idempotent monad

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###Context###
#### Idempotents
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[[!include idempotents - contents]]
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#### Algebra
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[[!include higher algebra - contents]]
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#### 2-Category theory
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[[!include 2-category theory - contents]]
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#### Modalities, Closure and Reflection
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[[!include modalities - contents]]
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# Idempotent monads
* table of contents
{: toc}

## Idea
 {#Idea}

An _idempotent monad_ is a [[monad]] that "squares to itself" in the evident [[category theory|category-theoretic]] sense. Idempotent monads hence serve as [[categorification|categorified]] [[projection]] operators, in that they encode [[reflective subcategories]] and the reflection/[[localization]] onto these.

In terms of [[type theory]] idempotent monads interpret (co-)[[modal operators]]; see [[modal type theory]].

## Definition

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

An __idempotent monad__  is a [[monad]] $(T,\mu,\eta)$ on a [[category]] $\mathcal{C}$ such that one (hence all) of the following equivalent statements are true:

1. $\mu\colon T T \to T$ is a [[natural isomorphism]].

2. All components of $\mu \,\colon\, T T \to T$ are [[monomorphisms]].

3. The [[maps]] $T\eta, \eta T \,\colon\, T \to T T$ are [[equality|equal]].

4. For every $T$-[[algebra over a monad|algebra]] (aka: $T$-module) $(M,u)$, the corresponding $T$-[[action]] $u\colon T M \to M$ is an [[isomorphism]] (i.e. every algebra is a [[fixed point]]).

5. The [[forgetful functor]] $\mathcal{C}^T \to \mathcal{C}$ (where $C^T$ is the [[Eilenberg-Moore category]] of $T$-algebras) is a [[full and faithful functor]].

6. There exists a pair of [[adjoint functors]] $F \dashv U$ such that the induced monad $(T \colon U F, \mu \coloneqq U \epsilon F, \eta)$ is [[isomorphism|isomorphic]] to $(T,\mu, \eta)$ and $U$ is a [[full and faithful functor]].

   {#SixPrime} 6'.

   There is a [[full subcategory]] $U \colon C^T \hookrightarrow C$ and a [[natural bijection]] of the form

   $$
     \array{
     Hom_{\mathcal{C}}\big(
       T(c)
       ,\,
       U(a)
     \big)
     &
     \overset{\sim}{\longrightarrow}
     &
     Hom_{\mathcal{C}}\big(
       c
       ,\,
       U(a)
     \big)
     \\
     \big(
       T(c) \overset{f}{\longrightarrow} U(a)
     \big)
     &\mapsto&
     \big(
       c
         \overset{\eta}{\longrightarrow}
       T(c)
         \overset{f}{\longrightarrow}
       U(a)
     \big)
     }
   $$

   > (This is the form in which [[modal operators]] tend to be axiomatized in [[modal type theory]], e.g. [UP13, Def. 7.7.1](modal+homotopy+type+theory#HoTTBook).)

7. The [[identity]] [[natural transformation]] $T T \Rightarrow T T$ is a [[distributive law]].

=--

e.g. ([Borceux, prop. 4.2.3](#Borceux)). For (7) see Lemma 3.4 of [RW95](#RosebrughWood95).

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

The conditons in Def. \ref{IdempotentMonad} are indeed equivalent.

=--

+-- {: .proof}
###### Proof (in more than one way).

* $1\Rightarrow 2$

  This is trivial.

* $2\Rightarrow 3$

  Compositions $\mu\circ T\eta$ and $\mu\circ\eta T$ are always the identity (unit axioms for the monad), and in particular agree; if $\mu$ has all components monic, this implies $T\eta = \eta T$.

* $3\Rightarrow 4$

  Compatibility of action and unit is $u \circ \eta_M = id_M$, hence also $T(u)\circ T(\eta_M) = id_{T M}$. If $T\eta = \eta T$ then this implies $id_{T M} = T(u)\circ \eta_{T M} = \eta_M\circ u$, where the naturality of $\eta$ is used in the second equality. Therefore we exhibited $\eta_M$ both as a left and a right inverse of $u$.

* $4\Rightarrow 1$

  If every action is iso, then the components of multiplication $\mu_M\colon T T M\to T M$ are isos as a special case, namely of the free action on $T M$.

* $4\Rightarrow 5$

  For any monad $T$, the forgetful functor from Eilenberg-Moore category $C^T$ to $C$ is faithful: a morphism of $T$-algebras is always a morphism of underlying objects in $C$. To show that it is also full, we consider any pair $(M,u)$, $(M',u')$ in $C^T$
and must show that any $f\colon M\to M'$ is actually
a map $f\colon (M,u)\to (M',u')$; i.e. $u'\circ T f = f\circ u$. But we know that $\eta_M, \eta_{M'}$ are inverses of $u,u'$ respectively and the naturality for $\eta$ says $\eta_{M'}\circ f = T f \circ \eta_M$. Compose that equation with $u$ on the right and $u'$ on the left with the result (notice that we used just the invertibility of $u$).

* $5\Rightarrow 6$

  Because the Eilenberg-Moore construction induces the original monad by the standard recipe.

* $6\Rightarrow 3$

  By $6$ the counit $\epsilon$ is iso, hence $U\epsilon F$ has a unique 2-sided inverse; by [[triangle identities]], $T\eta$ and $\eta T$ are both right inverses of $U\epsilon F$, hence 2-sided inverses, hence they are equal.

* $6 \Leftrightarrow 6'$

  This is the special case of the characterization ([here](adjoint+functor#CollectionOfUniversalArrowsEquivalentToAdjointFunctor)) of ([[adjunction units]] of) [[adjoint functors]] $F \dashv U$ as systems of [universal arrows](adjoint+functor#UniversalArrows) when $U$ is [[fully faithful functor|fully faithful]].

* $6\Rightarrow 1$

  If $F\dashv U$ is an adjunction with $U$ fully faithful,
then the counit $\epsilon$ is iso. Since $D(FU X,Y)\simeq C(UX,UY)\simeq D(X,Y)$ where the last equivalence holds since $U$ is full and faithful; hence by essential unicity of the representing object there is an isomorphism $FUX\stackrel{\sim}{\to} X$ ; let $X=Y$ then the adjoint of this identity is the counit of the adjunction; since the hom objects correspond bijectively, the counit is an isomorphism. Hence the multiplication of the induced monad $\mu = U\epsilon F$ is also an iso.

=--

Part 5 means that in such a case $C^T$ is, up to equivalence a full [[reflective subcategory]] of $C$.  Conversely, the monad induced by any reflective subcategory is idempotent, so giving an idempotent monad on $C$ is equivalent to giving a reflective subcategory of $C$.

In the language of [[stuff, structure, property]], an idempotent monad may be said to equip its algebras with _properties only_ (since $C^T\to C$ is fully faithful), unlike an arbitrary monad, which equips its algebras with _at most structure_ (since $C^T\to C$ is, in general, faithful but not full).

If $T$ is idempotent, then it follows in particular that an object of $C$ admits at most one structure of $T$-algebra, that this happens precisely when the  unit $\eta_X\colon X\to T X$ is an isomorphism, and in this case the $T$-algebra structure map is $\eta_X^{-1}\colon T X \to X$.  However, it is possible to have a non-idempotent monad for which any object of $C$ admits at most one structure of $T$-algebra, in which case $T$ can be said to equip objects of $C$ with [[property-like structure]]; an easy example is the monad on [[semigroups]] whose algebras are [[monoids]].

+-- {: .num_remark}
###### Remark
Let us be in a $2$-category $K$. Part of the structure of an idempotent monad $(C,T,\eta,\mu)$ in $K$ is of course an idempotent morphism $T:C\to C$. More precisely (Definition 1.1.9) considers $\mu$ as part of the structure such that an idempotent 1-cell has a 2-isomorphism $\mu:TT\to T$ such that $\mu T=T\mu$. Equivalently an idempotent morphism is a normalized pseudofunctor from the two object monoid $\{*,e\}$ with $e^2=e$ to $K$.

Recall that a *splitting of an idempotent* $(T,\mu)$ consists of a pair of 1-cells $I:D\to C$ and $R:C\to D$ and a pair of 2-isomorphisms $a:R I\to id_D$ and $b:T\to I R$ such that $\mu=b^{-1}(I\circ s\circ R)(b\circ b)$ where $\circ$ denotes horizontal composition of 2-cells. Equivalently an splitting of an idempotent is a limit or a colimit of the defining pseudofunctor. If $K$ has equalizers or coequalizers, then all its idempotents split.

Now if $(I,R,a,b)$ is a splitting of an idempotent monad, then $R\dashv I$ are adjoint. And in this case the splitting of an idempotent is equivalently an Eilenberg-Moore object for the monad $(C,T,\eta,\mu)$. In this case $D$ is called an *adjoint retract of $C$*.

=--

([[Sketches of an Elephant|Johnstone, B 1.1.9, p.248-249]])

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

Equivalences (resp. cores) in an allegory are precisely those symmetric idempotents which are idempotent monads (resp. comonads). In an allegory the following statements are equivalent: all symmetric idempotents split, idempotent monads split, idempotent comonads split. A similar statement holds at least for some 2-categories.

=--

+-- {: .num_remark}
###### Remark
Given an algebra $(X, u:TX\to X)$ , by (1) and (4) the action $u$ yields an isomorphism in $C^T$ between the free algebra $(TX, \mu_X)$ and $(X,u)$ i.e. for an idempotent monad the Eilenberg-Moore and the Kleisli categories coincide.
=--

## Properties
 {#Properties}

### Algebras for an idempotent monad and Localization
 {#AlgebrasForAnIdempotentMonad}

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

Let $(T, \eta, \mu)$ be an idempotent monad on a category $E$. The following conditions on an object $e$ of $E$ are equivalent:

1. The object $e$ carries an $T$-[[algebra for a monad|algebra structure]].

2. The [[unit of a monad|unit]] $\eta e\colon e \to T e$ is a [[split monomorphism]].

3. The [[unit of a monad|unit]] $\eta e$ is an [[isomorphism]].

(It follows from 3. that there is at most one algebra structure on $e$, given by $\xi = (\eta e)^{-1}\colon T e \to e$.)

