalization of [[locally connected geometric morphism|locally connected sublocales]].

## Definition

+-- {: .num_defn}
###### Definition
A [[sublocale]] $X_j$ of a [[locale]] $X$, given by a [[nucleus]] $j : \mathcal{O}(X) \to \mathcal{O}(X)$, is called *essential* (sometimes also *principal*) if and only if the following equivalent conditions are satisfied:

1. There is a monotone map $b : \mathcal{O}(X) \to \mathcal{O}(X)$ which is [[left adjoint]] to $j$.
2. The nucleus $j$ preserves arbitrary (not only finite) meets.
3. For any $u \in \mathcal{O}(X)$, there is a smallest $v \in \mathcal{O}(X)$ such that $u \preceq j(v)$.
4. The pullback functor $\mathrm{Sh}(X) \to \mathrm{Sh}(X_j)$ possesses a left adjoint (it always has a right adjoint).
5. The [[geometric embedding]] $\mathrm{Sh}(X_j) \hookrightarrow \mathrm{Sh}(X)$ is an [[essential geometric morphism]].
=--

+-- {: .proof}
###### Proof of equivalence
* The equivalence of (1) and (2) is by the [[adjoint functor theorem]] for complete lattices, which furthermore gives an explicit formula for the left adjoint $b$, namely
$$ b(u) \coloneqq inf \{ v \in \mathcal{O}(X) \,|\, u \preceq j(v) \}. $$
This also shows the equivalence of (1) and (3).
* The equivalence of (4) and (5) is by definition.
* Recall that the pullback of the representable sheaf $Hom_{\mathcal{O}(X)}(\cdot,u)$ is $Hom_{\mathcal{O}(X)}(\cdot,j(u))$. Therefore [[continuous functor|continuity]] of the pullback functor translates to continuity of $j$. This shows that (4) implies (2).
* Conversely, assume (3). Then the explicit description of the pullback functor given below (which is valid under this assumption) shows that the pullback functor preserves arbitrary limits. By the adjoint functor theorem for Grothendieck toposes, statement (4) follows.
=--

+-- {: .num_remark}
###### Remark
The left adjoint $b$ automatically satisfies $b(u) \preceq u$, $b(b(u)) = u$, $j(b(u)) = j(u)$, and $b(j(u)) = b(u)$ for all $u \in \mathcal{O}(X)$. These relations follow from playing around with the adjunction.
=--

## Examples

* Any [[open sublocale]] is essential: The nucleus for an open sublocale is of the form $j = (u \rightarrow -)$. Its left adjoint is $b = (- \wedge u)$.

* More generally, any [[locally connected geometric morphism|locally connected sublocale]] is essential.

## Properties

* For a general sublocale $i : X_j \hookrightarrow X$ and a sheaf $\mathcal{F}$ on $X$, the pullback $i^{-1} \mathcal{F}$ is the [[sheafification]] of the presheaf $u \mapsto \colim_{u \preceq j(v)} \mathcal{F}(v)$.
In the case that $X_j$ is an essential sublocale, this presheaf is simply given by the formula $u \mapsto \mathcal{F}(b(u))$
and is already a sheaf, so sheafification is not necessary.

* A sublocale of the one-point locale is essential if and only if it is [[open morphism|open]]. This is because the extra [[Frobenius reciprocity]] condition demanded by openness is automatically satisfied (in classical mathematics and in impredicative [[constructive mathematics]]).

* The lattice of essential sublocales of a given locale is [[complete lattice|complete]]. Suprema are calculated as in the lattice of all sublocales; infima, however, are not. There are counterexamples in ([Kelly and Lawvere (1989)](#KL89)), however in the slightly different context of essential [[localizations of categories]].

## In the internal language

Let $X_j \hookrightarrow X$ be a sublocale. From the [[internal language|internal point of view]] of the [[topos]] $Sh(X)$, this sublocale looks like a sublocale of the one-point locale and it's interesting to compare the properties of $X_j$ with this [[internal locale]].

+-- {: .num_prop}
###### Proposition
The following statements are equivalent:

1. $Sh(X) \models X_j \hookrightarrow 1 \,\text{is an essential sublocale}$.
2. $Sh(X) \models X_j \hookrightarrow 1 \,\text{is an open sublocale}$.
3. $X_j \hookrightarrow X$ is an open sublocale.

Note that none of these statements are equivalent to $X_j \hookrightarrow X$ being an essential sublocale.
=--

## References

* {#KL89}[[G. M. Kelly]], [[F. W. Lawvere]], _On the complete lattice of essential localizations_, Bull. Soc. Math. de Belgique **XLI** (1989) pp. 289–319 [[[Kelly-Lawvere_EssentialLocalizations.pdf:file]]]

* Guilherme Frederico Lima, _[From essential inclusions to local geometric morphisms](https://www.youtube.com/watch?v=YsoGN91Rh_s)_, talk at Topos à l'IHÉS in September 2015.

## Related concepts

* [[essential subtopos]]

* [[essential geometric morphism]]


essential subtopos

An [[essential geometric morphism|essential]] [[subtopos]], also called a [[level of a topos]].

Any subtopos is an [[internal locale|internal sublocale]] of the one-point locale. However, this sublocale is an [[essential sublocale]] from the [[internal language|internal point of view]] if and only if the subtopos is an _open_ subtopos (not an essential one). See _[[essential sublocale]]_ for details.

[[!redirects essential subtoposes]]


essential supremum

The __essential supremum__ of a [[measurable function]] is essentially the [[supremum]] of its [[image]].  But we ignore things that happen only on a [[null set]].

Until we write this article, see:

* [Wikipedia](http://en.wikipedia.org/wiki/Essential_supremum)
* [MathWorld](http://mathworld.wolfram.com/EssentialSupremum.html) (with a picture)
* [PlanetMath](http://planetmath.org/encyclopedia/EssentialSupremum.html)


essentially affine category

A category is **essentially affine** if its change of base functors for its [[fibration of points]] are [[equivalences]] of categories. An essentially affine category is both [[protomodular category|protomodular]] and [[Mal'cev category|Mal'cev]]. Every subobject is [[normal subobject|normal]].

Borceux and Bourn liken an essentially affine category to a non-pointed [[abelian category]] (see [[Borceux-Bourn]]).

##Example

* The category of abelian extensions of an arbitrary group $G$.


essentially algebraic (infinity,1)-theory

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## Definition

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######Definition

An **(finitary) essentially algebraic $(\infty,1)$-theory** is an [[(∞,1)-category]] $T$ with (finite) [[(∞,1)-limit]]s.

An **algebra** over an essentially algebraic $(\infty,1)$-theory in some [[(∞,1)-topos]] $\mathcal{X}$ is a (finite) $(\infty,1)$-limit preserving [[(∞,1)-functor]]

$$
  A : T \to \mathcal{X}
  \,.
$$

=--

## Properties

* An ordinary [[essentially algebraic theory]] is a 0-[[truncated]] essentially algebraic $(\infty,1)$-theory.

## Examples

* A [[geometry (for structured (∞,1)-toposes)]] is an essentially algebraic $(\infty,1)$-theory.

## Related entries

* [[algebraic theory]], [[essentially algebraic theory]]

* [[algebraic (∞,1)-theory]], **essentially algebraic $(\infty,1)$-theory**

## References

Algebras over essentially algebraic $(\infty,1)$-theories that play the role of [[structure sheaves]] of algebras are considered in

* [[Jacob Lurie]], _[[Structured Spaces]]_

[[!redirects essentially algebraic (∞,1)-theory]]

[[!redirects essentially algebraic (∞,1)-theories]]
[[!redirects essentially algebraic (infinity,1)-theories]]


essentially algebraic theory

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## Idea

A [[mathematical structure]] is _essentially algebraic_ if its definition involves [[functional relation|partially defined operations]] satisfying equational laws, where the [[domain]] of any given operation is a subset where various other operations happen to be equal.  An actual [[algebraic theory]] is one where all operations are total [[function]]s.

The most familiar example may be the ([[strict category|strict]]) notion of [[category]]: a [[small category]] consists of a set $C_0$ of objects, a set $C_1$ of morphisms, source and target maps $s,t : C_1 \to C_0$ and so on, but composition is only defined for pairs of morphisms where the source of one happens to equal the target of the other.

Essentially algebraic theories can be understood through [[category theory]] at least when they are finitary, so that all operations have only finitely many arguments.  This gives a generalisation of [[Lawvere theory|Lawvere theories]], which describe finitary [[algebraic theory|algebraic theories]].

As the domains of the operations are given by the solutions to equations, they may be understood using the notion of [[equalizer]].  So, just as a Lawvere theory is defined using a category with finite [[product]]s, a finitary essentially algebraic theory is defined using a category with [[finite limits]] — or in other words, finite products and also equalizers (from which all other finite limits, including [[pullback]]s, may be derived).

## Definition

As alluded to above, the most concise and elegant definition is through category theory.  The traditional definition is this:

+-- {: .un_def}
######Definition
An **essentially algebraic theory** or **finite limits theory** is a category that is [[finitely complete category|finitely complete]], i.e., has all finite limits.  A **model** of an essentially algebraic theory $T$ is a [[functor]]

$$F: T \to Set$$

that is [[left exact]], i.e., preserves all finite limits.  A **homomorphism** of models is a natural transformation

$$ \alpha : F \to F'$$

between left exact functors $F, F' : T \to Set$.  Models of an essentially algebraic theory $T$ and the homomorphisms between them form a category $Mod(T) = Lex(T, Set)$.

More generally, for any category with finite limits $X$, we can define the category of **models of $T$ in $X$**, $Lex(T, X)$, which has left exact functors $F: T \to X$ as objects and natural transformations between these as morphisms.
=--

However, the finiteness restriction on the limits above is not part of the core concept of an 'essentially algebraic' structure, so one may prefer to call a category with finite limits a **finitary** essentially algebraic theory.  We do this in what follows.

A more traditional syntactic definition (following Adámek--Rosicky; see the references below) is as follows:

+-- {: .un_def}
######Definition
An __essentially algebraic theory__ is a quadruple

$$\Gamma = (\Sigma, E, \Sigma_t, Def)$$

where $\Sigma$ is a many-sorted [[signature (in logic)|signature]] consisting only of operation symbols, $E$ is a set of $\Sigma$-equations, $\Sigma_t \subseteq \Sigma$ is a set of operation symbols called "total", and $Def$ is a function which assigns, to each operation $\sigma \in \Sigma - \Sigma_t$ of type $\prod_{i \in I} s_i \to s$, a set $Def(\sigma)$ of $\Sigma_t$-equations involving only variables $x_i \in Var(s_i)$.

A (set-theoretic) **model** $M$ of a theory $\Gamma$ assigns to each sort $s$ a set $M(s)$, to each operation symbol $\sigma: \prod_{i \in I} s_i \to s$ of $\Sigma$ a _partial_ function
$$M(\sigma): \prod_{i \in I} M(s_i) \to M(s)$$
such that

* For each $\sigma \in \Sigma_t$ the function $M(\sigma)$ is a total function;

* For each $\sigma \in \Sigma - \Sigma_t$ of type $\prod_{i \in I} s_i \to s$, and each $I$-tuple
$$a = \langle a_i \rangle_{i \in I} \in \prod_{i \in I} M(s_i),$$
$M(\sigma)(a)$ is defined if and only if all the equations in $Def(\sigma)$ are satisfied at the argument $a$.

* All the equations of $E$ are satisfied (i.e., are interpreted as equations between partial functions).
=--

Homomorphisms of models $\theta: M \to M'$ are defined in the standard way: a collection of functions $\theta(s): M(s) \to M'(s)$ for each sort of the signature $\Sigma$ which are compatible with the $M(\sigma), M'(\sigma)$ in the evident way.

## Properties

The point is that (in the finitary case) either notion of theory may be used to specify the same category of models, and that

+-- {: .standout}

Categories of models of finitary essentially algebraic theories are precisely equivalent to [[locally presentable category|locally finitely presentable categories]]. These are equivalent to categories of models of finite limit [[sketch]]es.

=--

A [[duality]] between essentially algebraic theories and their categories of models is given by [[Gabriel-Ulmer duality]].

## Examples

* A (multisorted) [[Lawvere theory]] $T$ is the same thing (has the same models) as a finitary essentially algebraic theory in which all operations are total. If $C_T$ is the opposite of the category of finitely presented $T$-algebras, then the category of models is equivalent to $Lex(C_T, Set)$.

* As mentioned above, categories are models of a finitary essentially algebraic theory.

* An equational [[Horn theory]] is essentially algebraic, but not all essentially algebraic theories are equational Horn theories. Perhaps surprisingly, $Cat$ is not the category of models of any equational Horn theory, nor is even the category $Pos$ of posets. See [this paper](ftp://132.206.150.195/pub/barr/pdffiles/horn.pdf) by Barr for a proof. Essentially algebraic theories are equivalent to *partial* Horn theories ([Palmgren, Vickers](#PalmgrenVickers)).

* An equivalent formulation is as a _cartesian theory_, a [[geometric theory]] in which disjunction $\bigvee$ is not used, and each use of existential quantification $\exists$ must be accompanied by a proof that existence is unique. See [[Elephant]].

## Related concepts

* [[algebraic theory]] / [[Lawvere theory]] / **essentially algebraic theory**

  * [[2-Lawvere theory]]

  *  [[algebraic (∞,1)-theory]] / [[essentially algebraic (∞,1)-theory]]

* [[cartesian logic]]

* [[monad]] / [[(∞,1)-monad]]

* [[operad]] / [[(∞,1)-operad]]

* [[generalized algebraic theory]]

[[!include categories and logic - table]]

## References
 {#References}

Freyd first introduced essentially algebraic theories here:

* [[Peter Freyd]], _Aspects of Topoi_, Bull. Austr. Math. Soc. 7, pp. 1--76, 467--80. 1972 ([pdf](https://www.cambridge.org/core/services/aop-cambridge-core/content/view/B65FB15DCD85C816F31D9C87D355AD24/S0004972700044828a.pdf/aspects-of-topoi.pdf))

* [[Jiří Adámek]],  M. Hébert,  [[Jiří Rosický]], _On essentially algebraic theories and their generalizations_, Algebra Universalis, August 1999, Volume 41, Issue 3, pp 213-227

* {#AdamekRosicky} [[Jiří Adámek]], [[Jiří Rosický]], section 3.D of _[[Locally presentable and accessible categories]]_, Cambridge University Press, (1994)

A nice equivalent formulation can be found in

* {#PalmgrenVickers}[[Erik Palmgren]], [[Steve Vickers]] Partial Horn logic and cartesian categories. Annals of Pure and Applied Logic, 145(2007), 314-355. ([pdf](http://www2.math.uu.se/~palmgren/partialalgebras_pre.pdf))

Cartesian theories were introduced under different names in the early seventies by [[John Isbell]], [[Peter Freyd]] and [[Michel Coste]] (cf. [Johnstone 1979](#Johnstone79)). A standard source is Johnstone ([2002](#Johnstone02)).

* {#Freyd02}[[Peter Freyd]], _Cartesian Logic_ , Theor. Comp. Sci. **278** (2002) pp.3-21.

* {#Johnstone79}[[Peter Johnstone]], _A Syntactic Approach to Diers' Localizable Categories_ , pp.466-478 in Springer LNM **753** Heidelberg 1979.

* {#Johnstone02}[[Peter Johnstone]], _Sketches of an [[Elephant]] II_ , Oxford UP 2002. (Around D1.3.4 p.833)

A short introductory/overview note:

* {#keml-diagrams} [[Andreas Nuyts]], _Understanding Universal Algebra Using Kleisli-Eilenberg-Moore-Lawvere Diagrams_, [note](https://anuyts.github.io/files/keml-diagrams.pdf)

[[!redirects essentially algebraic]]
[[!redirects essentially algebraic theory]]
[[!redirects essentially algebraic theories]]
[[!redirects finite limits theory]]
[[!redirects finite limits theories]]
[[!redirects finite limit theory]]
[[!redirects finite limit theories]]
[[!redirects cartesian theory]]


essentially bounded function

# Essentially bounded functions
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## Definition

Let $X$ be a [[measure space]], or at least a [[measurable space]] equipped with a $\delta$-[[delta-filter|filter]] of [[full sets]] (or a $\sigma$-[[sigma-ideal|ideal]] of [[null sets]]).  Let $Y$ be a [[bornological set]].

A [[function]] from $X$ (or from a full subset of $X$) to $Y$ is __essentially bounded__ if it is [[almost equal]] to a [[bounded function]] from $X$ to $Y$.  That is, for some full subset $E$ of $X$, the [[direct image]] $f(E)$ is a [[bounded subset]] of $Y$.  (Here we equip $X$, and indeed equip every full subset of $X$, with the [[final structure|trivial bornology]] according to which every set is bounded.)

If $Y$ also has the structure of a measurable space, then we typically require an essentially bounded function to be [[measurable function|measurable]], although technically this is an independent property.  (We can always equip $Y$ with the [[final structure|trivial measurable structure]] according to which only the [[empty subset]] and the [[improper subset]] are measurable; then every function to $Y$ is measurable.)

The [[quotient set]] of essentially bounded measurable functions from $X$ to $Y$ modulo [[almost equality]] is denoted $L^\infty(X,Y)$.  With $L^\infty(X)$, we typically take $Y$ to be the [[real line]] equipped with the [[Borel-measurable sets]], or occasionally the [[complex plane]].  With $L^\infty$, we typically take $X$ to be the real line equipped with the [[Lebesgue-measurable sets]].

## Properties

Given any measure space $X$, $L^\infty(X,\mathbb{R})$ and $L^\infty(X,\mathbb{C})$ are (real and complex) [[Lebesgue spaces]] with $p = \infty$, hence [[Banach spaces]].  In fact, they are [[Banach algebras]], indeed $C^*$-[[C-star-algebras|algebras]].  At least when $X$ is [[localizable measure space|localizable]], they are in fact $W^*$-[[W-star-algebra|algebras]].  (Note that the [[involution]] $*$ is trivial when the target is $\mathbb{R}$.)  Of course, they are [[commutative algebra|commutative]]; $L^\infty(X,\mathbb{R})$ may also be viewed as an associative [[JB-algebra]] (thus a [[JBW-algebra]] if $X$ is localizable).

Conversely, every commutative complex $W^*$-algebra is, up to [[isomorphism]] of $W^*$-algebras, of the form $L^\infty(X,\mathbb{C})$ for some localizable measure space $X$; the same goes for $L^\infty(X,\mathbb{R})$ and commutative real $W^*$-algebras with trivial involution.  In fact, we have more: a [[dual equivalence]] between the [[category]] of localizable measure spaces (or rather, [[localizable measurable spaces]], as the [[morphisms]] ---taken up to almost equality, of course--- need not respect the measure, except for the full and null sets) and the category of commutative complex $W^*$-algebras (or the category of commutative real $W^*$-algebras with trivial involution, or the category of associative real $JBW$-algebras).

## Uselessness

Arguably, we do not need the concept of essentially bounded function; it is sufficient to consider bounded [[partial functions]] with full domains, that is [[almost functions]].  Traditionally, [[measure theory]] uses [[total functions]], but it cares about them only up to [[almost equality]], so almost functions are a natural concept.  In [[constructive analysis]], one must use almost functions, since then not every partial function can necessarily be extended to a total function.

Nevertheless, essentially bounded functions are traditional.

[[!redirects essentially bounded function]]
[[!redirects essentially bounded functions]]
[[!redirects essentially-bounded function]]
[[!redirects essentially-bounded functions]]


essentially injective functor

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# Essentially injective functors
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## Idea

A [[functor]] $F\colon C \to D$ is **essentially injective** if it is [[injective]] on objects "up to isomorphism", namely on [[isomorphism classes]] of objects.

## Definition

$F\colon C \to D$ is **essentially injective** or **essentially injective on objects** if, for any two objects $c$ and $c'$ of $C$, if there is an [[isomorphism]] $F(c) \cong F(c')$ in $D$, then $c \cong c'$ in $C$. (The converse holds trivially, since functors preserve isomorphisms.) Equivalently, a functor is essentially injective if and only if it is [[injective]] on [[isomorphism classes]].

## Examples

* A functor between [[discrete categories]] (or, more generally, [[skeletal categories]]) is essentially injective iff it is a [[injective function]] between the classes of objects.

* An [[injective-on-objects functor]] is essentially injective if its image is [[skeletal category|skeletal]], but not generally otherwise.

* A [[composite]] of any two essentially injective functors is essentially injective.

* If $g f$ is essentially injective, then $f$ is essentially injective.

* A [[conservative functor]] is essentially injective when it is [[full functor|full]]. More generally, any [[pseudomonic functor]] is essentially injective. More generally still, any [[fully faithful functor]] is essentially injective.

* An [[isocofibration]] is a functor that is (strictly) injective on objects, and is hence essentially injective.

## Remark on terminology

Some sources call this property "isomorphism [[reflected limit|reflecting]]" or "isomorphism [[created limit|creating]]". However, such terminology more accurately refers to [[conservative functors]].

## Related concepts

[[!include properties of functors -- contents]]

## References

* [[Jiří Rosický]]. _Generalized Brown representability in homotopy categories_. Theory and Applications of Categories 20 (2008): 18-24.

[[!redirects essentially injective functors]]
[[!redirects essentially injective]]
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[[!redirects essentially injective on objects functor]]
[[!redirects essentially injective on objects functors]]
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essentially pointwise isomorphic > history

< [[unnaturally isomorphic]].


essentially small (infinity,1)-category

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# Contents
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## Idea

The notion of _essentially small $(\infty,1)$-category_ is the generalization of the notion of [[essentially small category]] from [[category theory]] to [[(∞,1)-category]] theory.

## Definitions

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

A [[quasi-category]] $C$ is **essentially $\kappa$-small** for some [[regular cardinal]] $\kappa$ if

1. the collection of [[equivalence in a quasi-category|equivalence classes]] in $C$ is $\kappa$-small;

2. for every [[morphism]] $f : x \to y$ in $C$ the [[simplicial homotopy group|homotopy sets]] of the [[hom-object in a quasi-category|hom ∞-groupoid]] at $f$ (that is, the  sets $\pi_i(Hom^R(x,y),f)$) are $\kappa$-small.

$C$ is **essentially small** if the above conditions hold "absolutely," i.e. with "$\kappa$-small" replaced by "small."

=--

This appears as [[Higher Topos Theory|HTT, def. 5.4.1.3, prop. 5.4.1.2]].

In the presence of the [[regular extension axiom]] (which follows from the [[axiom of choice]]), essential smallness is equivalent to being essentially $\kappa$-small for some small [[regular cardinal]] $\kappa$.

## Properties

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

Let $C$ be an [[(∞,1)-category]] and $\kappa$ an [[countable set|uncountable]] [[regular cardinal]]. The following are equivalent:

1. $C$ is $\kappa$-small.

2. $C$ is a $\kappa$-[[compact object in an (infinity,1)-category|compact object]] in [[(∞,1)Cat]].

3. $C$ is equivalently given by a [[quasi-category]] whose underlying [[simplicial set]] is a $\kappa$-[[small set]].