=--

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

The implication 1. $\Rightarrow$ 2. is immediate. Next, if $\xi\colon T e \to e$ is any retraction of $\eta e$, we have both $\xi \circ \eta e = 1_e$ and
$$\array{
\eta e \circ \xi & = & (T \xi)(\eta T e) & & \text{naturality of}\, \eta \\
 & = & (T \xi)(T \eta e) & & \text{see definitions above} \\
 & = & T(\xi \circ \eta e) & & \text{functoriality} \\
 & = & 1_{T e} & &
}$$
so 2. implies 3. Finally, if $\eta e$ is an isomorphism, put $\xi = (\eta e)^{-1}$. Then $\xi \circ \eta e = 1_e$ (unit condition), and the associativity condition for $\xi$,

$$\xi \circ \mu e = \xi \circ T \xi,$$

follows by inverting the naturality equation $\eta T e \circ \eta e = T \eta e \circ \eta e$. Thus 3. implies 1.

=--

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

This means that the [[Eilenberg-Moore category]] of an [[idempotent monad]] is equivalently the [[reflective subcategory]] (a "[[localization]]" of the ambient category) whose embedding-reflection [[adjunction]] gives the idempotent monad.

=--

See also ([Borceux, volume 2, corollary 4.2.4](#Borceux)).

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

Hence [[duality|dually]] the co-algebras over an idempotent [[comonad]] form a [[coreflective subcategory]], hence a "co-localization" of the ambient category.

=--

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

In [[modal type theory]] one thinks of a (idempotent) (co-)monad as a (co-)[[modal operator]] and of its algebras as (co-)[[modal types]]. In this terminology the above says that categories of (co-)modal types are precisely the (co-)reflective [[localizations]] of the ambient type system.

=--

### The associated idempotent monad of a monad
 {#AssociatedIdemopotentMonad}

We discuss here how under suitable conditions, for every [[monad]] $T$ there is a "completion" to an idempotent monad $\tilde T$ in that the completion construction is [[right adjoint]] to the inclusion of idempotent monads into the category of all monads on a given category, exhibiting the subcategory of idempotent monads as a [[coreflective subcategory]]. Here $\tilde T$ inverts the same morphisms that $T$ does and hence exhibits the [[localization]] ([[reflective subcategory]]) at the $T$-equivalences, and in fact the factorization of any [[adjunction]] inducing $T$ through that localization ([Fakir 70](#Fakir70), [Applegate-Tierney 70](#ApplegateTierney70), [Day 74](#Day74) [Casacuberta-Frei 99](#CasacubertaFrei99). [Lucyshyn-Wright 14](#Lucyshyn-Wright14)).

\begin{theorem}\label{ExistenceOfIdempotentCore}
**([Fakir 70](#Fakir70))**
\linebreak
Let $C$ be a [[complete category|complete]], [[well-powered category]], and let $M\colon C \to C$ be a [[monad]] with [[unit of a monad|unit]] $u\colon 1 \to M$ and multiplication $m\colon M M \to M$. Then there is a universal idempotent monad, giving a [[right adjoint]] to the inclusion

$$IdempotentMonad(C) \hookrightarrow Monad(C)$$

\end{theorem}

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

Given a [[monad]] $M$, define a [[functor]] $M'$ as the [[equalizer]] of $M u$ and $u M$:

$$M' \hookrightarrow M \stackrel{\overset{u M}{\longrightarrow}}{\underset{M u}{\longrightarrow}} M M.$$

This $M'$ acquires a unique monad structure such that $M' \hookrightarrow M$ is a morphism of monads (see this [MathOverflow thread](http://mathoverflow.net/questions/147264/regarding-a-difficulty-in-the-fakir-article-about-associated-idempotent-triple/147272#147272) for some detailed discussion). It might not be an idempotent monad (although it will be if $M$ is [[left exact functor|left exact]]). However we can apply the process again, and continue transfinitely. Define $M_0 = M$, and if $M_\alpha$ has been defined, put $M_{\alpha+1} = M_{\alpha}'$; at limit ordinals $\beta$, define $M_\beta$ to be the inverse limit of the chain

$$\ldots \hookrightarrow M_{\alpha} \hookrightarrow \ldots \hookrightarrow M$$

where $\alpha$ ranges over ordinals less than $\beta$. This defines the monad $M_\alpha$ inductively; below, we let $u_\alpha$ denote the [[unit of a monad|unit]] of this monad.

Since $C$ is [[well-powered category|well-powered]] (i.e., since each object has only a small number of [[subobjects]]), the large limit

$$E(M)(c) = \underset{\alpha \in Ord}{\lim} M_\alpha(c)$$

exists for each $c$. Hence the large limit $E(M) = \underset{\alpha \in Ord}{\lim} M_\alpha$ exists as an endofunctor. The underlying functor

$$Monad(C) \to Endo(C)$$

reflects limits (irrespective of size), so $E = E(M)$ acquires a monad structure defined by the limit. Let $\eta\colon 1 \to E$ be the unit and $\mu\colon E E \to E$ the multiplication of $E$. For each $\alpha$, there is a monad map $\pi_\alpha\colon E \to M_\alpha$ defined by the limit projection.

=--

+-- {: .num_lemma}
###### Lemma
$E$ is idempotent.
=--

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

For this it suffices to check that $\eta E = E \eta\colon E \to E E$. This may be checked objectwise. So fix an object $c$, and for that particular $c$, choose $\alpha$ so large that $\pi_\alpha (c)\colon E(c) \to M_\alpha(c)$ and $\pi_\alpha E(c)\colon E E(c) \to M_{\alpha} E(c)$ are isomorphisms. In particular, $\pi_\alpha \pi_\alpha(c)\colon E E (c) \to M_\alpha M_\alpha(c)$ is invertible.

Now $u_\alpha M_\alpha(c) = M_{\alpha} u_{\alpha} c$, since $\pi_\alpha\colon E \to M_\alpha$ factors through the equalizer $M_{\alpha + 1} \hookrightarrow M_\alpha$. Because $\pi_\alpha$ is a monad morphism, we have

$$\array{
\eta E(c) & = & (\pi_\alpha \pi_\alpha (c))^{-1} (u_\alpha M_\alpha(c))\pi_\alpha(c) \\
       & = & (\pi_\alpha \pi_\alpha (c))^{-1} (M_\alpha u_\alpha(c))\pi_\alpha(c) \\
       & = & E \eta(c)
}$$

as required.

Finally we must check that $M \mapsto E(M)$ satisfies the appropriate universal property. Suppose $T$ is an idempotent monad with unit $v$, and let $\phi\colon T \to M$ be a monad map. We define $T \to M_\alpha$ by induction: given $\phi_\alpha\colon T \to M_\alpha$, we have

$$(u_\alpha M_\alpha)\phi_\alpha = \phi_\alpha \phi_\alpha (v T) = \phi_\alpha \phi_\alpha (T v) = (M_\alpha u_{\alpha})\phi_\alpha$$

so that $\phi_{\alpha}$ factors uniquely through the inclusion $M_{\alpha + 1} \hookrightarrow M_\alpha$. This defines $\phi_{\alpha + 1}\colon T \to M_{\alpha + 1}$; this is a monad map. The definition of $\phi_\alpha$ at limit ordinals, where $M_\alpha$ is a limit monad, is clear. Hence $T \to M$ factors (uniquely) through the inclusion $E(M) \hookrightarrow M$, as was to be shown.
=--

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

For
$
  (L \dashv R)
  \;\colon\;
  \mathcal{C}
  \stackrel{\overset{L}{\longrightarrow}}{\underset{R}{\longleftarrow}}
  \mathcal{D}
$
a pair of [[adjoint functors]] with induced [[monad]] $T = R\circ L$ on the complete and well-powered category $\mathcal{C}$, then the idempotent monad $\tilde T$ of theorem \ref{ExistenceOfIdempotentCore} corresponds via remark \ref{ReflectiveSubcategoryOfIdempotentMonad} to a [[reflective subcategory]] inclusion $\mathcal{C}_T \stackrel{i}{\hookrightarrow} \mathcal{C}$ which factors the original adjunction

$$
  (L\dashv R)
  \;\colon\;
  \mathcal{C}
    \stackrel{\overset{}{\longrightarrow}}{\underset{i}{\longleftarrow}}
  \mathcal{C}_T
    \stackrel{\overset{L'}{\longrightarrow}}{\underset{}{\longleftarrow}}
  \mathcal{D}
$$

such that $L'$ is a [[conservative functor]].

=--

([Lucyshyn-Wright 14, theorem 4.15](#LucyshynWright14))

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

The factorization in theorem \ref{FactorizationOfIdempotentCore} has its analog in [[homotopy theory]] in the concept of [[Bousfield localization of model categories]]: given a [[Quillen adjunction]]

$$
  (L \dashv R)
  \;\colon\;
  \mathcal{C}
    \stackrel{\longrightarrow}{\longleftarrow}
  \mathcal{D}
$$

then (if it exists) the [[Bousfield localization of model categories|Bousfield localized]] [[model category]] structure $\mathcal{C}_W$ obtained from $\mathcal{C}$ by adding the $L$-weak equivalences factors this into two consecutive Quillen adjunctions of the form

$$
  \mathcal{C}
   \stackrel{\overset{id}{\longrightarrow}}{\underset{id}{\longleftarrow}}
  \mathcal{C}_{W}
    \stackrel{\overset{L}{\longrightarrow}}{\underset{R}{\longleftarrow}}
  \mathcal{D}
  \,.
$$

On the [[(∞,1)-categories]] [[presentable (∞,1)-category|presented]] by these model categories this gives a factorization of the [[derived functor|derived]] [[(∞,1)-adjunction]] through [[localization of an (∞,1)-category|localization]] onto a [[reflective sub-(∞,1)-category]] followed by a [[conservative (∞,1)-functor]].

=--

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

Let $A$ be a commutative ring, and let $f\colon A \to B$ be a flat (commutative) $A$-algebra. Then the [[forgetful functor]]

$$f^\ast = Ab^f\colon Ab^B \to Ab^A$$

from $B$-modules to $A$-modules has a [[left exact functor|left exact]] [[left adjoint]] $f_! = B \otimes_A -$. The induced monad $f^\ast f_!$ on the category of $B$-modules preserves [[equalizers]], and so its associated idempotent monad $T$ may be formed by taking the equalizer

$$T(M) \to B \otimes_A M \stackrel{\overset{f^\ast f_! \eta M}{\longrightarrow}}{\underset{\eta f^\ast f_! M}{\longrightarrow}} B \otimes_A B \otimes_A M$$

=--

+-- {: .query}

(To be continued. This example is based on how Joyal and Tierney introduce effective descent for commutative ring homomorphisms, in An Extension of the Galois Theory of Grothendieck. I would like to consult that before going further -- Todd.)