=--

This is [[Higher Topos Theory|HTT, prop. 5.4.1.2]]

The analogous statement holds for [[∞-groupoids]].

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

For $X$ an [[∞-groupoid]] and $\kappa$ an [[countable set|uncountable]]
[[regular cardinal]], the following are equivalent

1. For each [[object]] $x \in C$ the [[simplicial homotopy group|homotopy sets]]
   $\pi_n(X,x)$ are $\kappa$-[[small sets]].

1. $X$ is presented by a $\kappa$-[[small set|small]] [[simplicial set]]/[[Kan complex]].

1. $X$ is a $\kappa$-[[compact object in an (∞,1)-category|compact object]] in [[∞Grpd]].

=--

This is ([[Higher Topos Theory|HTT, corollary 5.4.1.5]]).

Notice that this proposition really requires that $\kappa$ be uncountable. When $\kappa = \omega$ it is not true: the $\omega$-compact objects of ∞Grpd are the homotopy retracts of finite CW-complexes, while the $\omega$-small ∞-groupoids are just the finite CW-complexes. Not every retract of a finite CW-complex has the homotopy type of a finite CW-complex: there is an obstruction, defined for a retract $X$ of a finite CW-complex, which is an element of $\tilde{K}_0(\mathbb{Z}[\pi_1(X)])$, is called Wall's finiteness obstruction, and vanishes if and only if $X$ has the homotopy type of a finite CW-complex. See Wall's paper in the references.

## Related concepts

* [[essentially small category]]

* [[compact object in an (∞,1)-category]]

## References

This is the topic of section 5.4.1 of

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

Wall's finiteness obstruction was defined in

* [[C. T. C. Wall]], _Finiteness conditions for CW-complexes_.

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

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

[[!redirects essentially small ∞-groupoid]]
[[!redirects essentially small ∞-groupoids]]
[[!redirects essentially small infinity-groupoid]]
[[!redirects essentially small infinity-groupoids]]

[[!redirects small ∞-groupoid]]
[[!redirects small ∞-groupoids]]
[[!redirects small infinity-groupoid]]
[[!redirects small infinity-groupoids]]


essentially small category > history

< [[essentially small category]]
[[!redirects essentially small category -- history]]


essentially small type

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Universes
+-- {: .hide}
[[!include universe - contents]]
=--
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition ##

In [[dependent type theory]], given a [[Russell universe]] $U$, a [[type]] $A$ is **purely essentially $U$-small** if the universe has an element $B:U$ and an equivalence $p:A \simeq B$.

A [[type]] $A$ is **merely essentially $U$-small** if [[existential quantifier|there merely exists]] an element $B:U$ such that $A \simeq B$:
$$\mathrm{isEssentiallySmall}_U(A) \coloneqq \exists B:U.A \simeq B$$

Similarly, given a [[Tarski universe]] $(U, T)$, a [[type]] $A$ is **purely essentially $U$-small** if the universe has an element $B:U$ and an equivalence $p:A \simeq T(B)$.

A [[type]] $A$ is **merely essentially $U$-small** if [[existential quantifier|there merely exists]] an element $B:U$ such that $A \simeq T(B)$:
$$\mathrm{isEssentiallySmall}_U(A) \coloneqq \exists B:U.A \simeq T(B)$$

## Examples

For a single universe $U$, [[propositional resizing]] states that the type of all [[h-propositions]] in a universe is essentially $U$-small.

## Making all essentially small types small

Given a [[Russell universe]] $U$, one could make all essentially $U$-small types actually small types by the following [[inference rules]]:

$$\frac{\Gamma \vdash A \; \mathrm{type} \quad \Gamma \vdash \mathrm{witnSmall}_A:\exists B:U.A \simeq B}{\Gamma \vdash A:U}$$

## See also ##

* [[type universe]]
* [[locally small type]]
* [[propositional resizing]]
* [[type theoretic axiom of replacement]]

## References ##

* [[Marc Bezem]], [[Ulrik Buchholtz]], [[Pierre Cagne]], [[Bjørn Ian Dundas]], [[Daniel R. Grayson]]: Section 2.19 of: *[[Symmetry]]* (2021) $[$[pdf](https://unimath.github.io/SymmetryBook/book.pdf)$]$

[[!redirects essentially small type]]
[[!redirects essentially small types]]

[[!redirects purely essentially small type]]
[[!redirects purely essentially small types]]

[[!redirects merely essentially small type]]
[[!redirects merely essentially small types]]


essentially surjective (infinity,1)-functor

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

#Contents#
* table of contents
{:toc}

## Definition

An $(\infty,1)$-[[(infinity,1)-functor|functor]] $F : C \to D$ is **essentially surjective** if the induced functor of the [[core]] [[infinity-groupoids]]
$$
  core(F_0) : core(C_0) \to core(D_0)
$$
is an [[effective epimorphism]].

An $(\infty,1)$-[[(infinity,1)-functor|functor]] $F : C \to D$ is **essentially surjective** if the induced functor of the [[homotopy category of an (infinity,1)-category|homotopy categories]] of the $(\infty,1)$-categories
$$
  h F_0 : h C_0 \to h D_0
$$
is [[essentially surjective functor|essentially surjective]]

## Properties

An [[(∞,1)-functor]] which is both essentially surjective as well as [[full and faithful (∞,1)-functor]] is precisely an [[equivalence of (∞,1)-categories]].

## Related concepts

[[!include properties of functors -- contents]]

[[!redirects (∞,1)-essentially surjective functor]]
[[!redirects (infinity,1)-essentially surjective functor]]
[[!redirects essentially surjective (∞,1)-functor]]


essentially surjective and full functor

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

#Contents#
* table of contents
{:toc}

## Idea

An _essentially surjective and full functor_ is a [[functor]] which is both [[essentially surjective functor|essentially surjective]] and [[full functor|full]].

Sometimes this condition is abbreviated _eso and full_, where "eso" is short for "essentially surjective on objects".

## Properties

### As 0-connected morphisms

+-- {: .num_prop #EsoAndFullIs0Connected}
###### Proposition

Let $f \colon X \longrightarrow Y$ be a [[functor]] between [[small categories]] that happen to be [[groupoids]], and write  $i \;\colon\; Grpd \hookrightarrow \infty Grpd$ for the [[fully faithful (infinity,1)-functor|full inclusion]] of groupoids into the [[(∞,1)-topos]] [[∞Grpd]] of [[∞-groupoids]]. Then the following are equivalent:

* $F$ is essentially surjective and full

* $i(F)$ is a [[0-connected morphism]]

=--

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

As discussed there, an [[effective epimorphism in an (∞,1)-category|effective epimorphism]] in [[∞Grpd]] between 1-groupoids is precisely an
[[essentially surjective functor]].

So it remains to check that for an essentially surjective $f$, being
0-connected is equivalent to being full.

The [[homotopy pullback]] $X \times_Y X$ is given by the groupoid
whose objects are triples $(x_1, x_2 \in X, \alpha : f(x_1) \to f(x_2))$
and whose morphisms are corresponding tuples of morphisms in $X$ making the
evident square in $Y$ commute.

By prop. \ref{CharacterizationByDiagonal}
it is sufficient to check that the [[diagonal]] functor $X \to X \times_Y X$ is
(-1)-connected, hence, as before, essentially surjective, precisely if $f$ is full.

First assume that $f$ is full.
Then for $(x_1,x_2, \alpha) \in X \times_Y X$
any object, by [[full functor|fullness]] of $f$
there is a morphism $\hat \alpha : x_1 \to x_2$ in $X$,
such that $f(\hat \alpha) = \alpha$.

Accordingly we have a morphism
$(\hat \alpha,id) : (x_1, x_2) \to (x_2, x_2)$ in $X \times_Y X$

$$
  \array{
     f(x_1) &\stackrel{f(\hat \alpha)}{\to}& f(x_2)
     \\
     \downarrow^{\mathrlap{\alpha}} && \downarrow^{\mathrlap{id}}
     \\
     f(x_2) &\stackrel{id}{\to}& f(x_2)
  }
$$

to an object in the diagonal.

Conversely, assume that the diagonal is essentially surjective. Then
for every pair of objects $x_1, x_2 \in X$ such that there is a morphism
$\alpha : f(x_1) \to f(x_2)$ we are guaranteed morphisms
$h_1 : x_1 \to x_2$ and $h_2 : x_2 \to x_2$ such that

$$
  \array{
    f(x_1) &\stackrel{f(h_1)}{\to}& f(x_2)
    \\
    \downarrow^{\mathrlap{\alpha}} && \downarrow^{\mathrlap{id}}
    \\
    f(x_2) &\stackrel{f(h_2)}{\to}& f(x_2)
  }
  \,.
$$

Therefore $h_2^{-1}\circ h_1$ is a preimage of $\alpha$ under $f$,
and hence $f$ is full.

=--

### Eso+full/faithful factorization system

In the [[2-topos]] [[Cat]], the [[pair]] of [[classes]] of [[morphisms]] consisting of

* left class: essentially surjective and full functors

* right class: [[faithful functors]]

form a [[factorization system in a 2-category]] -- the [[(eso and full, faithful) factorization system]].

When restricted to the [[(2,1)-topos]] [[Grpd]] and in view of Prop. \ref{EsoAndFullIs0Connected}, this is the special case of the [[n-connected/n-truncated factorization system]] in the [[(∞,1)-topos]] [[∞Grpd]] for the case that $(n = 0)$ and restricted to [[1-truncated]] [[objects]]. More on this is at _[infinity-image --  Of Functors between groupoids](infinity-image#NImagesOf1FunctorsBetweenGroupoids)_.

[[!redirects essentially surjective and full functors]]

## Related concepts

[[!include properties of functors -- contents]]

[[!redirects eso and full functor]]
[[!redirects eso and full functors]]


essentially surjective functor

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

# Essentially surjective functors
* table of contents
{: toc}

## Idea

A [[functor]] $F\colon C \to D$ is **essentially surjective**, or **essentially surjective on objects** (sometimes abbreviated to **eso**), if it is surjective on objects "up to isomorphism".

## Definition

$F\colon C \to D$ is **essentially surjective** if for every [[object]] $y$ of $D$, there exists an object $x$ of $C$ and an [[isomorphism]] $F(x) \cong y$ in $D$.

### In homotopy type theory

A [[functor]] $F : C \to D$ is **essentially surjective** if for all $y:D$ there *merely* exists an $x:C$ such that $F(x) \cong y$.

## Examples

* A functor between [[discrete categories]] (or, more generally, [[skeletal categories]]) is essentially surjective iff it is a [[surjective function]] between the classes of objects.

* Any [[bijective-on-objects functor]] or [[surjective-on-objects functor]] is essentially surjective.

* A [[composition]] of any two essentially surjective functors is essentially surjective.

* If $g f$ is essentially surjective, then $g$ is essentially surjective.

* An essentially surjective functor is additionally [[fully faithful functor|fully faithful]] precisely when it is an [[equivalence of categories]].

* The [[inclusion functor]] of a [[subcategory]] is essentially surjective precisely when the subcategory is [[essentially wide subcategory|essentially wide]].

* Every [[split essentially surjective functor]] is essentially surjective. The converse is true for [[strict functors]] in the presence of the [[axiom of choice]].

## Properties

* Strengthening the last example, there is an [[factorization system in a 2-category|orthogonal factorization system]] (in the up-to-isomorphism [[strict 2-category|strict]] sense) on $Cat$, in which eso functors are the left class and fully faithful functors are the right class.

  This is an "up-to-isomorphism" version of the [[bo-ff factorization system]], which is a 1-categorical orthogonal factorization system on $Cat$ in which the left class consists of [[bijective-on-objects functors]].  Thus the notion of essential surjectivity is a version of "bijective on objects" which does respect the [[principle of equivalence]], i.e. the version which views [[Cat]] as a [[bicategory]].

  In particular, while a functor factors uniquely-up-to-isomorphism as a b.o. functor followed by a fully faithful one, it factors only uniquely-up-to-equivalence as an e.s.o. functor followed by a fully faithful one.  Since b.o. functors are also e.s.o., any (eso,ff) factorization of some functor is equivalent to its (bo,ff) factorization.

* In any 2-category there is a notion of [[eso morphism]] which generalizes the essentially surjective functors in [[Cat]].  In a [[regular 2-category]], these form a [[factorization system in a 2-category]] together with the [[ff morphisms]].

## Related concepts

[[!include properties of functors -- contents]]

[[!redirects essentially surjective functor]]
[[!redirects essentially surjective functors]]
[[!redirects essentially surjective on objects functor]]
[[!redirects essentially surjective on objects functors]]
[[!redirects eso functor]]
[[!redirects eso functors]]
[[!redirects essentially surjective]]


essentially unique

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

#Contents#
* tale of contents
{:toc}

## Definition

An [[object]] in a [[category]] with a given property is **essentially unique** with this property if it is [[isomorphic]] to any other object with that property.

## Examples

An object that is the [[limit]] or [[colimit]] over a given diagram is essentially unique. There is a _unique_ isomorphism to any other limiting (colimit) object.


eta invariant

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Functional analysis
+-- {: .hide}
[[!include functional analysis - contents]]
=--
#### Arithmetic geometry
+--{: .hide}
[[!include arithmetic geometry - contents]]
=--
#### Theta functions
+--{: .hide}
[[!include theta functions - contents]]
=--
#### Physics
+--{: .hide}
[[!include physicscontents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Definition

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

Given a [[self-adjoint operator]] $D$ (usually first-order, such as a [[Dirac operator]] acting on [[sections]] of a [[vector bundle]] on a [[closed manifold|closed]] [[Riemannian manifold]]) with [[eigenvalues]] with multiplicities $\{\lambda_n\}$, then its _eta function_ is given by the [[series]]

$$
  \begin{aligned}
    \eta(s)
     & \coloneqq
    \underoverset{n = -\infty}{^\infty }{\sum} sgn(\lambda_n) \frac{1}{ {\vert \lambda_n\vert}^s}
  \end{aligned}
$$

expression wherever this [[convergence|converges]], and extended by [[analytic continuation]] from there.

=--

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

At the [[special values of L-functions|special value]] $s = 1$ the series in def. \ref{SeriesDefinition} does not converge, but if $D$ is indeed a [[Dirac operator]] then it is the expression of the [[Dirac propagator]]. Indeed the definition of $\eta$ by [[analytic continuation]] at $s = 1$ is the [[regularization (physics)|regularized]] [[Dirac propagator]].

=--

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

The eta function of $D$ is related to the [[zeta function of an elliptic differential operator]] $H = D^2$ (regarding $D$ as a [[Dirac operator]]/[[supersymmetric quantum mechanics]]-like square root of $H$) see [below](RelationToTheZetaFunction).

=--

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

The _eta invariant_ of $D$ is the [[special values of L-functions|special value]] $\eta(0)$.

=--

(e.g. [Richardson, first page](#Richardson))

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

Def. \ref{EtaInvariant} means that $\eta_0$ is the
[[zeta function regularization|regularized]] number of positive minus negative eigenvalues of $D$.

=--

(Notice that if $D$ itself happens to have only positive [[eigenvalues]], then its eta function already is on the notre the [[zeta function of an elliptic differential operator]].)

## Properties

### Relation to the theta-function / Mellin transform

The eta function is a kind of odd version of the [[Mellin transform]] of an odd version of the [[theta function]]:

$$
  \eta_D(s)
  =
  \frac{1}{\Gamma((s+1)/2)}
  \underoverset{0}{\infty}{\int}
  t^{(s-1)/2}
  Tr(D\,\exp(-t D^2))
  \,
  d t
  \,.
$$

e.g. ([Müller 94 (0.2)](#Müller94)).

### Relation to the zeta function
 {#RelationToTheZetaFunction}

#### Generally

Let $D$ be a [[self-adjoint operator]] such that

1. its eta function $\eta(s)$ is defined and [[analytic function|analytic]] at $s= 0$;

1. for $c \in I \subset \mathbb{R}$ in an interval such that no $-c$ is an [[eigenvalue]] of $D$ such that both the eta series $\eta_{D+c}$ and the [[zeta function of an elliptic differential operator|zeta function]] series $\zeta_{(D+c)^2}((s+1)/2)$ have a common lower bound $s \gt B$ for the values on wich the series converges

then

$$
  \frac{d}{d c} \eta_{D+c}(s) = s \zeta_{(D + c)^2}((s+1)/2)
  \,,
$$

where on the left we have the [[zeta function of an elliptic differential operator]] for $(D+c)^2$.

(e.g. [Richardson prop. 2](#Richardson)).

In particular this means that under the above assumptions the [[functional determinant]] of $D^2$ is given by

$$
  det (D^2)
   =
  \exp\left(
    \frac{\partial}{\partial s}\frac{\partial}{\partial c} \eta_{D}(0)
  \right)
  \,.
$$

#### On odd-dimensional manifolds
 {#OnOddDimensionalManifolds}

Under suitable conditions the exponentiated $\eta$-invariant $\exp(\pi i \eta(0))$ equals the [[Selberg zeta function]] of odd type.  ([Millson 78](#Millson78), [Park 01, theorem 1.2](#Park01), [Guillarmou-Moroianu-Park 09](#GuillarmouMoroianuPark09)).

### Relation with L-function
 {#AnalogyWithLSeries}

Relation of eta functions to [[Dirichlet L-functions]] includes ([Atiyah-Donelly-Singer 83](#AtiyahDonellySinger83), [Podesta 14](#Podesta14))

[[!include zeta-functions and eta-functions and theta-functions and L-functions -- table]]

### Boundaries, determinant line bundles and perturbative Chern-Simons
 {#OnManifoldsWithBoundary}

Let $\pi \colon X \to Z$ be a $Z$-parameterized collection of [[spin structure|spin]] [[Riemannian manifold]] of odd [[dimension]] with [[boundary]].

Equip the corresponding collection of [[Dirac operators]] $D_X$ with the [[boundary condition]] given by a choice of [[isometry]]

$$
  Ker^+ D_{\partial X} \simeq Ker^- D_{\partial X}
  \,.
$$

e.g. ([Müller 94, below (0.3)](#Müller94))

Define then the exponentiated eta-invariant to be the element

$$
  \tau_X
   \coloneqq
  \exp\left(
    \pi i
    \left(
        \eta_X(0)  + dim\left(ker D_X\right)
    \right)
  \right)
  \in
  Det^{-1}_{\partial X}
$$

in the inverse of the [[determinant line]]

$$
  Det_{\partial X}
    \coloneqq
  \left(
    det\left(ker^- D_{\partial X}\right)
  \right)
  \otimes
  \left(
    det\left(ker^+ D_{\partial X}\right)
  \right)^{-1}
  \,.
$$

(Here it is maybe noteworthy that, by the [above](#OnOddDimensionalManifolds), $\zeta_S = \exp(i \pi\, \eta_X(0))$ is the [[Selberg zeta function]].)

In fact this is a smooth [[section]] of the [[determinant line bundle]] as $X$ varies.

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

These sections given by the exponentiated eta invariant satisfy the [[sewing law]].

=--

This is due to ([Dai-Freed 94](#DaiFreed94)), reviewed in ([Freed 95a](#Freed95a)). See also ([Witten 15](#Witten15)) for discussion in relation to [[anomaly cancellation]] of [[fermions]] (specifically for the eta invariant in the [[Green-Schwarz mechanism]] see [Witten 99, section 2.2](Green-Schwarz+mechanism#Witten99)).

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

Prop. \ref{ExponentiatedEtaSatisfiesSewing} means that the eta-invariant satisfies something like the Atiyah-axioms for [[TQFT]] (but of course $\eta$ depends on a [[Riemannian metric|metric]]), a point of view highlighted in ([Bunke 94](#Bunke94)).
Indeed, this exponentiated eta invariant is one factor (together with [[analytic torsion]] and the classical CS invariant) of the [[perturbation theory|perturbative]] [[path integral quantization]] of [[Chern-Simons theory]] ([Witten 89 (2.17) (2.23)](#Witten89)), see at _[Chern-Simons theory -- Perturbative path integral quantization](Chern-Simons+theory#PerturbativePathIntegralQuantization)_.

=--

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

Also the [[theta function]] is a section of, up to [[isomorphism]], this determinant line bundle (or maybe its inverse) ([Freed 95b, p. 31](#Freed95b)).

> (and hopefully it coincides with the section given by the exponentiated $\eta$ under suitable conditions?)

=--

## Examples

### For Dirac operator on Riemann surfaces

For the [[Dirac operator]] on a [[Riemann surface]]/[[complex curve]] the eta function was discussed in ([Millson 78](#Millson78)).

See at _[[zeta function of a Riemann surface]]_ for more on this case.

## Related concepts

* [[e-invariant]]

* [[index of an operator]]

* [[Spin Chern-Simons theory]]

## References

The $\eta$-invariant was introduced by Atiyah-Patodi-Singer in the series of articles

* [[Michael Atiyah]], V. K. Patodi and [[Isadore Singer]], _Spectral asymmetry and Riemannian geometry I_ Proc. Cambridge Philos. Soc. 77 (1975), 43-69.

   _Spectral asymmetry and Riemannian geometry II. Proc. Cambridge Philos. Soc.

  _Spectral asymmetry and Riemannian geometry III_, Proc. Cambridge Philos. Soc. 79 (1976), 71-99.

as the [[boundary]] correction term  for the [[analytic index|index formula]] on a [[manifold with boundary]].