[[Mike Shulman]]: How about some examples of monads and their associated idempotent monads?

Do 2-monads have associated lax-, colax-, or pseudo-idempotent 2-monads?

=--

### Idempotent strong monads are commutative

\begin{theorem}
Let $(\mathcal{C}, \otimes, I)$ be a monoidal category and let $(T, \mu, \eta)$ be an idempotent monad on $\mathcal{C}$ equipped with compatible left and right [[strong monad|strengths]] $t_{A, B} : TA \otimes B \to T(A \otimes B)$ and $t'_{A, B} : A \otimes TB \to T(A \otimes B)$. Then $T$ is a [[commutative monad]].
\end{theorem}

\begin{proof}
The commutativity equation is obtained using the strength axioms, the monad axioms, and the fact that $\eta_T = T \eta$ for an idempotent monad:
\begin{tikzcd}[column sep=2em]
	{TA\otimes TB} && {TA \otimes TB} & {T(TA \otimes B)} & {T^2(A\otimes B)} && {T(A\otimes B)} \\
	& {T^2A\otimes TB} & {T(TA \otimes TB)} & {T^2(TA \otimes B)} & {T^3(A\otimes B)} & {T^2(A \otimes B)} \\
	{TA \otimes TB} & {T(A \otimes TB)} && {T^2(A\otimes B)} & {T^2(A \otimes B)} & {T^2(A \otimes B)} & {T(A \otimes B)}
	\arrow[Rightarrow, no head, from=1-1, to=3-1]
	\arrow[Rightarrow, no head, from=1-1, to=1-3]
	\arrow["t"', from=3-1, to=3-2]
	\arrow["\mu"', from=3-6, to=3-7]
	\arrow["{t'}", from=1-3, to=1-4]
	\arrow["{Tt}", from=1-4, to=1-5]
	\arrow["\mu", from=1-5, to=1-7]
	\arrow[Rightarrow, no head, from=1-7, to=3-7]
	\arrow["{T\eta \otimes TB}"{description}, from=3-1, to=2-2]
	\arrow["{\eta_T \otimes TB}"', from=1-3, to=2-2]
	\arrow["t"', from=2-2, to=2-3]
	\arrow["\eta"{description}, from=1-3, to=2-3]
	\arrow["{Tt'}", from=2-3, to=2-4]
	\arrow["{T\mu}", from=2-5, to=2-6]
	\arrow["\mu"', from=2-6, to=3-7]
	\arrow["{T^2t}", from=2-4, to=2-5]
	\arrow["\mu"{description}, from=2-5, to=3-6]
	\arrow["\eta"', from=1-7, to=2-6]
	\arrow["{T(\eta \otimes TB)}"{description}, from=3-2, to=2-3]
	\arrow["{Tt'}"', from=3-2, to=3-4]
	\arrow["{T^2(\eta \otimes B)}", from=3-4, to=2-4]
	\arrow["{T^2(\eta)}"{description}, from=3-4, to=2-5]
	\arrow[Rightarrow, no head, from=3-4, to=3-5]
	\arrow["{T(\eta_T)}"{description}, from=3-5, to=2-5]
	\arrow[Rightarrow, no head, from=3-5, to=3-6]
\end{tikzcd}
\end{proof}

Alternatively we can deduce this from the fact that that [[thunk-force category|thunkable]] morphisms are [[premonoidal category|central]], since a monad is idempotent iff every Kleisli map is thunkable, and a monad is commutative iff every Kleisli map is central. (This fact is proved by Paul Levy [there](https://www.cs.bham.ac.uk/~pbl/papers/thunkcentral.pdf).)

## Related concepts

* [[idempotent monoid in a monoidal category]]

* [[idempotent adjunction]]

* [[reflective subcategory]]

* [[closure operator]], [[modality]]

* [[lax-idempotent 2-monad]]

* [[idempotent (∞,1)-monad]]

* The analog in [[model category]] theory of the localization at idempotent monad is the content of the [[Bousfield-Friedlander theorem]] ("[[Quillen idempotent monad]]").

## References
 {#References}

* [[Harry Appelgate]], [[Myles Tierney]], Section 6 in: *Categories with models*, p. 156-244 In: [[Beno Eckmann]] (ed.) *Seminar on Triples and Categorical Homology Theory* Lecture Notes in Mathematics, **80**, Springer  1969  ([doi:10.1007/BFb0083086](https://doi.org/10.1007/BFb0083086), [pdf](https://link.springer.com/content/pdf/10.1007/BFb0083086))

Textbook accounts:

* {#Borceux} [[Francis Borceux]], _[[Handbook of Categorical Algebra]]_, vol.2, p. 196.

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_, A.4.3.11, p.194, B1.1.9, p.249

See also:

* [[Pierre Gabriel]], [[Michel Zisman]], _[[Calculus of fractions and homotopy theory]]_

The idempotent monad which exhibits the [[localization]] at the $T$-equivalences for a given monad $T$ is discussed in

* {#ApplegateTierney70} [[Harry Applegate]], [[Myles Tierney]], _Iterated cotriples_, Lecture Notes in Math. **137** (1970) 56-99 ([doi:10.1007/BFb0060440](https://doi.org/10.1007/BFb0060440))

* {#Fakir70} S. Fakir, _Monade idempotente associée à une monade_, C. R. Acad. Sci. Paris Ser. A-B 270 (1970), A99-A101. ([gallica](http://gallica.bnf.fr/ark:/12148/bpt6k480298g/f103.image))

* {#Day74} [[Brian Day]], _On adjoint-functor factorisation_, Lecture Notes in Math. 420 (1974), 1-19.

* {#CasacubertaFrei99} [[Carles Casacuberta]], Armin Frei, _Localizations as idempotent approximations to completions_, Journal of Pure and Applied Algebra 142 (1999), 25-33 ([pdf](http://atlas.mat.ub.es/personals/casac/articles/cfre1.pdf))

and for [[enriched category theory]] in

* {#LucyshynWright14} [[Rory Lucyshyn-Wright]], _Completion, closure, and density relative to a monad, with examples in functional analysis and sheaf theory_, Theory Appl. Cat. 29 (2014) 31, 896-928 [abs](http://emis.ams.org/journals/TAC/volumes/29/31/29-31abs.html) [arXiv:1406.2361](http://arxiv.org/abs/1406.2361)

Extension of idempotent monads along subcategory inclusions is discussed in

* [[Carles Casacuberta]], Armin Frei, Tan Geok Choo, _Extending localization functors_ ,  Journal of Pure and Applied Algebra 103 (1995), 149-165 ([pdf](http://atlas.mat.ub.es/personals/casac/articles/cft.pdf))

* A. Deleanu, A. Frei, [[Peter Hilton|P. Hilton]], _Idempotent triples and completion_, Math. Z. **143** (1975) pp.91-104. ([pdf](http://gdz-lucene.tc.sub.uni-goettingen.de/gcs/gcs?&&action=pdf&metsFile=PPN266833020_0143&divID=LOG_0014&pagesize=original&pdfTitlePage=http://gdz.sub.uni-goettingen.de/dms/load/pdftitle/?metsFile=PPN266833020_0143%7C&targetFileName=PPN266833020_0143_LOG_0014.pdf&))

Discussion of idempotent comonads and their relation to adjoint strings is contained in

* {#RosebrughWood95} [[Robert Rosebrugh]], and [[Richard Wood]] *Distributive adjoint strings.* Theory and applications of categories (1995), pages 119-145.

[[!redirects idempotent monad]]
[[!redirects idempotent monads]]
[[!redirects idempotent comonad]]
[[!redirects idempotent comonads]]


idempotent monoid in a monoidal category

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###Context###
#### Categorical algebra
+-- {: .hide}
[[!include categorical algebra -- contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### Monoid theory
+-- {: .hide}
[[!include monoid theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _idempotent monoid_ is a [[monoid]] which is [[idempotent]] in that it “squares to itself” in the evident [[category theory|category-theoretic]] sense.

## Definition

An **idempotent monoid** $(A,\mu,\eta)$ in a [[monoidal category]] $\mathcal{C}$ is a [[monoid in a monoidal category|monoid in]] $\mathcal{C}$ whose multiplication morphism $\mu\colon A\otimes_{\mathcal{C}}A\to A$ is an [[isomorphism]].

Similarly, an **idempotent semigroup in $\mathcal{C}$** (also called a _non-unital idempotent monoid in $\mathcal{C}$_) is a semigroup $(A,\mu)$ in $\mathcal{C}$ with $\mu$ an isomorphism.

**Note.** This is different from the usual algebraic notion of an idempotent monoid (namely, one in which $a \cdot a = a$). One can make sense of that notion in any [[monoidal category with diagonals]] $\Delta_A : A \to A \otimes_{\mathcal{C}} A$ by requiring that $\mu \circ \Delta_A = \id_A$.

We write $\mathsf{IdemMon}(\mathcal{C})$ for the [[full subcategory]] of $\mathsf{Mon}(\mathcal{C})$ spanned by the idempotent monoids in $\mathcal{C}$.

### Strict idempotency

An idempotent monoid $(A,\mu,\eta)$ or semigroup $(A,\mu)$ is **strict** if $\mu\colon A\otimes_{\mathcal{C}}A\to A$ is not only an [[isomorphism]], but in fact the [[identity morphism]] of $A$.

## Properties

### Preservation by strong monoidal functors

A [[strong monoidal functor]] (resp. [[strict monoidal functor]]) $F\colon\mathcal{C}\to\mathcal{D}$ induces a functor
\[\mathsf{IdemMon}(F)\colon\mathsf{IdemMon}(\mathcal{C})\to\mathsf{IdemMon}(\mathcal{D}),\]
and hence “preserves” idempotent monoids (resp. strict idempotent monoids).