Introductions and surveys include

* [[Jean-Michel Bismut]], _Local index theory, eta invariants and holomorphic torsion: a survey_, pp. 1-76, in: Surveys in diff. geom. 3 (C.-C. Hsiung, S.-T. Yau, eds.) 1996. International Press

* {#Richardson} [[Ken Richardson]], _Introduction to the Eta invariant_ ([pdf](http://faculty.tcu.edu/richardson/Seminars/etaInvariant.pdf))

* [[Xianzhe Dai]], _Eta invariant and holonomy_ Chern Centennial (2011) ([pdf slides](http://www.nim.nankai.edu.cn/activites/conferences/Chern-Centennial-20111024/ppt_pdf/1027pm-XZDai.pdf))

* Wikipedia, _[Eta invariant](http://en.wikipedia.org/wiki/Eta_invariant)_

Formulation in the broader context of bordism theory is in

* [[Ulrich Bunke]], _On the topological contents of eta invariants_ ([arXiv:1103.4217](http://arxiv.org/abs/1103.4217))

Further discussion of the relation to [[holonomy]] is in

* [[Xianzhe Dai]], [[Weiping Zhang]], _Eta invariant and holonomy, the even dimensional case_, [arXiv:1205.0562](http://arxiv.org/abs/1205.0562)

* _Eta invariant and Selberg  zeta function of odd type over convex co-compact hyperbolic manifolds_ ([pdf](http://www.imar.ro/~sergium/fisiere/Eta-Selberg2501.pdf))

Discussion of relation to [[L-functions]] includes

* {#AtiyahDonellySinger83} [[Michael Atiyah]],  H. Donnelly; , [[Isadore Singer]], _Eta invariants, signature defects of cusps, and values of L-functions_, Annals of Mathematics. Second Series 118 (1): 131–177 (1983) doi:10.2307/2006957, ISSN 0003-486X, MR 707164

* {#Podesta14} Ricardo A. Podestá, _The eta function and eta invariant of Z2r-manifolds_ ([ariv:1407.7454](http://arxiv.org/abs/1407.7454))

Discussion of the case over [[Riemann surfaces]] includes

* {#Millson78} [[John Millson]], _Closed geodesic and the $\eta$-invariant_, Ann. of Math., 108, (1978) 1-39 ([jstor](http://www.jstor.org/stable/1970928))

* {#Park01} Jinsung Park, _Eta invariants and regularized determinants for odd dimensional hyperbolic manifolds with cusps_ ([arXiv:0111175](http://arxiv.org/abs/math/0111175))

* {#GuillarmouMoroianuPark09} Colin Guillarmou, Sergiu Moroianu, Jinsung Park, _Eta invariant and Selberg Zeta function of odd type over convex co-compact hyperbolic manifolds_ ([arXiv:0901.4082](http://arxiv.org/abs/0901.4082))

Discussion in relation to [[analytic torsion]] and [[perturbation theory|perturbative]] quantum [[Chern-Simons theory]] goes back to

* {#Witten89} [[Edward Witten]] _Quantum Field Theory and the Jones Polynomial_ Commun. Math. Phys. 121 (3) (1989) 351–399. MR0990772 ([project EUCLID](http://projecteuclid.org/euclid.cmp/1104178138))

with more in

* {#Lott90} [[John Lott]], _Eta and torsion_, 1990 ([pdf](http://math.berkeley.edu/~lott/lhouches.pdf))

* [[Lisa Jeffrey]], _Symplectic quantum mechanics and Chern-Simons gauge theory I_,  ([arxiv/1210.6635](http://arxiv.org/abs/1210.6635))

* [[Lisa Jeffrey]], Brendan McLellan, _Eta-Invariants and Anomalies in U(1) Chern-Simons Theory_ ([pdf](http://www.math.toronto.edu/mclellan/eta%20invariants%20and%20anomalies.pdf))

* {#Young} M. B. Young, section 2 of _Chern-Simons theory, knots and moduli spaces of connections_ ([pdf](http://www.math.sunysb.edu/~myoung/CS.pdf))

Discussion of the eta-invariant on manifolds with boundary is in

* {#Müller94} Werner Müller, _Eta invariants and manifolds with boundary_, J. Diff. Geom. 40 (1994) 311-377 ([pdf](http://www.maths.ed.ac.uk/~aar/papers/mueller.pdf))

* {#Bunke94} [[Ulrich Bunke]], _The $\eta$-Invariant as a Lagrangian of a Topological Quantum Field Theory_ ([arXiv:hep-th/9408162](http://arxiv.org/abs/hep-th/9408162))

* {#Witten15} [[Edward Witten]], _Anomalies revisited_, talk at [Strings2015](https://strings2015.icts.res.in) ([pdf slides](https://strings2015.icts.res.in/talkDocuments/6-2.00-2.30-Edward-Witten.pdf))

and regarding the result as taking values in the determinant line over the boundary is due to

* {#DaiFreed94} [[Xianzhe Dai]] [[Daniel Freed]], _$\eta$-Invariants and Determinant Lines_, J. Math. Phys. 35 (1994),
5155–5194 and C. R. Acad. Sci. Paris (1995),
585–592 ([arXiv:hep-th/9405012](http://arxiv.org/abs/hep-th/9405012))

with review and streamlined results in

* {#Freed95a} [[Daniel Freed]], _Determinant line bundles revisited_ ([dg-ga/9505002](http://arxiv.org/abs/dg-ga/9505002))

* {#Freed95b} [[Daniel Freed]], _On determinant line bundles_, Math. aspects of [[string theory]], ed. S. T. Yau, World Sci. Publ. 1987,  ([revised pdf](http://www.math.utexas.edu/~dafr/Index/determinants.pdf))

[[!redirects eta-invariant]]
[[!redirects Eta invariant]]

[[!redirects eta invariants]]
[[!redirects eta-invariants]]
[[!redirects Eta invariants]]

[[!redirects eta function]]
[[!redirects eta functions]]

[[!redirects eta function of a self-adjoint operator]]


eta-conversion

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

# $\eta$-conversion
* table of contents
{: toc}

## Idea

In [[type theory]], $\eta$-conversion is a [[conversion rule]] for "computation" which is "dual" to [[beta-conversion]].

Whereas $\beta$-reduction tells us how to simplify a term that involves an *eliminator* applied to a *constructor*, $\eta$-reduction tells us how to simplify a term that involves a *constructor* applied to an *eliminator*.

In contrast to $\beta$-reduction, whose "directionality" is universally agreed upon, the directionality of $\eta$-conversion is not always the same.  Sometimes one may talk about $\eta$-**reduction**, which (usually) simplifies a constructor--eliminator pair by removing it, but we may also talk about $\eta$-**expansion**, which (usually) makes a term more complicated by introducing a constructor--eliminator pair.  Although one might expect that of course we always want to use reduction to simplify, it is possible to put bounds on $\eta$-expansion, and $\eta$-reduction is ill-defined for the [[unit type]] (the exception prompting 'usually' above).

However, from a category-theoretic POV, it is more natural to consider $\eta$ to be an expansion, which paired with $\beta$ as a reduction can be used as a syntax for a [[lax 2-adjunction]] [(Seely)](#Seely).

The [[equivalence relation]] generated by $\eta$-reduction/expansion is called $\eta$-**equivalence**, and the whole collection of processes is called $\eta$-**conversion**.

## Examples

### For function types

The most common example is for a [[function type]] $A \to B$.

In this case, the constructor of $A \to B$ is a *$\lambda$-expression*: given a term $b$ of type $B$ containing a free variable $x$ of type $A$, then $\lambda x.b$ is a term of type $A \to B$.

The eliminator of $A \to B$ says that given a term $f$ of type $A \to B$ and a term $a$ of type $A$, we can *apply* $f$ to $a$ to obtain a term $f(a)$ of type $B$.

An $eta$-[[redex]] (a term that can be reduced by $\eta$-reduction) is then of the form $\lambda x.\, f(x)$ -- the constructor (lambda expression) applied to the eliminator (application).  Eta reduction reduces such a redex to the term $f$.

Conversely, $\eta$-expansion expands any bare function term $f$ to the form $\lambda x.\, f(x)$.  If $\eta$-expansion is applied again to this, we get $\lambda x.\, (\lambda y.\, f(y))(x)$, but $\beta$-reduction returns this to $\lambda x.\, f(x)$; therefore, this last form is considered to be fully $\eta$-expanded.  In general, the rule when applying $\eta$-expansion is to use it only when the result is not a $\beta$-redex.

### For product types

Although function types are the most publicized notion of $\eta$-reduction, basically all types in type theory can have a form of it.  For instance, in the [[negative type|negative]] presentation of a [[product type]] $A \times B$, the constructor is an [[ordered pair]] $(a,b)\colon A\times B$, while the eliminators are projections $\pi_1$ and $\pi_2$ which yield elements of $A$ or $B$.

The $\eta$-expansion rule then says that for a term $p\colon A\times B$, the term $(\pi_1 p, \pi_2 p)$ --- the constructor applied to the eliminators --- is equivalent to $p$ itself.  (Again, we do not repeat the $\eta$-expansion, as this would produce a $\beta$-redex.)  If we use $\eta$-reduction instead, then we simplify any subterm of the form $(\pi_1 p, \pi_2 p)$ to $p$ (and leave anything not of that form alone).

### For unit types

Above we did a product type with two factors, although it\'s easy to generalise to any [[natural number]] of factors.  The case with zero factors is known as the [[unit type]], and $\eta$-conversion behaves a bit oddly there; let us examine it.

In the [[negative type|negative]] presentation of the unit type $1$, the constructor is an [[empty list]] $()\colon 1$, while there are no eliminators.  The $\eta$-expansion rule then says that any term $p\colon 1$ is equivalent to the term $()$ --- the constructor applied to no eliminators.  In this case, if we repeat the $\eta$-expansion, this does not produce a $\beta$-redex (indeed, there is *no* $\beta$-reduction for the unit type), but simply makes no change.  If we try to apply $\eta$-reduction to $()$, then this is ill-defined; we could 'simplify' this to *any* term $p\colon 1$ that we might be able to construct.

The [[positive type|positive]] presentation of the unit type does have a well-defined $\eta$-reduction, however; see [[unit type]].

## Implementation

Eta-reduction/expansion is not as well-behaved formally as beta-reduction, and its introduction can make computational equality undecidable.  For this reason and others, it is not always implemented in computer proof assistants.

[[Coq]] versions 8.3 and prior do not implement $\eta$-equivalence (definitionally), but versions 8.4 and higher do implement it for [[dependent product types]] (which include [[function types]]).  Even in Coq v8.4, $\eta$-equivalence is not implemented for other types, such as inductive and coinductive types.  This is a good thing for [[homotopy type theory]], since $\eta$-equivalence for [[identity types]] forces us into [[set-level type theory]].

When $\eta$-equivalence is not an implemented as a direct identity, it may be derived for a defined (coarser than identity) equality.  For example, if $f =_{A \to B} g$ is defined to mean $\forall x.\, f(x) =_B g(x)$ (where $=_B$ is assumed to have been previously defined) and $(\lambda x.\, b)(a)$ is taken to be identical to $b[a/x]$ (implementing $\beta$-reduction), then $f$ and $\lambda x.\, f(x)$ are provably equal even if not identical.  Thus, eta-equivalence for function types follows from [[function extensionality]] (relative to any appropriate notion of equality).

Similarly, if "equality" refers to a Martin-Löf [[identity type]] in [[dependent type theory]], then a suitable form of $\eta$-equivalence is provable for inductively defined types (with $\beta$-reduction and a dependent eliminator).  This includes the identity types themselves, but this form of $\eta$-equivalence does not imply the identity types are valued in propositions because the identity type itself must be incorporated in stating the equivalence.  See the next section.

## Propositional $\eta$-conversion
 {#Propositional}

In [[dependent type theory]], an important role is played by *propositional $\eta$-conversions* which "compute to identities" along constructors.  For example, consider binary products with $\beta$-reduction, but not (definitional) $\eta$-conversion.  We say that **$\eta$-conversion holds propositionally** if

1. For any $p\colon A\times B$ we have a term $\eta_p \colon Id_{A\times B}(p, (\pi_1 p, \pi_2 p))$, and

2. For $a\colon A$ and $b\colon B$ we have a [[definitional equality]] $\eta_{(a,b)} = 1_{(a,b)}$ (where $1_{(a,b)}$ denotes the reflexivity constructor of the [[identity type]]).

Similar definitions apply for any other type.

The reason this notion is important is that it is "equivalent" to the ability to extend the eliminator of non-dependent type theory to a *dependent* eliminator, where the type being eliminated into is dependent on the type under consideration.

For instance, in the case of the binary product, suppose that $\eta$-conversion holds propositionally as above, and that we have a dependent type $z\colon A\times B \vdash C(z)\colon Type$ along with a term $x\colon A, y\colon B \vdash c(x,y) \colon C((x,y))$ defined over the constructor.  Then for any $p\colon A\times B$ we can "transport" along $\eta_p$ to obtain a term defined over $p$, yielding the dependent eliminator.  The rule $\eta_{(a,b)} = 1_{(a,b)}$ ensures that this dependent eliminator satisfies the appropriate $\beta$-reduction rule.

Conversely, if we have a dependent eliminator, then $\eta_p$ can be defined by eliminating into the dependent type $z\colon A\times B \vdash id_{A\times B}(z,(\pi_1 z, \pi_2 z))$, since when $z$ is $(x,y)$ we have $1_{(x,y)}$ inhabiting this type.

Note that this "equivalence" is itself only "propositional", however; if we go back and forth, we should not expect to get literally the same dependent eliminator or propositional $\eta$ term, only a propositionally-equal one.

The same principle applies to other types, particularly [[dependent sum types]] and [[dependent product types]]/[[function types]] (although the latter are a bit trickier).

## Related concepts

* [[lambda-calculus]]

* [[beta-reduction]]

## References ##

* Seely, _Modelling computations: a 2-categorical framework_, [pdf](http://www.math.mcgill.ca/~rags/WkAdj/LICS.pdf)
 {#Seely}

[[!redirects eta conversion]]
[[!redirects eta-conversion]]
[[!redirects ∞-conversion]]

[[!redirects eta reduction]]
[[!redirects eta-reduction]]
[[!redirects ∞-reduction]]

[[!redirects eta expansion]]
[[!redirects eta-expansion]]
[[!redirects ∞-expansion]]

[[!redirects eta rule]]
[[!redirects eta-rule]]
[[!redirects ∞-rule]]

[[!redirects eta equivalent]]
[[!redirects eta-equivalent]]
[[!redirects ∞-equivalent]]
[[!redirects eta equivalence]]
[[!redirects eta-equivalence]]
[[!redirects ∞-equivalence]]

[[!redirects η-expansion]]
[[!redirects η-reduction]]
[[!redirects η-conversion]]


eta-meson

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Fields and quanta
+-- {: .hide}
[[!include fields and quanta - table]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[meson]]:

The [[Lorentz group|Lorentz]] [[pseudoscalar]] and [[isospin]] [[scalar]], [[bound state|bound]] from [[up-quark|up]]/[[down-quarks]] in the [[first generation of fundamental particles]] and a [[strange quark]] in the [[second generation of fundamental particles]]. Hence a [[heavy meson]] with respect to 2-flavor [[chiral perturbation theory]].

\begin{imagefromfile}
    "file_name": "LightAndChiralPartnerMesonFieldsPinIII.jpg",
    "width": 600,
    "unit": "px",
    "margin": {
        "top": 0,
        "right": 10,
        "bottom": 0,
        "left": 20
    }
\end{imagefromfile}

[[!include flavors of fundamental particles -- table]]

## References

### General

See also:

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

### Decays

* B. M. K. Nefkens, J. W. Price, _The Neutral Decay Modes of the Eta-Meson_, Phys. Scripta T99:114-122, 2002 ([arXiv:nucl-ex/0202008](https://arxiv.org/abs/nucl-ex/0202008))

* Andrzej Kupsc, _What is interesting in $\eta$ and $\eta'$ Meson Decays?_,  AIPConf. Proc. 950:165-179, 2007 ([arXiv:0709.0603](https://arxiv.org/abs/0709.0603))

* Florian Jonas, _Measurement of $\omega$ and $\eta$ mesons via their three pion decay with ALICE in pp collisions at $\sqrt{s} = t TeV$_, 2018 ([cds:2653176](https://cds.cern.ch/record/2653176))

On the [[Dalitz decay]] of the $\eta$-meson:

* Sergi González-Solís, _Single and double Dalitz decays of $\pi^0$, $\eta$ and $\eta'$ mesons_, Nuclear and Particle Physics Proceedings Volumes 258–259, January–February 2015, Pages 94-97 ([doi:10.1016/j.nuclphysbps.2015.01.021](https://doi.org/10.1016/j.nuclphysbps.2015.01.021))

[[!redirects eta-mesons]]

[[!redirects η-meson]]
[[!redirects η-mesons]]


etale morphism of schemes

> This entry is about étale morphisms between [[schemes]]. The term _[[étale map]]_ is preferred in the context of [[topology]] and [[differential geometry]], see [[étalé space]] for the topological version.

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Étale morphisms
+--{: .hide}
[[!include etale morphisms - contents]]
=--
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The notion of _étale morphism of schemes_ is the realization of the general notion of [[étale morphism]] for maps between [[schemes]], hence it captures roughly the idea of a map of schemes which is a [[local homeomorphism]]/[[local diffeomorphism]].

A central use of étale morphisms of schemes is that they appear as [[coverings]] in the [[Grothendieck topology]] of the [[étale site]]. The [[abelian sheaf cohomology]] with respect to these étale covers of schemes is accordingly called _[[étale cohomology]]_.

## Definition

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

A [[morphism]] of [[schemes]] is an **étale morphism** if the following equivalent conditions hold:

1. it is

   1. [[smooth morphism of schemes|smooth]]

   1. [[unramified morphism|unramified]]

1. it is

   1. [[smooth morphism of schemes|smooth]]

   1. of [[relative dimension]] $0$.

1. it is

   1. [[flat morphism|flat]]

   1. [[unramified]];

1. it is

   1. [[formally étale morphism of schemes|formally étale]];

   1. [[locally of finite presentation]]

=--

(A number of other equivalent definitions are listed at [wikipedia](http://en.wikipedia.org/wiki/Etale_morphism).)

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

For morphisms $f \colon X \longrightarrow Y$ between [[algebraic varieties]] over an [[algebraically closed field]] this means that for all points $p \in X$ the induced morphism on [[tangent cones]]

$$
  T_p X \longrightarrow T_{f(p)} Y
$$

is an [[isomorphism]]. This is analogous to the corresponding characterization of [[local diffeomorphisms]] of [[smooth manifolds]].

=--

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

Relaxing the finiteness condition in item 4 of \ref{EquivalentConditionsForEtale} yields the notion of _[[weakly étale morphism]]_.

[[étale morphism of schemes|étale morphism]] $\Rightarrow$ [[pro-étale morphism of schemes|pro-étale morphism]] $\Rightarrow$ [[weakly étale morphism of schemes|weakly étale morphism]] $\Rightarrow$ [[formally étale morphism of schemes|formally étale morphism]]

=--

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

A jointly surjective collection of étale morphisms $\{U_i \to X\}$ is called an [[étale cover]].

=--

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

Most of the pairs of conditions in def. \ref{EquivalentConditionsForEtale} can be read as constraining the fiber of the morphism to be first suitably surjective/[[bundle]]-like ([[smooth morphism of schemes|smooth]], [[flat morphism|flat]]) and second suitably locally injective ([[unramified]]).

Specifically the first condition has an [[infinitesimal object|infinitesimal anlog]]: a [[formally étale morphism]] is a [[formally smooth morphism|formally smooth]] and [[formally unramified morphism]]. These notions also have an interpretation in [[synthetic differential geometry]] and there they correspond to the statement that a [[local diffeomorphism]] is a [[submersion]] which is also an [[immersion of smooth manifolds]].

=--

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

A morphism is **[[formally étale morphism]]** if it is

* [[formally smooth]] (satisfying an infinitesimal lifting property)

* and [[formally unramified]].

=--

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

These are sheaf-like properties, which can be formalized in the language of [[Q-categories]] ([[monopresheaf]] and [[epipresheaf]] properties on the $Q$-category of nilpotent thickenings).

See at _[[differential cohesion]]_ and at _[[infinitesimal shape modality]]_.
=--

## Properties

### Closure properties

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

* A [[composite]] of two étale morphism is itself étale.

* The [[pullback]] of an étale morphism is étale.

* If $f_1 \circ f_2$ is étale and $f_1$ is, then so is $f_2$.

=--

(e.g. [Milne, prop. 2.11](#Milne))

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

Use that an étale morphism is a [[formally étale morphism]] with finite fibers, and that $f \colon X \to Y$ is formally étale precisely if the [[infinitesimal shape modality]] [[unit of a monad|unit]] [[natural transformation|naturality square]]

$$
  \array{
    X &\longrightarrow& \Pi_{inf}(X)
    \\
    \downarrow && \downarrow
    \\
    Y &\longrightarrow& \Pi_{inf}(Y)
  }
$$

is a [[pullback]] square. Then the three properties to be shown are equivalently the [[pasting law]] for pullback diagrams.

=--

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

A [[smooth morphism of schemes]] is étale iff there is an [[étale cover]] of the base  scheme by [[open subschemes]] such that the pullback of the projection to each of them is an open local isomorphism of [[locally ringed spaces]] (and in particular, the pullback of the projection morphism is an [[étalé space|étale map]] of the corresponding underlying topological spaces).

=--

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

This disjointness picture of étale covers make them suitable for having nontrivial [[cohomology]] in situations where Zariski covers give vanishing cohomology.

=--

### Classes of examples

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

Let $k$ be a [[field]]. A morphism of [[scheme]]s $Y \to Spec k$ is étale precisely if $Y$ is a [[coproduct]] $Y \simeq \coprod_i Spec k_i$ for each $k_i$ a finite and separable [[field extension]] of $k$.

=--

This appears for instance as [de Jong, prop. 3.1 i)](#deJong).

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

Such étale morphisms are classified by the classical [[Galois theory]] of field extensions.

=--

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

A [[ring]] homomorphism of [[affine varieties]] $Spec(A) \to Spec(B)$ for $Spec(B)$ non-singular and for $A \simeq B[x_1, \cdots, x_n]/(f_1, \cdots, f_n)$  with [[polynomials]] $f_i$ is étale precisel if the [[Jacobian]] $det(\frac{\partial f_i}{\partial x_j})$ is invertible.

=--

This appears for instance as ([Milne, prop. 2.1](#Milne)).

### As locally constant sheaves {#AsLocallyConstantSheaves}

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

A [[sheaf]] $F$ on a scheme $X$ corresponds to an étale morphism $Y \to X$ precisely if there is an [[étale cover]] $\{U_i \to X\}$ such that each restriction

$$
  F|_{U_i} \simeq LConst K_i
$$

is [[isomorphic]] to a [[constant sheaf]] on a [[set]] $K_i$.

=--

A proof is in ([Deligne](#Deligne)).

## Examples
 {#Examples}

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

A finite separable [[field extension]] $K \hookrightarrow L$ corresponds dually to an étale morphism $Spec L \to Spec K$. These are the morphisms classified by classical [[Galois theory]].

=--

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

Every [[open immersion of schemes]] is an étale morphism of schemes.
In particular a standard open inclusion (a [[cover]] in the [[Zariski topology]]) induced by the [[localization of a commutative ring]]

$$
  Spec(R[S^{-1}]) \longrightarrow Spec(R)
$$

is étale.

=--

(e.g. [[The Stacks Project|Stacks Project, lemma 28.37.9]])

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

By one of the equivalent characterizations of [[étale morphism]] it is sufficient to check that the map $Spec(R[S^{-1}]) \longrightarrow Spec(R)$ is a [[formally étale morphism]] and [[locally of finite presentation]].

The latter is clear, since the very definition of

$$
  R[S^{-1}] = R[s_1^{-1}, \cdots, s_n^{-1}](s_1 s_1^{-1} - 1, \cdots , s_n s_n^{-1} - 1)
$$

exhibits a [[finitely presented algebra]] over $R$.

To see that it is formally étale we need to check that for every [[commutative ring]] $T$ with [[nilpotent ideal]] $J$ we have a [[pullback]] diagram

$$
  \array{
    Hom(R[S^{-1}], T) &\longrightarrow& Hom(R[S^{-1}],T/J)
    \\
    \downarrow && \downarrow
    \\
    Hom(R, T) &\longrightarrow& Hom(R, T/J)
  }
  \,.
$$

Now by the [[universal property]] of the [[localization of a commutative ring|localization]], a homomorphism $R[S^{-1}] \longrightarrow T$ is a homomorphism $R \longrightarrow T$ which sends all elements in $S \hookrightarrow R$ to invertible elements in $T$. But no element in a [[nilpotent ideal]] can be invertible, Therefore the fiber product of the bottom and right map is the set of maps from $R$ to $T$ such that $S$ is taken to invertibles, which is indeed the top left set.