Similarly, strong (strict) semigroupal functors ("non-unital strong monoidal functors") “preserve” (strict) idempotent semigroups.

### As strong monoidal functors from the punctual category

(Strict) idempotent monoids in $\mathcal{C}$ are the same as (strict) strong monoidal functors from the punctual monoidal category $\mathsf{pt}$.

Similarly, (strict) idempotent semigroups in $\mathcal{C}$ may be identified with (strict) strong semigroupal functors functors $(F,F^\otimes)\colon\pt\to\mathcal{C}$.

### As strictly unitary strong monoidal functors from the Boolean monoid

(Strict) idempotent semigroups in $\mathcal{C}$ are also the same as _strictly unitary_ strong monoidal functors (resp. strict monoidal functors) from $\mathbb{B}_\mathsf{disc}$ to $\mathcal{C}$, where $\mathbb{B}=(\{0,1\},\vee,1)$ is the “Boolean monoid”, the initial monoid with an idempotent element.

## Examples

* An idempotent semigroup in $\left(A_{\mathsf{disc}},\cdot_A,1_A\right)$ for $A$ an ordinary [[monoid]] (in [[Set|$\mathsf{Set}$]]) is an [[idempotent]] [[element]] of $A$, i.e. an element $a\in A$ such that $a^2=a$.

* An idempotent semigroup in $\left(\mathrm{End}_{\mathcal{C}}(X)_{\mathsf{disc}},\circ^{\mathcal{C}}_{X,X,X},\mathrm{id}_X\right)$ with $\mathrm{End}_{\mathcal{C}}(X)$ the monoid of [[endomorphisms]] of $\mathcal{C}$ at $X$ is an idempotent morphism $f\colon X\to X$ of $\mathcal{C}$, satisfying $f\circ f=f$.

* An idempotent monoid in [[abelian groups]] $\left(\mathsf{Ab},\otimes_{\mathbb{Z}},\mathbb{Z}\right)$ is a [[solid ring]].

* An idempotent monoid in an [[endomorphism]] [[functor category]] $\left(\mathsf{Fun}(\mathcal{C},\mathcal{C}),\circ,\mathrm{id}_\mathcal{C}\right)$ is an [[idempotent monad]].

* An idempotent monoid in the monoidal category $\mathsf{End}_{\mathcal{C}}(X)$  of endomorphisms of a bicategory $\mathcal{C}$ at $X\in\mathrm{Obj}(\mathcal{C})$ is an idempotent $1$-morphism $f\colon X\to X$ of $\mathcal{C}$, satisfying $f\circ f\simeq f$ up to a coherent $2$-isomorphism.

* An idempotent monoid in [[Spectra]] $\left(\mathsf{Sp},\otimes_{\mathbb{S}},\mathbb{S}\right)$ is a "solid ring spectrum" as in [Gutierrez 2013, Section 4](#Gutierrez13). See also [MO #298435](https://mathoverflow.net/questions/298435).

* An idempotent monoid in the category $\mathsf{Fun}(\mathcal{C},\mathsf{Sets})$ equipped with the [[Day convolution]] monoidal structure is a [[strong monoidal functor]]. Similarly, strict idempotent monoids in $\mathsf{Fun}(\mathcal{C},\mathsf{Sets})$ recover [[strict monoidal functors]].

## Related concepts

* [[monoid]]

* [[commutative monoid in a symmetric monoidal category]]

* [[idempotent monad]]

## References

* Peter Hines, _The categorical theory of self-similarity_, 1999. [[Theory and Applications of Categories]]. ([abstract](http://www.tac.mta.ca/tac/volumes/6/n3/6-03abs.html), [pdf](http://www.tac.mta.ca/tac/volumes/6/n3/n3.pdf), [dvi](http://www.tac.mta.ca/tac/volumes/6/n3/n3.dvi), [ps](http://www.tac.mta.ca/tac/volumes/6/n3/n3.ps).)

* {#Gutierrez13} [[Javier J. Gutiérrez]], *On solid and rigid monoids in monoidal categories*, Applied Categorical Structures 23, no. 2 (2015), 575-589 ([arXiv:1303.5265](https://arxiv.org/abs/1303.5265))


idempotent morphism > history

< [[idempotent morphism]]
[[!redirects idempotent morphism -- history]]


idempotent relation

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#### Relations
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# Contents
* table of contents
{: toc}

## Definition

A [[binary endorelation]] $R$ on a set $A$ is **idempotent** if it is a [[transitive]] [[dense relation]]

* for all $x \in A$, $y \in A$, and $z \in A$, $R(x, y)$ and $R(y, z)$ implies that $R(x, z)$.

* for all $x \in A$ and $y \in A$ such that $R(x, y)$ there exists an element $z \in A$ such that $R(x, z)$ and $R(z, y)$.

Idempotent relations are [[idempotent]] in the sense that in [[Rel]] the composition of the relation $R$ with itself is $R$ again, $R \circ R = R$.

Every [[preorder]] is a reflexive idempotent relation

## Related concepts

* [[dense relation]]

* [[transitive relation]]

* [[DLO]]

* [[idempotent element]]

## References

See also:

* Wikipedia, *[Idempotent relation](https://en.wikipedia.org/wiki/Idempotent_relation)*

[[!redirects idempotent relation]]
[[!redirects idempotent relations]]
[[!redirects idempotent binary relation]]
[[!redirects idempotent binary relations]]
[[!redirects idempotent binary endorelation]]
[[!redirects idempotent binary endorelations]]


idempotent semifield

## Idea

An idempotent semifield is an [[idempotent semiring]] that has a zero and in which every non-zero element is invertible. Or, in other words, a [[skewfield]] in which an element may not have a negative but is always idempotent with respect to addition.

## Definition

An __idempotent semifield__ $K$ is [[idempotent semiring]] that has a zero $0$ and in which for every non-zero element $x$ there is an element $y$ such that $x \cdot y = 1$. It is said to be __commutative__ if the multiplication is [[commutative magma|commutative]].

## Properties as a lattice

As in the case of an [[idempotent semiring#Properties|idempotent semiring]] there is a partial order given by

$$ x \leq y \;\;{:\!\!\Longleftrightarrow}\;\; x + y = y $$

which has a [[join]] given by addition. In case of an idempotent semifield this partial order also has a [[meet]]. To each semifield there is a dual semifield $K^*$ given on the same set of elements as $K$ and with the same zero, identity, and multiplication but with addition given by interchanging join and meet, i.e. the meet becomes the new addition on non-zero elements and the zero element behaves neutral.

## Equations in semifields

A version of the fundamental theorem of algebra can be formulated for semifields:
A semifield is said to be __algebraically closed__ if the equation $x^n = y$ has a solution for all $x\in K$ and $n=1,2,\ldots$.

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

In an algebraically closed commutative idempotent semifield $ K $ with $ 0 \cdot a = 0 $ for all $ a \in K $ the equation
$$
  p_0 + p_1 \cdot x^1 + p_2 \cdot x^2 + \ldots + p_n x^n = y
$$

with $y,p_0, \ldots, p_n \in K$ and $n=1,2,\ldots$ has a solution in $K$ if and only if $y \geq p_0$. If $p_0 = 0$, then the solution is unique and can be expressed in terms of the coefficients with help of [[radical|radicals]], multiplication, inverse and meet (i.e. a polynomial in the radicals of $y, p_0, \ldots, p_n$ over the dual semifield).

=--

## Examples

* The [[max-plus algebra]] $[-\infty, +\infty)$ with addition given by maximum and multiplication given by ordinary addition.

* The two element semifield $ \{ 0, 1 \} $ with $ 0 \leq 1 $
 and $ + = \cdot = \vee $ is commutative and algebraically closed but does not fulfill the assumptions of theorem \ref{fundamentalThm}. Indeed, the equation $ 1 \cdot x = 1 $ does not have a unique solution.

## Related concepts

* [[rig]]

* [[rng]] ([[nonunital ring]])

* [[ring]], [[near-ring]]

* [[tropical geometry]]

* [[tropical semiring]]

* [[idempotent semiring]]

## References

In the reference the assumption that $0 \cdot a = 0$ is not explicit (otherwise the base step in the induction in the proof of Proposition 2 therein doesn't work).

* Shpiz, _Solution of algebraic equations in idempotent semifields_, Communications of the Moscow Mathematical Society (1960), [russian](http://www.mathnet.ru/php/archive.phtml?wshow=paper&jrnid=rm&paperid=331&option_lang=eng) [english](https://iopscience.iop.org/article/10.1070/RM2000v055n05ABEH000331/meta)


idempotent semiring

# Idempotent semirings
* table of contents
{: toc}

## Idea

Recall that a [[semiring]] is a set $R$ equipped with two binary operations, denoted $+$, and $\cdot$ and called _addition_ and _multiplication_, satisfying the ring (or [[rng]]) axioms except that there may or may not be either be a zero nor a negative nor an inverse, for which reason do check.

## Definition

An idempotent semiring (also known as a _dioid_) is one in which addition is [[idempotent]]: $x + x = x$, for all $x\in R$.

## Terminology

The term _dioid_ is sometimes used as an alternative name for idempotent semirings.

From now on we will assume that the semiring, $S$, has a neutral element $\varepsilon$ for + and one $e$ for $\cdot$.  Moreover we assume that for all $s\in S$, $s\cdot \varepsilon =\varepsilon s = \varepsilon$.

## Properties

On an idempotent semiring, $S$, there is a partial order given by

$$ x \leq y : \iff x + y = y. $$

To check transitivity observe that $ x \leq y $ and $ y \leq z$ imply $ z \stackrel{y \leq z}{=} y + z \stackrel{x \leq y}{=} (x + y) + z = x + (y + z) \stackrel{y \leq z}{=} x + z $. Due to idempotence the addition is a [[join]] with respect to this partial order: take a $z$ such that $x\leq z$ and $y\leq z$, then $z = z + z = x + z + y + z = x + y + z$. The partial order is preserved by multiplication.

## Examples

*  Any [[quantale]] is an idempotent semiring, or dioid, under [[join]] and multiplication.

* The set of [[language|languages]] over a given alphabet $A$ forms an idempotent semiring in which $L + L' = L \cup L'$ and multiplication is given by concatenation. In fact this is a quantale $P(A^\ast)$ where the multiplication is the $\mathbf{2}$-[[enriched category|enriched]] [[Day convolution]] product induced from the monoid multiplication of the free monoid $A^\ast$.