=--

## Related concepts

* [[étale morphism]]

* **étale morphism of schemes**

  * [[étale site]], [[étale cohomology]]

  * [[étale (∞,1)-site]]

  * [[étale morphism of E-∞ rings]]

## References

The classical references are

* [[Pierre Deligne]] et al., _Cohomologie étale_ , Lecture Notes in Mathematics, no. 569, Springer-Verlag, 1977.
{#Deligne}

* [[James Milne]], _[[Étale Cohomology]]_, Princeton Mathematical Series __33__, 1980. xiii+323 pp.

Lecture notes include

* [[James Milne]], section 2 of _[[Lectures on Étale Cohomology]]_
 {#Milne}

* [[Aise Johan de Jong]], _Étale cohomology_ ([pdf](http://math.columbia.edu/~pugin/Teaching/Etale_files/EtaleCohomology.pdf))
{#deJong} --- link broken, couldn't find another copy online

The local structure theorems are discussed in

* Leovigildo Alonso, Ana Jeremias, Marta Perez, _Local structure theorems for smooth maps of formal schemes_ ([arXiv:math/0605115](http://arxiv.org/abs/math/0605115))

Discussion of etale morphisms between [[E-infinity rings]]/[[spectral schemes]] is in

* [[Jacob Lurie]], section 8.5 of _[[Higher Algebra]]_

and generally in [[E-∞ geometry]] in

* [[Jacob Lurie]], section 1.2 of _[[Quasi-Coherent Sheaves and
Tannaka Duality Theorems]]_

[[!redirects étale morphism of schemes]]

[[!redirects etale morphisms of schemes]]
[[!redirects étale morphisms of schemes]]


etale morphisms - contents

**[[étale morphism]]** ([[submersion]], [[immersion]])

generally in **[[differential cohesion]]**

* [[reduction modality]] $\dashv$ [[infinitesimal shape modality]]

* [[formally étale morphism|formally étale]] ([[formally smooth morphism|formally smooth]], [[formally unramified morphism|formally unramified]])

* [[étale topos]]

* [[étale homotopy]]

* [[étale cohomology]]

* [[étale ∞-groupoid]]

in **[[topology]]**

* [[local homeomorphism]], [[étale space]]

* [[étale groupoid]]

in **[[differential geometry]]**

* [[local diffeomorphism]], ([[submersion]], [[immersion of smooth manifolds]])

* [[étale Lie groupoid]]

in **[[algebraic geometry]]**

* [[étale morphism of schemes]], [[formally étale morphism of schemes]]

* [[étale site]], [[pro-étale site]], [[étale (∞,1)-site]]

* [[Deligne-Mumford stack]]

* [[ℓ-adic cohomology]]


etale scheme

# Étale schemes
* table of contents
{: toc}

## Definition

Let $k$ be a field.

An _étale $k$-scheme_ is defined to be a directed colimit of $k$-[[ring spectrum|spectra]] $Sp_k k'$ of finite separable field-extensions $k'$ of $k$.

An _étale formal $k$-scheme_ is defined to be a directed colimit of [[formal spectrum|formal k-spectra]] $Spf_k k'$ of finite separable field-extensions $k^'$ of $k$.

## Properties

We give a characterization of étale $k$-schemes and étale formal $k$-schemes in terms of [[constant scheme|constant schemes]]:

The category $Sch_k$ of $k$-schemes is [[copower|copowered (= tensored)]] over $Set$. We define the _constant $k$-scheme_ on a set $E$ by

$$E_k \coloneqq E \otimes Sp_k k = \coprod_{e\in E} Sp_k k$$

For a scheme $X$ we compute $M_k(E_k,E) = Set(Sp_k k,X)^E = X(k)^E = Set(E,X(k))$ and see that there is an adjunction

$$((-)_k \dashv (-)(k))\colon Sch_k \to Set$$

A _constant formal scheme_ is defined to be a completion of constant scheme. The completion functor induces an equivalence between the category of constant schemes and the category of constant formal schemes.

+-- {: .num_remark}
###### Remark
Let $X$ be a $k$-scheme or a formal $k$-scheme. Then the following statements are equivalent:

1. $X$ is étale.

2. $X \otimes_k \overline k$ is constant.

3. $X \otimes_k k_s$ is constant.
where $\overline k$ denotes an algebraic closure of $k$, $k_s$ denotes the subextension of $\overline k$ consisting of all separable elements of $\overline k$ and $\otimes_k$ denotes skalar extension.
=--

+-- {: .proof}
###### Proof
$X$ is étale iff its scalar extension $X\otimes_k k_s$ is étale. And a $k_s$-scheme is étale iff it is constant.
=--

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

$$\begin{cases}
Sch_{et}\to Gal(k\hookrightarrow k_s)-Set
\\
X\mapsto X(k_s)
\end{cases}$$

from the category of [[étale scheme|étale schemes]] to the category of sets equipped with an [[action]] of the absolute Galois group is an equivalence of categories.

This statement is an instance of the main theorem of [[Grothendieck's Galois theory]] in the classical case of fields.
=--

Since this functor preserves products we have the analogue statement for [[group schemes]]:

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

The functor

$$\begin{cases}
GrSch_{et}\to Gal(k\hookrightarrow k_s)-Mod
\\
X\mapsto X(k_s)
\end{cases}$$

from the category of [[étale group scheme|étale group schemes]] to the category of [[Galois module|Galois modules]] of the [[absolute Galois group]] of $k$ is an equivalence of categories.
=--

If now the [[characteristic]] of $k$ is a prime number $p$ there is a relation of étale [[formal scheme|formal schemes]] resp. étale [[group scheme|group schemes]] and the [[Frobenius morphism]]:

+-- {: .num_prop}
###### Proposition
Let $X$ be a k-[[formal scheme]] resp. a [[locally algebraic scheme]].

Then $X$ is étale iff the [[Frobenius morphism]] $F:X\to X^{(p)}$ is a monomorphism resp. an isomorphism.
=--

## References

Michel [[Demazure, lectures on p-divisible groups]], sections I.8 and II.2, [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

[[!redirects etale scheme]]
[[!redirects etale schemes]]
[[!redirects étale scheme]]
[[!redirects étale schemes]]


ETCC

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

# Contents
* table of contents
{: toc}

[[!redirects category of categories]]

## Idea

> The explicit formulation of the principles of category theory ... is still in need of improved axiomatization. I will be overjoyed when some young person responds to that need.  ([[William Lawvere|Lawvere]], [2016](#Law16), p.1)

{#IsInABroadSense} **Elementary theory of the category of categories**, or _ETCC_ for short, is in a broad sense an appropriate name for any [[first order logic|first order]] [[theory]] axiomatizing the metacategory [[Cat|CAT]] of [[categories]] that forms the intuitive background for naive [[category theory]] with a view to the categorical [[foundations of mathematics]]. This is in [[categorification]] of the analogous first-order axiomatization of the metacategory [[Set|SET]] of [[sets]], for which see at *[[ETCS]]*.

Historically, the first attempt to formulate such a system of axioms has been undertaken by [[William Lawvere|F. W. Lawvere]] in his dissertation ([Lawvere 63](#Lawvere63)) and a subsequent publication ([Lawvere 1966](#Lawvere66)), and it is this system that is sometimes referred to as _the_ ETCC. We will discuss it together with subsequent attempts to repair its technical flaws under this heading in the next section.

Taking into account that the intuitively given category of categories is a [[2-category]], from an _[[nPOV|n-categorical perspective]]_ one would prefer to axiomatize CAT directly as a 2-category, and the resulting _elementary theory of the 2-category of categories_, or **ET2CC**, is the subject of a subsequent [section](#ET2CC).

## ETCC: Lawvere (1963) and beyond {#ETCC}

>In 1964, F. W. Lawvere proposed to [[foundations of mathematics|found]] mathematics on the [[category of categories]]. When he lectured on this at an international conference in Jerusalem, [[Alfred Tarski]] objected: "But what is a category if not a [[set]] of [[objects]] together with a [[set]] of [[morphisms]]?" Lawvere replied by pointing out that set theory axiomatized the binary relation of membership, while category theory axiomatized the ternary relation of [[composition]]. [^LS]

[^LS]: [Lambek&Scott 2009](#LS11), pp.171-172.

W. Lawvere launched in the early 60s an ambitious long-term project to replace membership based mathematics by more invariant concepts in order to arrive at a renewal of analysis and geometry adapted for the needs of modern (continuum) physics.

His 1963 [[Functorial Semantics of Algebraic Theories|dissertation]] takes issue with set-theoretic notions at various levels, e.g. redefining [[adjoint functors]] via [[comma categories]] in order to avoid reference to Hom-sets, but the most radical step is the attempt at a first-order axiomatization of the category of categories that he proposes to bypass [[set theory]] completely at a foundational level.

The system of axioms that he published in a more elaborate form in 1966, is layered in three parts:

* The _elementary theory of [[category|abstract categories]]_ (ETAC) has besides equality three primitive predicates for domain, codomain, and composition and as axioms the usual axioms for identity, associativity etc. (See at [[theory of categories]] for a variant of such an axiomatisation!)

* The _basic theory_ (BT) adds axioms for e.g. **1**, a one-morphism category, **2**, a category with one non-identity morphism between two objects, **3**, a category with a commutative triangle of morphisms, and various exactness properties among which is the existence of exponentials. This permits then to define e.g. sets as discrete objects $A$ with $A^1\cong A^2$ i.e. as categories whose morphisms are all identity morphisms. The final axiom of the basic theory tries to express the _[[dense subcategory|density]]-adequacy_ of the simple categories like **1** etc. within the metacategory.

* The _stronger theory_ (ST) then adds two further axioms to the basic theory concerning existence of completions and an 'inaccessible' category (p.18).

As pointed out by [[John Isbell|J. Isbell]] in 1967, one of Lawvere's results (namely, the theorem on the 'construction of categories  by description' on p.14) was mistaken, which left the axiomatics dangling with insufficient power to construct models for categories. Several ways to overcome these problems where suggested in the following but no system achieved univocal approval (cf. Blanc-Preller(1975), Blanc-Donnadieu(1976), Donnadieu(1975),  McLarty(1991)).

As ETCC also lacked the simplicity of [[ETCS]], it rarely played a role in the practice of category theory in the following and was soon eclipsed by [[topos theory]] in the attention of the research community that generally preferred to hedge their foundations with appeals to Gödel-Bernays set-theory or Grothendieck universes.

Nevertheless, the axiomatic treatment of 1963/66 contained also the seminal idea to internally define discrete set-theoretic 'foundations' within a bigger category, that will later on play a central role in Lawvere's approach to [[cohesion]]. So when one can view the 1964 ETCS as a forerunner of the Lawvere-Tierney axioms for an elementary topos, one could view ETCC as an anticipation of the 2007 axioms for a [[cohesive topos]].

## ET2CC: an elementary theory of the 2-category of categories{#ET2CC}

For Lawvere ETCS and ETCC have a different status, the first provides abstract constant sets as convenient carriers for structured objects, whereas the latter provides the overall ontology of concepts, the intellectual working space.

Yet from an **n-categorical perspective**, the categories they axiomatize differ in level only and so it is tempting to think of the second as a sort of [[categorification]] of the first: this suggests to posit an abstract concept 'foundation' that at the level of ETCS shows up as set theory, gets refined to 'category theory' one step higher, then as 2-category theory, and so on.