*  The [[tropical semiring|tropical algebra]] and the [[max-plus algebra]] are idempotent semirings.

* Given an idempotent semiring $R$ one can form the __weak interval extension__ $I(R)$. This is the idempotent semiring defined on all close [[interval|intervals]] $[x, y] = \{ z \in R \mid x \leq z \leq y \}$ by the operations $[x, y] + [x', y'] = [x + x', y + y']$ and $[x, y] \cdot [x', y'] = [x \cdot x', y \cdot y']$. The multiplicative unit is given by $[1,1]$ and, if $R$ has an additive unit $0$, an additive unit is given by $[0,0]$.

## Related concepts

* [[ring]]
* [[rig]]
* [[idempotent semifield]]
* [[tropical semiring]]
* [[max-plus algebra]]
* [[residuated idempotent semiring]]

[[!redirects idempotent semiring]]
[[!redirects idempotent semirings]]
[[!redirects dioid]]
[[!redirects dioids]]


idempotents - contents

**[[idempotent]]**,

* [[split idempotent]], [[retract]]

**[[completion]]**

* [[idempotent complete category]], [[Cauchy complete category]], [[idempotent complete (∞,1)-category]]

* [[Karoubi envelope]], [[Cauchy completion]]

### Examples {#sidebar_examples}

* [[projector]]

* [[idempotent monad]], [[idempotent adjunction]]


identification

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#Contents#
* table of contents
{:toc}

## Idea

In [[dependent type theory]], and particularly [[homotopy type theory]], an **identification** is a word sometimes used for an inhabitant of an [[identity type]]. Alternatives to the term identification include **[[identity]]**, **[[path]]**, or **[[equality]]**, though those phrases are also used in different contexts.

Thus an identification $p:a=b$ provides a "reason", a "witness", or a "proof" that $a$ and $b$ "are equal", or more precisely a *way in which to identify them*.  The distinguishing feature of homotopy type theory is that in general, there may be more than one way to identify two things, i.e. more than one identification between two given elements.

## See also

* [[function application to identities]]
* [[transport]]

## References ##

* [[Univalent Foundations Project]], [[Homotopy Type Theory – Univalent Foundations of Mathematics]] (2013)

[[!redirects identifications]]


identity

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* [[equality]]

* [[identity element]]

* [[identity morphism]]

* [[identity functor]]

* [[identity type]]

* [[identity of indiscernibles]]

[[!redirects identities]]


identity > history

< [[identity]]
[[!redirects identity -- history]]


identity among the relations

##Idea

We consider a presentation, $\mathcal{P} = (X : R)$, of a group $G$. We thus have a
short exact sequence,
$$1\to N \to F \to G \to 1,$$

where $F = F(X)$, the free group on the set $X$, $R$ is a
subset of $F$ and $N = N(R)$ is the normal closure in $F$ of the set $R$.  The group,
$F$, acts on $N$ by conjugation: ${}^u c = u c u^{-1}$,  $c\in N$, $u \in F$, and the elements of $N$ are words in the conjugates of the elements of $R$:

$$c = {}^{u_1}(r_1^{\varepsilon_1}){}^{u_2}(r_2^{\varepsilon_2})\ldots
{}^{u_n}(r_n^{\varepsilon_n}),$$
often says such elements are _consequences_ of
$R$.  Heuristically, an **identity among the relations** of $\mathcal{P}$ is such an element, $c$, which equals 1.

The problem of what this actually _means_ is analogous to that of working with a relation, since, for example, in the presentation, $( a : a^3)$, of $C_3$, the cyclic group of order 3, if $a$ is thought of as being an element of $C_3$, then $a^3 = 1$! Doesn't that say the presentation is $( a : 1)$?
Why is this then different from the situation with the 'presentation', $( a : a = 1)$?

To get around that mess, the
free group on the generators, $F(X)$, was introduced and, of course, in $F(\{a\})$, $a^3$ is not 1. The relation is thus thought of as being a piece of data that is giving the instruction to rewrite that element to 1.

 An analogous free algebraic device, namely a [[free crossed module]] on the presentation can be introduced  to handle the identities.

Before giving the formal definition we will look at some examples.

##Example (Proper powers)

Suppose $r\in R$, but it is a power of some element $s\in F$, i.e., $r = s^m$.
Of course, $ r s = s r$ and

$${}^s r r^{-1} = 1$$

so ${}^s r . r^{-1}$ is an identity.  In fact, there will be a unique $z\in F$
with $r = z^q$, $q$ maximal with this property.  This $z$ is called the
_root of_ $r$ and if $q \gt 1$, $r$ is called a _proper power_.

##Example (Standard presentation of $S_3$)

Here we take up the example from [[Cayley graph]]:

Consider one of the standard presentations of $S_3$,

$$ (a,b  :  a^3, b^2, (ab)^2).$$

Write $ r = a^3$, $s = b^2$, $t = (ab)^2$.  Here the
presentation leads to $F= F(a,b)$, free of rank 2, and  $N(R) \subset F$, so it must be
free as well, by the [[Nielsen-Schreier theorem]].  Its rank will be 7, given by the Schreier index formula or,
geometrically, it will be the fundamental group of the Cayley quiver, also called the  [[Cayley graph]], of the
presentation.  This group is free on generators corresponding to edges outside
a maximal tree.

The set of normal generators of $N(R)$ has 3 elements; $N(R)$ is free on 7
elements (corresponding to the edges not in the tree), yet it is specified as consisting of products of conjugates of $r$, $s$
and $t$, and there are infinitely many of these conjugates.  Clearly there must be some
_slight_ redundancy, so there must be some identities among the relations!

Note that the Cayley graph is [[planar graph|planar]].

A path around the first triangle, $1 \to a \to a^2$, corresponds to the relation $r$; each other
region corresponds to a conjugate of one of $r$, $s$ or $t$.  (It may help in what follows to think of the graph being embedded on a 2-sphere, so 'outer' and 'outside' mean 'round the back face'.) Consider a loop
around a region.

Pick a path to a start vertex of the loop, a path starting at 1.
For instance the path that leaves 1 and goes along edges labelled $a$, $b$ and then goes
around $a a a$ before returning by $b^{-1} a^{-1}$ gives
$a b r b^{-1} a^{-1}$. Now the path around the outside can be written as
a product of paths around the inner parts of the graph,
e.g. $(a b a b) b^{-1} a^{-1} b^{-1} (b b) (b^{-1}a^{-1} b^{-1} a^{-1}) \ldots $ and so
on, thus $r$ can be written in a non-trivial way as a product of conjugates
of $r$, $s$ and $t$. (An explicit identity constructed like this is given in the paper by Brown and Huebschmann (see below).)

##Example (Free Abelian group $\mathbb{Z}^3$)

In a presentation of the free Abelian group on 3 generators, one would expect
the commutators,  $[x,y]$, $ [x,z]$ and $[y,z]$.  There is a well-known identity,
that expands out to give an identity among these
relations (again see Brown-Huebschman (below), p.154, or Loday, (again below) for a complete treatment.)

In a bit more  detail, look at this as a _presented group_,
$$G=\mathbb{Z}^{3}=\langle x,y,z\mid [y,z],[z,x],[x,y]\rangle$$
then label the relators $r_x= [y,z]$, $r_y=[z,x]$, $r_z= [x,y]$.

 There is an identity:
$$[[x,y],{}^y z].[[y,z],{}^z x].[[z,x],{}^x y] =1$$
 which is easily checked to hold just by expanding it out and cancelling terms. This is one of the (many) forms of the well-known identity known variously as the _Hall-Witt identity_, the _Jacobi-Hall-Witt identity_, the _Jacobi identity_, and various variants of these.
 There is thus a word,

$$[r_z,{}^y z].[r_x,{}^z x].[r_y,{}^x y] $$

which can be written out as a product of conjugates of the three relations, $r_x$, $r_y$ and $r_z$, and which evaluates to 1 when considered as an element of $F(x,y,z)$. It is thus another case of an _identity among the relations_ for this presented group.

Note the [[Cayley quiver]] of this presentation is infinite.

##Definition

We can now formally define the module of identities of a presentation, $\mathcal{P} = (X : R)$.

We first form the [[free crossed module]] on the mapping $R\to F(X)$, which
we will denote by $\partial : C(\mathcal{P}) \to F(X)$.

+-- {: .num_defn #module of identities}
###### Definition
The _module of identities_ , $\kappa(\mathcal{P})$,
of $\mathcal{P}$ is $Ker\, \partial$.
=--

An _identity among the relations_ of $\mathcal{P}$ is then an element of $\kappa(\mathcal{P})$.

By construction, the group presented by $\mathcal{P}$ is $G
\cong F(X)/Im\, \partial$, where $Im \,\partial$ is just the normal closure of
the set, $R$, of relations and we know  that $Ker\, \partial$ is a $G$-module.  In fact we have an exact sequence:

$$0\to \kappa(\mathcal{P}) \to C(\mathcal{P}) \xrightarrow{\partial} F(X)\to G\to 1.$$

##Related entries

* [[Cayley graph]]

* [[homotopical syzygy]]

* [[homological syzygy]]

##References:

* [[R. Brown]] and [[J. Huebschmann]],  _Identities among relations_, in R.Brown and T.L.Thickstun, eds., Low Dimensional Topology, London Math. Soc Lecture Notes, Cambridge University Press, (1982).

* [[M. Kapranov]] and [[M. Saito]], 1999, _Hidden Stasheff polytopes in algebraic K-theory and in the space of Morse functions_, in _Higher homotopy structure in topology and mathematical physics_ (Poughkeepsie, N.Y. 1996), volume 227 of Contemporary Mathematics, 191 – 225, AMS

*  [[J.-L. Loday]], 2000, _Homotopical Syzygies_, in _Une dégustation topologique: Homotopy theory in the Swiss Alps_, volume 265 of Contemporary Mathematics, 99 – 127, AMS.

[[!redirects identities among relations]]
[[!redirects identities among the relations]]


identity anafunctor

The __identity anafunctor__ on a [[category]] $C$ is the [[anafunctor]] $id_C: C \to C$ that, as a [[span]] of [[functor]]s, looks like $C \stackrel{id_C}\leftarrow C \stackrel{id_C}\rightarrow C$, where now $id_C$ is the [[identity functor]] on $C$.  Of course, this is the same as the identity functor on $C$ interpreted as an anafunctor.

The identity anafunctors are the [[identity morphism|identities]] for [[composition]] of anafunctors in [[Cat]].


identity assigning morphism > history

< [[identity assigning morphism]]
[[!redirects identity assigning morphism -- history]]


identity component

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## Definition

If $G$ is a [[topological group]], then the **identity component** is the [[connected space|connected component]] of the [[identity element]] $e$ in $G$.

## Basic results

+-- {: .un_prop}
###### Proposition
The identity component $G_0$ is a [[closed set|closed]] [[normal subgroup]] of $G$.
=--

+-- {: .proof}
###### Proof
It is clearly closed (indeed, any connected component is closed). If $g, h \in G_0$, then $g h$ is in the same connected component as $g$ (since $h$ is in the same connected component as $e$ and left multiplication by $g$ is a [[homeomorphism]]), which in turn is in the same connected component as $e$. Using similar reasoning, if $g$ is in the connected component as $e$, then $e$ is in the same connected component as $g^{-1}$. Hence $G_0$ is a subgroup.

If $\phi$ is any [[automorphism]] of $G$, then $\phi(G_0) = G_0$. (Indeed, $\phi(G_0)$ is a connected set containing $e$ and therefore $\phi(G_0) \subseteq G_0$. Replacing $\phi$ by its [[inverse]] $\phi^{-1}$, we similarly have $\phi^{-1}(G_0) \subseteq G_0$ and therefore $G_0 \subseteq \phi(G_0)$.) Applying this to [[inner automorphism|inner automorphisms]] $\phi$, we conclude that $G_0$ is a normal subgroup of $G$.
=--

* **Remark:** $G_0$ need not be [[open set|open]] in $G$; for example, for the group of $p$-[[p-adic integer|adic integers]], $G_0$ is the (non-open) [[singleton]] $\{e\}$. However, if $G$ is [[locally connected space|locally connected]], for example if $G$ is a [[Lie group]], then $G_0$ is open (and therefore also [[clopen]]. In this case, $G/G_0$ is discrete (because $G \to G/G_0$ is an open map, implying that the identity and therefore every point in $G/G_0$ is open).

+-- {: .un_prop}
###### Proposition
The group $G/G_0$, equipped with the [[quotient space|quotient space topology]], is a [[Hausdorff space|Hausdorff]] [[topological group]].
=--

+-- {: .proof}
###### Proof
Given the fact that $p \colon G \to G/G_0$ is an [[open map|open]] [[surjection]], the [[product]] $p \times p \colon G \times G \to G/G_0 \times G/G_0$ is also an open surjection and therefore a quotient map. It follows easily from the [[universal property]] of quotient maps that the multiplication $G \times G \to G$ therefore descends to a continuous multiplication $G/G_0 \times G/G_0 \to G/G_0$, so that $G/G_0$ is a topological group.

Because a topological group is a [[uniform space]], the Hausdorff condition follows from a weaker [[separation axiom]] such as $T_1$ (points are closed). It suffices that the identity of $G/G_0$ be closed. Its [[complement]] $C$ is the image under $p$ of the complement of $G_0$ in $G$ (just by examining [[coset]] decompositions), which is open. Since $p$ is an open map, it follows that $C$ is open, so that $\{e\}$ is closed, as desired.
=--


identity element

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# Identity elements
* table of contents
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## Definitions

Given an operation ([[function]]) $*\colon X \times Y \to Y$, an element $e$ of $X$ is called a __left identity__ for $*$ if $e * a = a$ for every element $a$ of $Y$.  That is, the map $Y \to Y$ given by $e * -$ is the [[identity function]] on $Y$.

If $*\colon Y \times X \to Y$, then there is a similar concept of __right identity__.

If $*\colon X \times X \to X$, then $e$ is a __two-sided identity__, or simply __identity__, if it is both a left and right identity.

Identity elements are sometimes also called __neutral elements__.

## Related notions

Historically, identity elements (as above) came first, then [[identity functions]], and then [[identity morphisms]].  These are all the same basic idea, however: an identity morphism is an identity element for the operation of [[composition]].

An identity is also called a [[neutral element]] or sometimes a __[[unit]]__ (although that term also has a broader meaning, and an operation that has an identity element is called __unital__ or __unitary__).  In particular, a [[magma]] whose underlying operation $*\colon X \times X \to X$ has an identity is called a __[[unital magma]]__ or a __unitary magma__.

Similarly, a __unit law__ is the statement that a given operation has an identity element.  In [[higher category theory]], we generalise from the [[property]] of uniticity/unitality to the [[structure]] of a [[unitor]].

## Examples

* The identity element in a [[matrix algebra]] is the corresponding [[identity matrix]].

## Related concepts

* [[unit]], [[tensor unit]]

* [[associativity]], [[co-associativity]]

* [[co-unitality]]

* [[unitization]]

* [[equivalence]]

* [[identity type]]

* [[identity morphism]]

* [[identity functor]]

* [[identity natural transformation]]

[[!redirects identity element]]
[[!redirects identity elements]]

[[!redirects neutral element]]
[[!redirects neutral elements]]

[[!redirects unital operation]]
[[!redirects unital operations]]
[[!redirects unitary operation]]
[[!redirects unitary operations]]

[[!redirects unit law]]
[[!redirects unit laws]]
[[!redirects uniticity]]
[[!redirects unitality]]

[[!redirects unital]]

[[!redirects identity operation]]
[[!redirects identity operations]]


identity function

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## Definition

Given a [[set]] $S$, the __identity function__ on $S$ is the [[function]] $id_S\colon S \to S$ that maps any element $x$ of $S$ to itself:
$$ id_S = (x \mapsto x) = \lambda x.\; x ;$$
or equivalently,
$$ id_S(x) = x .$$

Thus, the identity function is the function for which every element $x$ of $S$ is a [[fixed point]].

The identity functions are the [[identity morphisms]] in the category [[Set]] of sets.

More generally, in any [[concrete category]], the identity morphism of each object is given by the identity function on its [[underlying set]].

\begin{definition}
In ([[dependent type theory|dependent]]) [[type theory]], the identity function on a [[type]] $T$ may be declared as
\[
  \label{IdentityFunctionInTypeTheory}
  \mathrm{id}_T
  \;\coloneqq\;
  \big(\lambda(t \colon T).t \big)
  \;\colon\;
  T \to T
  \,.
\]
\end{definition}

## Properties

\begin{definition}
In [[dependent type theory]], the type of identity functions on a type $A$ is the type

$$\sum_{f:A \to A} \prod_{x:A} f(x) =_A x$$

of functions with a witness that every element $x:A$ is a [[fixed point]] of the function.
\end{definition}

In general, an identity function is an element of this type. However, any element of the type of identity functions is unique up to identification:

\begin{theorem}
For all types $A$, the type

$$\sum_{f:A \to A} \prod_{x:A} f(x) =_A x$$

is a proposition.
\end{theorem}

\begin{proof}
Suppose we have $i:A \to A$ with $I:\prod_{x:A} i(x) =_A x$ and $j:A \to A$ with $J:\prod_{x:A} j(x) =_A x$. Then we have a [[homotopy]]

$$\lambda x:A.I(x) \bullet J(x)^{-1}:\prod_{x:A} i(x) =_A j(x)$$

and by [[function extensionality]], an [[identification]]

$$\mathrm{homToId}(i, j, \lambda x:A.I(x) \bullet J(x)^{-1}):i =_{A \to A} j$$

and by dependent function application on this identification, a dependent identification

$$\mathrm{apd}(i, j, \mathrm{homToId}(i, j, \lambda x:A.I(x) \bullet J(x)^{-1}, I, J):I =_{f:A \to A.\prod_{x:A} f(x) =_A j(x)}^{i, j, \lambda x:A.I(x) \bullet J(x)^{-1}} J$$

Then by the extensionality principle for dependent pair types, we have

$$\mathrm{pairIdsToId}(i, j, I, J)(\mathrm{homToId}(i, j, \lambda x:A.I(x) \bullet J(x)^{-1}), \mathrm{apd}(i, j, \mathrm{homToId}(i, j, \lambda x:A.I(x) \bullet J(x)^{-1}, I, J))$$

in the identity type

$$(i, I) =_{\sum_{f:A \to A} \prod_{x:A} f(x) =_A x} (j, J)$$

By induction on dependent pair types, this implies that for all elements
$$F:\sum_{f:A \to A} \prod_{x:A} f(x) =_A x \; \mathrm{and} \; G:\sum_{f:A \to A} \prod_{x:A} f(x) =_A x \; ,$$
$$F =_{\sum_{f:A \to A} \prod_{x:A} f(x) =_A x} G$$

Thus, the type

$$\sum_{f:A \to A} \prod_{x:A} f(x) =_A x$$

is a proposition.
\end{proof}

\begin{theorem}
For all types $A$, the type

$$\sum_{f:A \to A} \prod_{x:A} f(x) =_A x$$

is contractible.
\end{theorem}

\begin{proof}
Since the type

$$\sum_{f:A \to A} \prod_{x:A} f(x) =_A x$$

is a proposition, it suffices to construct an element of the above type. Given a type $A$, by the weakening rules and the generic variable rule in dependent type theory, one gets the identity family of elements $x:A \vdash x:A$. Then by lambda abstraction, one gets a function $\lambda x:A.x:A \to A$, and by the typal computation rule for [[weak function types]] one gets the dependent function $\beta_{A \to A}^{x:A.x}:\prod_{x:A} (\lambda x:A.x)(x) =_A x$. For [[strict function types]], we have $\beta_{A \to A}^{x:A.x} \equiv \mathrm{refl}_A$. Thus, one has the identity function
$$(\lambda x:A.x, \beta_{A \to A}^{x:A.x}):\sum_{f:A \to A} \prod_{x:A} f(x) =_A x$$

Since the type
$$\sum_{f:A \to A} \prod_{x:A} f(x) =_A x$$
is a proposition and has element $(\lambda x:A.x, \beta_{A \to A}^{x:A.x})$, the type is contractible.
\end{proof}

Thus, it makes sense to refer to the canonical identity function $\lambda x:A.x$ as *[[generalized the|the]]* identity function on type $A$.