Now what comes closest to such a vision at the moment is probably [[homotopy type theory]] but truncating such a tower of 'foundations' might be still relevant for applications where concepts of [[2-topos|2-toposes]] are considered. It would also provide an axiomatisation for the 'familiar' meta 2-category CAT suggestive for the generalization to n-CAT.

At the time no such axiomatisation is fully worked out, but in 2009 [[Mike Shulman|M. Shulman]] gave a rough sketch of what such an **elementary theory of the 2-category of categories** (ET2CC) should look like (see at at *[[formal category theory]]*):

> Firstly, some exactness axioms amounting to its being a "2-pretopos" in the sense I described here:[[michaelshulman:2-categorical logic]] . This gives you an "internal logic" like that of an ordinary (pre)topos.

> Secondly, the existence of certain exponentials (this is optional).

> Thirdly, the existence of a "classifying discrete opfibration" el→set in the sense introduced by [[Mark Weber]] ("Yoneda structures from 2-toposes") which serves as "the category of sets," and internally satisfies some suitable axioms.

> Finally, a "well-pointedness" axiom saying that the terminal object is a generator, as is the case one level down with in ETCS. This is what says you have a 2-category of categories, rather than (for instance) a 2-category of stacks.

> Once you have all this, you can use finite 2-categorical limits and the "internal logic" to construct all the usual concrete categories out of the object "set".      ([Shulman 2009](#MO-CC))

We can see here how at the end Lawvere's idea of internal set-theoretic 'foundations' resurfaces. Such a level of _discreteness_ seems to be pervasive in this context and e.g. the approach of Przybylek (2014) to the concept of internal 2-cartesian closure takes discreteness as a starting point.

### Remark

Note that already B. Walters and R. Street conceived of their work on [[Yoneda structure|Yoneda structures]] in the early 1970s as a 2-dimensional version of ETCC (cf. Walters [1971](#W71)).

## Related entries

* [[ETCS]]

* [[ETCR]]

* [[Yoneda structure]]

* [[Functorial Semantics of Algebraic Theories]]

* [[foundations of mathematics]]

* [[theory of categories]]

* [[formal category theory]]

## References

* {#Lawvere63} [[William Lawvere]], *[[Functorial Semantics of Algebraic Theories]]*, Ph.D. thesis, Columbia University New York 1963. (With an author's comment and a supplement in: Reprints in Theory and Applications of Categories, no.5 (2004) pp.1-121 ([pdf](http://www.tac.mta.ca/tac/reprints/articles/5/tr5.pdf)))

A revised version of the axioms appears as

* {#Lawvere66} [[William Lawvere]], *The Category of Categories as a Foundation for Mathematics*, pp. 1-20 in: [[Samuel Eilenberg|S. Eilenberg]], [[D. K. Harrison]], [[S. MacLane]], [[H. Röhrl]] (eds.): *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

Important reviews of the paper pointing out technical problems appeared here

* [[John Isbell|J. R. Isbell]], *Review of Lawvere 1966*, Mathematical Reviews **34** (1967) #7332.

* G. Blanc. A. Preller, *Lawvere's basic theory of the category of categories*, JSL **40** (1975) pp.14-18.

A more recent view of Lawvere exposing in particular his 'Münchhausen' approach to set-theoretic 'foundations' is in

* [[William Lawvere|F. W. Lawvere]], _Foundations and Applications: Axiomatization and Education_, Bulletin of Symbolic Logic **9** no.2 (2003) pp.213-224. ([ps-preprint](https://www.math.ucla.edu/~asl/bsl/0902/0902-006.ps))

Lawvere briefly comments on his 1965 system 50 years later p.6 in

* {#Law16}F. W. Lawvere, *Alexander Grothendieck and the Concept of Space*, address CT15 Aveiro 2016. ([link](http://www.acsu.buffalo.edu/~wlawvere/GrothendiekAveiro15.htm))

An insightful and non-partisan view of ETCC can be found in a section of:

* [[Andreas Blass]], [[Yuri Gurevich]], *Why Sets?*, Bull. Europ. Assoc. Theoret. Comp. Sci. **84** (2004) 139-156.  [[doi:10.1007/978-3-540-78127-1_11](https://doi.org/10.1007/978-3-540-78127-1_11), [pdf](https://web.eecs.umich.edu/~gurevich/Opera/172.pdf), [pdf](http://www.math.lsa.umich.edu/~ablass/set.pdf)]

Alternative or complementary approaches are

* [[J. Bénabou]], *Fibered Categories and the foundations of naive category theory*, JSL **50** (1985) pp.10-37. ([preprint](http://www.cwru.edu/artsci/phil/Benabou%20Fibered.pdf))

* G. Blanc, *Propriétés du Premier Ordre et Catégories*, Cah. Top. Géom. Diff. Cat. **16** (1975) pp.222-224. ([Colloque Amiens 1975 proceedings](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1975__16_3/CTGDC_1975__16_3_217_0/CTGDC_1975__16_3_217_0.pdf),6.81 MB)

* G. Blanc, M. R. Donnadieu, *Axiomatisation de la catégorie des catégories*, Cah.Top.Géom.Diff.Cat. **XVII** no. 2 (1976) pp.1-35. ([pdf](http://archive.numdam.org/ARCHIVE/CTGDC/CTGDC_1976__17_2/CTGDC_1976__17_2_135_0/CTGDC_1976__17_2_135_0.pdf))

* M. W. Bunder, *Category Theory Based on Combinatory Logic*, Arch. Math. Logik **24** (1984) pp.1-15. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002045680))

* N. C. A. da Costa, O. Bueno, A. Volkov, *Outline of a Paraconsistent Category Theory*, pp.95-114 in Weingartner (ed.), *Alternative Logics. Do Science Need Them?*, Springer Heidelberg 2004. ([draft](http://homepage.mac.com/otaviobueno/.Public/ParacCategTheory.pdf))

* M. R. Donnadieu, *Démonstration du théorème de construction de catégories d'après description de Lawvere sous certains axiomes*, C. R. Acad. Sc. Paris **280** série A (1975) pp.307-308. ([gallica](http://gallica.bnf.fr/ark:/12148/bpt6k6228238v/f59.image))

* E. Engeler, H. Röhrl, *On the problem of foundations of category theory*, Dialectica **23** (1969) pp.58-66.

* [[Colin McLarty|C. McLarty]], *Axiomatizing a category of categories*, JSL **56** (1991) pp.1243-1260. ([preprint](http://www.cwru.edu/artsci/phil/AxiomatizingCategoryCategories.pdf))

* L. Oubiña, *Teoría axiomatica de categorías*, Cah. Top. Géom. Diff. Cat., **X** no. 3 (1968) pp.375-394. ([numdam](http://www.numdam.org/numdam-bin/item?id=CTGDC_1968__10_3_375_0))

* J. Sonner, *On the formal definition of categories*, Math. Zeitschr. **80**  (1962) pp.163-176. ([gdz](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002392232))

* H. Tsukuda, *Category theory not based upon set theory*, Sci. Papers
College Gen. Ed. Univ. Tokyo **31** (1981) pp.1-24.

A critical view of purely categorical foundations can be found in a series of [papers](http://math.stanford.edu/~feferman/papers.html) by S. Feferman e.g. :

* [[Solomon Feferman|S. Feferman]], *Categorical Foundations and Foundations of Category Theory*, pp.149-169 in Butts&Hintikka (eds.), _Logic, Foundations of Mathematics, and Computability Theory_, Reidel Dordrecht 1977. ([pdf](http://math.stanford.edu/~feferman/papers/Cat_founds.pdf))

An inconsistency in a recent formalisation by Feferman is discussed in

* M. Ernst, *The Prospects of Unlimited Category Theory: Doing What Remains to be Done*, Review Symbolic Logic **8** no.2 (2015) pp.306-327.

The above remarks from [[Mike Shulman|M. Shulman]] come from the following MO discussion that also offers additional perspectives on ETCC and ET2CC:

* {#MO-CC}mathoverflow, *Category of categories as a foundation for mathematics*, December 2009 . ([link](http://mathoverflow.net/questions/9269/category-of-categories-as-a-foundation-of-mathematics))

See also

* {#Blass84}[[Andreas Blass]], _The interaction of category theory and set theory_ , Cont. Math. **30** (1984) pp.5-29. ([draft](http://www.math.lsa.umich.edu/~ablass/interact.pdf))

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

* [[Colin McLarty|C. McLarty]], [[Andrei Rodin|A. Rodin]], _A Discussion between Colin McLarty and Andrei Rodin about Structuralism and Categorical Foundations of Mathematics_, n.d. ([pdf](http://canoe.ens.fr/~rodin/spip/IMG/pdf/colin.pdf))

* M. R. Pryzbylek, _Logical systems I: Lambda calculi through discreteness_, arxiv:1306.3703v3 (2014). ([pdf](http://arxiv.org/pdf/1306.3703v3))

* {#W71}[[Bob Walters]], _Yoneda 2Categories_ , talk at the University of New South Wales December 1971. ([manuscript](https://dl.dropboxusercontent.com/u/92056191/Archive/temporary_new_material/yoneda.pdf))

[[!redirects elementary theory of the category of categories]]

[[!redirects elementary theory of the 2-category of categories]]


ETCR

## Idea

A model of the **Elementary Theory of the Category of Relations** (ETCR) is the [[dagger 2-poset]] whose [[category of maps]] is a model of [[ETCS]].

## Definition

A model of ETCR is a [[dagger 2-poset]] $C$ such that:

* Singleton: there is an object $\mathbb{1} \in Ob(C)$ such that for every morphism $f \in Hom(\mathbb{1},\mathbb{1})$, $f \leq 1_\mathbb{1}$, and for every object $A \in Ob(C)$ there is an [[onto morphism in a dagger 2-poset|onto morphism]] $u_A \in Hom(A,\mathbb{1})$.

* Tabulations: for every object $A \in Ob(C)$ and $B \in Ob(C)$ and morphism $R \in Hom(A,B)$, there is an object $\vert R \vert \in Ob(C)$ and [[jointly monic]] [[map in a dagger 2-poset|maps]] $f \in Hom(\vert R \vert, A)$, $g \in Hom(\vert R \vert, B)$, such that $R = f^\dagger \circ g$.

* Power sets: for every object $A \in Ob(C)$, there is an object $\mathcal{P}(A)$ and a morphism $\in_A:Hom(A, \mathcal{P}(A))$ such that for each morphism $R \in Hom(A,B)$, there exists a [[map in a dagger 2-poset|map]] $\chi_R \in Hom(A,P(B))$ such that $R = (\in_B^\dagger) \circ \chi_R$.

* Function extensionality: for every object $A \in Ob(C)$ and $B \in Ob(C)$ and [[map in a dagger 2-poset|maps]] $f \in Hom(A, B)$, $g \in Hom(A, B)$ and $x \in Hom(\mathbb{1}, A)$, $f \circ x = g \circ x$ implies $f = g$.

* Natural numbers: there is an object $\mathbb{N} \in Ob(C)$ with maps $0 \in Hom(\mathbb{1},\mathbb{N})$ and $s \in Hom(\mathbb{N},\mathbb{N})$, such that for each object $A$ with maps $0_A \in Hom(\mathbb{1},A)$ and $s_A \in Hom(A,A)$, there is a map $f \in Hom(\mathbb{N},A)$ such that $f \circ 0 = 0_A$ and $f \circ s = s_A \circ f$.

* Choice: for every object $A \in Ob(C)$ and $B \in Ob(C)$, every [[epic map in a dagger 2-poset|epic map]] $R \in Hom(A,B)$ has a section.

## See also

* [[Rel]]
* [[ETCS]]
* [[SEAR]]