Now, we show that the identity function is an [[equivalence of types]]:

\begin{theorem}
Any identity function is a (non-coherent) [[involution]].
\end{theorem}

\begin{proof}
Given a function $i:A \to A$ and a dependent function $I:\prod_{x:A} i(x) =_A x$, by function application on $i$, we have the family of identifications
$$x:A \vdash \mathrm{ap}(i, i(x), x, I(x)):i(i(x)) =_A i(x)$$
Thus by concatenation of identifications and lambda abstraction, we have the dependent function
$$\lambda x:A.\mathrm{ap}(i, i(x), x, I(x)) \bullet I(x):\prod_{x:A} i(i(x)) =_A x$$
which indicates that the identity function $i:A \to A$ is a non-coherent involution.
\end{proof}

By definition of [[involution]], it follows that:

\begin{corollary}
The [[inverse function]] of the identity function is the identity function itself.
\end{corollary}

\begin{theorem}
Any identity function is bi-invertible.
\end{theorem}

\begin{proof}
Given a function $i:A \to A$ and a dependent function $I:\prod_{x:A} i(x) =_A x$, we have an element
$$((i, \lambda x:A.\mathrm{ap}(i, i(x), x, I(x)) \bullet I(x)), (i, \lambda x:A.\mathrm{ap}(i, i(x), x, I(x)) \bullet I(x)))$$

of type

$$\left(\sum_{j:A \to A} \prod_{x:A} i(j(x)) =_A x\right) \times \left(\sum_{k:A \to A} \prod_{x:A} k(i(x)) =_A x\right)$$

indicating that $i:A \to A$ is bi-invertible.
\end{proof}

By definition, it follows that:

\begin{corollary}
Any [[identity function]] on a type $A$ is an [[equivalence of types]].
\end{corollary}

Hence, in [[dependent type theory]], the identity function is also called the **identity equivalence**.

There are other ways to prove that the identity function is an equivalence of types. One of them is given below:

\begin{theorem}
Suppose that [[function types]] are defined using judgmental [[computation rules]]. Then for all [[terms]] $x \colon T$, the [[fiber]] of the identity function $\lambda(t \colon T).t \,\colon\, T \to T$ (eq:IdentityFunctionInTypeTheory) at $x$ is a [[contractible type]].
\end{theorem}

\begin{proof}
The [[fiber type]] in question is the [[dependent type]]
$$
  \sum_{y \colon T}
  \big(
    \lambda(t:T).t
  \big)(y)
  =_T x
  \,.
$$
By the judgmental [[computation rule]] of [[function types]], the term $\big(\lambda(t:T).t\big)(x)$ reduces down to $x$, and the fiber type becomes
$$
  \sum_{y \colon T} y
  \;=_T\;
  x
$$
This is always a [[contractible type]]:

* the [[center of contraction]] is given by the [[dependent pair]] $(x, \mathrm{refl}_T(x))$,

* for any $z \colon \sum_{y \colon T} y \,=_T\, x$, there is an [[identification]]

$$
  z \;=_{\sum_{y:T} y =_T x}\;
  \big(
    x, \mathrm{refl}_T(x)
  \big)
  \,.
$$

By [[induction]] on [[dependent sum types]], it suffices to construct an identification
$$
  (w, p)
  \;=_{\sum_{y:T} y =_T x}\;
  \big(
    x, \mathrm{refl}_T(x)
  \big)
$$
for any $w \colon T$ and $p \;\colon\; w =_T x$, and by [[induction]] on [[identity types]], it suffices to construct an identification
$$
  \big(x, \mathrm{refl}_T(x)\big)
  \;=_{\sum_{y:T} y =_T x}\;
  \big(x, \mathrm{refl}_T(x)\big)
$$
but that's just reflexivity on $(x, \mathrm{refl}_T(x))$
$$
  \mathrm{refl}_{\sum_{y:T} y =_T x}
  \big(
    (x, \mathrm{refl}_T(x))
  \big)
  \;\colon\;
  \big(x, \mathrm{refl}_T(x)\big)
  \;=_{\sum_{y:T} y =_T x}\;
  \big(x, \mathrm{refl}_T(x)\big)
  \,.
$$

Thus, for all $x \colon T$, the [[fiber type]] of the identity function $\lambda(t:T).t:T \to T$ at $x$ is [[contractible type|contractible]].
\end{proof}

## Related concepts

* [[neutral element]]

* [[identity morphism]]

* [[identity functor]]

* [[identity natural transformation]]

* [[identity type]]

* [[transport]]

* [[fixed point]]

[[!redirects identity function]]
[[!redirects identity functions]]
[[!redirects identity map]]
[[!redirects identity maps]]

[[!redirects identity equivalence]]
[[!redirects identity equivalences]]

[[!redirects type of identity functions]]
[[!redirects types of identity functions]]
[[!redirects type of identity maps]]
[[!redirects types of identity maps]]

[[!redirects type of identity equivalences]]
[[!redirects types of identity equivalences]]


identity functor

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## Idea

The __identity functor__ on a [[category]] $C$ is the [[functor]] $id_C: C \to C$ that maps each [[object]] and [[morphism]] of $C$ to itself.  The identity functors are the [[identity morphism|identities]] for [[composition]] of functors in [[Cat]].

## Related concepts

* [[identity function]]

* [[inverse functor]]

* [[identity natural transformation]]

* [[Goodwillie derivatives of the identity functor]]

[[!redirects identity functors]]


identity matrix

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## Idea

An _identiy matrix_ is a [[diagonal matrix]] with all diagonal entries equal to the [[identity element]] $1 \in R$ of the [[ground ring]].

The identiy matrices are the [[identity elements]] in their [[matrix algebra]], whence the name.

## Related concepts

* [[permutation matrix]]

* [[unit]]

## References

See also

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

[[!redirects identity matrices]]


identity monad

Given a [[category]] $C$, the __identity monad__ on $C$ is simply the [[identity functor]] on $C$, made into a [[monad]] in a rather trivial way.


identity morphism

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## Idea

The **identity morphism**, or simply **identity**, of an [[object]] $x$ in some [[category]] $C$ is the morphism $1_x: x \to x$, or $\id_x: x \to x$, which acts as a two-sided identity for [[composition]].

Given a [[small category]] $C$ with set of objects $C_0$ and set of morphisms $C_1$, the **identity assigning function** of $C$ is the function $i: C_0 \to C_1$ that maps each object in $C_0$ to its identity morphism in $C_1$.

For the generalisation to an [[internal category]] $C$, see [[identity-assigning morphism]].

In [[Set]], the identity morphisms are the [[identity functions]].

## Related concepts

* [[equality]], [[equivalence]]

* [[identity function]]

* [[identity functor]]

* [[identity natural transformation]]

* [[identity type]]

* [[identity of indiscernibles]]

[[!redirects identity morphisms]]


identity natural transformation

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## Definition

The __identity natural transformation__ on a [[functor]] $F: C \to D$ is the [[natural transformation]] $id_F: F \to F$ that maps each [[object]] $x$ of $C$ to the [[identity morphism]] $id_{F(x)}$ in $D$.

The identity natural transformations are themselves the [[identity morphisms]] for [[vertical composition]] of natural transformations in the [[functor category]] $D^C$ and in the [[2-category]] [[Cat]].

One must be aware that when we say that a natural transformation $\alpha_{A}:F(A) \rightarrow G(A)$ is the identity, it doesn't mean only that $F(A)=G(A)$ for every object $A$ but also that $F(f)=G(f)$ for every morphism $f$ ie. that the two functors $F$ and $G$ are equal.

Not taking care of this can lead to redundant definitions  as for [[permutative categories]].

## Related concepts

* [[equality]], [[equivalence]]

* [[identity function]], [[identity morphism]]

* [[identity functor]]

* [[identity type]]

* [[identity of indiscernibles]]

[[!redirects identity natural transformations]]

[[!redirects identity transformation]]
[[!redirects identity transformations]]


identity of Bol-Moufang type

**Identities of Bol-Moufang type** are 60 fourth-order identities for a binary algebraic operation which include left and right _Bol identities_ and the three _Moufang identities_. The identities follow from associativity but are strictly weaker than associativity; in the case of [[loops]] some of them do imply associativity.

Four general elements $x,y,z,w$ in the fixed order can be bracketed in five ways:
\[
(xy)(zw), x((yz)w), (x(yz))w, ((xy)z)w, x(y(zw)).
\]
If the two of the four elements are the same (hence we have $a,a,b,c$ than we can order them in six ways such that $b$ and $c$ are in fixed order. Bol-Moufang identities are obtained when we take general elements $a,a,b,c$ with $b$ before $c$ (for definiteness) and form a fourth-order monomial in any bracketing and equate it with a monomial with a different bracketing,
for example $(ab)(ac) = a((ba)c)$. We can do this in $\binom{5}{2} \times 6= 60$ ways. Requiring different sets among Bol-Moufang identities in a [[quasigroup]] or in a loop one gets some interesting classes of quasigroups and loops, for example, [[Bol loop]]s, [[Moufang loop]]s and some others.

The notion is introduced in

* F. Fenyves, _Extra loops II. On loops with identities of Bol–Moufang type_, Publ. Math. Debrecen __16__ (1969), pp. 187–192

Important clarifications are introduced in

* [[Kenneth Kunen]], _Quasigroups, loops and associative laws_, J. Algebra __185__ (1) (1996), pp. 194–204 [doi](http://dx.doi.org/10.1006/jabr.1996.0321) (for all  identities of Bol-Moufang type); _Moufang quasigroups_, J. Algebra __183__:1 (1996) 231–234 [doi](http://dx.doi.org/10.1006/jabr.1996.0216) (the question of loop vs. quasigroup in presence of Moufang identities)
* A. Pavlů, A. Vanžurová, _On identities of Bol-Moufang type_,  Bul. Acad. Ştiinţe Repub. Mold. Mat., 2005, no. 3, 88–100 [pdf](http://www.mathnet.ru/php/getFT.phtml?jrnid=basm&paperid=139&what=fullt&option_lang=rus)

A systematic exposition is also in

* Vladimir Volenec, _Grupoidi, kvazigrupe i petlje_ (book in Croatian), Zagreb, Školska knjiga, 1982

category: algebra
[[!redirects identities of Bol-Moufang type]]


identity of indiscernibles

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## Idea

The principle of _identity of indiscernibles_ states that two objects are [[identical]] if they have all the same [[properties]].

This is also known as "Leibniz's Law", in honor of [[Gottfried Leibniz]] (not to be confused with his [[product rule]], also so-called).

\begin{imagefromfile}
    "file_name": "LeibnizOnIndiscernibles-Phil7SecXIX.jpg",
    "float": "right",
    "width": 490,
    "unit": "px",
    "margin": {
        "top": -40,
        "bottom": 20,
        "right": 0,
        "left": 40
    },
    "caption": "G. W. Leibniz around 1700, from [Gerhard 1890, p. 288](#Gerhard1890)"
\end{imagefromfile}

{#LeibnizWroteTheFollowing} Concretely, Leibniz wrote the following,  (in translation from [Lewis 1918, p. 373](#Lewis18) with original Latin terms in parenthesis; see also [Cartwright 1971, p. 119](#Cartwright71) and [Gries & Schneider 1998](#GriesSchneider98)):

> Two terms are the *same* (*eadem*) if one can be substituted for the other without altering the truth of any statement (*[[salva veritate]]*). If we have $A$ and $B$, and $A$ enters into some true proposition, and the substitution of $B$ for $A$ wherever it appears results in a new proposition that is likewise true, and if this can be done for every proposition, then $A$ and $B$ are said to be the same; ...

This sentence of Leibniz in fact closes with a statement of the converse, often known as the "[[principle of substitutivity]]" but which may deserve to be called the *indiscernibility of identicals*:

> ... conversely, if $P$ and $Q$ are the same, they can be substituted for one another.

The paragraph ends with the assertion that

> $A$ and $A$ are, of course, said to be the same

This is what [[Grundlage der gesamten Wissenschaftslehre|Fichte 1794]] later called the "first, absolutely unconditioned principle" and [[Science of Logic|Hegel 1812-]] [called](https://ncatlab.org/nlab/show/Science+of+Logic#875) the "[[first law of thought]]".

From the perspective of [[homotopy type theory]], the "[[first law of thought]]" is the [[term introduction]]-rule for [[identity types]], and "indiscernability of identicals" is the [[transport]]-rule, which together with "identiy of indiscernible" is implied by the [[induction]]-rule of the [[identity type]] (the "[[J-rule]]"). For more on this see [below](#InHomotopyTypeTheory).

In [[set theory]] with [[power sets]], the [[inference rule]] for the identity of indiscernables is given by

$$\frac{\Gamma \vdash A \; \mathrm{set}}{\Gamma, x \in A, y \in A \vdash \left(\forall P \in \mathcal{P}(A).P(x) \iff P(y)\right) \implies x =_A y \; \mathrm{true}}$$

## In dependent type theory
 {#InHomotopyTypeTheory}

### Leibniz's identity of indiscernibles

In [[dependent type theory]], a direct translation of Leibniz's identity of indiscernibles requires a way to quantify over all [[propositions]] and [[predicates]]. The only way this is possible is if the dependent type theory comes with a [[type of all propositions]] and thus [[power sets]], the latter of which are the type of all [[predicates]] on a type. In addition, unlike with [[set theory]], one cannot simply say that there is a dependent function

$$\mathrm{idOfInd}_A:\prod_{x:A} \prod_{y:A} \left(\prod_{P:\mathcal{P}(A)} P(x) \simeq P(y)\right) \to (x =_A y)$$

This is because the [[identity type]] has more than one element, and thus indiscernibles could be identified in more than one way. Instead, the identity of indiscenibles for type $A$ is given by the dependent function

$$\mathrm{idOfInd}_A:\prod_{x:A} \prod_{y:A} (x =_A y) \simeq \left(\prod_{P:\mathcal{P}(A)} P(x) \simeq P(y)\right)$$

More concretely, for all $x:A$ and $y:A$, the [[principle of substitution]] is always true because by induction on reflexivity, one could define a function
$$\mathrm{idToInd}_A(x, y):(x =_A y) \to \left(\prod_{P:\mathcal{P}(A)} P(x) \simeq P(y)\right)$$
The identity of indiscernibles is simply the statement that $\mathrm{idToInd}_A(x, y)$ is an [[equivalence of types]]. This is the same concept that motivates the definition of [[partial orders]], [[Kolmogorov topological spaces]], and [[skeletal categories]] in terms of equivalences between identity types and a [[binary endorelation]].

If we define the local membership relation $x \in_A P$ as $x \in_A P \coloneqq P(x)$, then Leibniz's identity of indiscenibles becomes

$$\mathrm{idOfInd}_A:\prod_{x:A} \prod_{y:A} (x =_A y) \simeq \left(\prod_{P:\mathcal{P}(A)} (x \in_A P) \simeq (y \in_A P)\right)$$

This is similar to but not identical to the [[axiom of extensionality]]: the axiom of extensionality is given by
$$\mathrm{extax}_A:\prod_{P:\mathcal{P}(A)} \prod_{Q:\mathcal{P}(A)} (P =_{\mathcal{P}(A)} Q) \simeq \left(\prod_{x:A} (x \in_A P) \simeq (x \in_A Q)\right)$$

If Leibniz's identity of indiscenibles is true for $A$, then $A$ is a set, because the type
$$\prod_{P:\mathcal{P}(A)} P(x) \simeq P(y)$$
is always a proposition, in the context of [[weak function extensionality]], and thus by the [[equivalence of types]], $x =_A y$ is also a proposition for all $x:A$ and $y:A$, which means that $A$ is a set. Thus, if Leibniz's identity of indiscernibles is true for all types $A$, the [[dependent type theory]] becomes a [[set-level type theory]] with Leibniz's identity of indiscernibles as an [[axiom of set truncation]].

On the other hand, one does not necessarily need Leibniz's identity of indiscernibles to be true for all types $A$, because the type family
$$x:A, y:A \vdash \mathrm{ind}_A(x, y) \coloneqq \prod_{P:\mathcal{P}(A)} P(x) \simeq P(y)$$
is always an [[equivalence relation]], and thus one could construct the [[quotient set]]

$$A/\mathrm{ind}_A$$

as the type

$$A/\mathrm{ind}_A \coloneqq \sum_{Q:\mathcal{P}(A)} \left[\sum_{x:A} \prod_{y:A} Q(x) \simeq \mathrm{ind}_A(x, y)\right]$$

In [[topology|topological]] terms, this means that every type $A$ with its power set $\mathcal{P}(A)$ is an [[R0-space]] and the quotient $A/\mathrm{ind}_A$ is a [[T1-space]].

### Type-theoretic identity of indiscernibles

For a version of the identity of indiscernibles which applies to not just sets, but other non-set types, one would have to go from quantifying over predicates to quantifying over [[type families]]: For this, one has to use a [[type universe]] $U$, upon which the identity of indiscenibles for type $A$ is given by the dependent function

$$\mathrm{idOfInd}_U:\prod_{x:A} \prod_{y:A} (x =_A y) \simeq \left(\prod_{P:A \to U} P(x) \simeq P(y)\right)$$

The type-theoretic identity of indiscernibles is similar to but not identical to the [[univalence axiom]] for [[Tarski universes]]: univalence for Tarski universe $(U, T)$ quantifies over $U$-small types and fixes the type reflector type family $A:U \vdash T(A)$ for Tarski universes, while the identity of indiscernables for type $A$ quantifies over $U$-small type families and fixes the type $A$.

### Other versions

In addition, extensionality principles like [[function extensionality]], [[propositional extensionality]], and [[univalence]] ("typal extensionality") are naturally regarded as a stronger form of identity of indiscernibles.
In particular, the [[consistency]] of [[univalence]] means that in [[Martin-Löf type theory]] without univalence, one cannot define any [[predicate]] that provably distinguishes [[equivalence in homotopy type theory|equivalent]] [[types]]; thus equivalent types are "externally indiscernible", and univalence incarnates that principle internally by making them identical.

"There are several ways to think about the axiom of [[univalence]]. One can see it as a sophisticated updating of Leibniz's principle of the identity of indiscernibles." --[[John Baez]] [nCafé](http://golem.ph.utexas.edu/category/2013/11/categories_for_the_working_phi.html)

On the other hand, the converse substitution/substitutivity principle of "indiscernibility of identicals" is expressd by *[[transport]]* in [[homotopy type theory]].

## In point-set topology
 {#InPointSetTopology}

A [[topology|topological]] or geometrical version of the idea of identity of indiscernibles is [[separation axioms|separation]]: if two points are distinct, then they are separated in some sense. This means in turn that if two points in a space subject to a given separation axiom can not be separated by any admissible separation condition then they are identical.

{#InTopologicalSpacesMoreInDetail} More in detail:

Given a [[topological space]] one might declare that "topological observations" about the space are questions of the form: "Which points are contained in a given open subset?", and hence that two points are "discernible", namely "topologically distinct", if there exists an [[open subset]] that contains one but not the other.
With this convention, the condition that "indiscernibles be identical" is equivalently the condition that the topological space
satisfies the $T_0$-[[separation axiom]], hence that it is a [[Kolmogorov space]]. Generally, one could regard the [$T_0$-reflection](Kolmogorov+topological+space#KolmogorovQuotient) as the [[modality]] under which identity becomes indiscernibility, in this sense.

In [[dependent type theory]], given a [[dominance]] $\Sigma \subseteq \Omega$, let $\mathcal{O}(A)$ denote the type of $\Sigma$-[[open sets]] of the type $A$, the type of functions $A \to \Sigma$. The $T_1$-[[separation axiom]] for type $A$ is given by the dependent function

$$T_0:\prod_{x:A} \prod_{y:A} (x =_A y) \simeq \left(\prod_{P:\mathcal{O}(A)} P(x) \simeq P(y)\right)$$

Leibniz's identity of indiscernibles is simply the $T_1$-separation axiom where the [[type of all propositions]] $\Omega$ is the dominance and where the [[predicates]]/[[subsets]] are the $\Omega$-[[open sets]].

In addition, given an type universe $U$, Leibniz's identity of indiscernibles for $U$ is simply the $T_1$-separation axiom where the type of $U$-small propositions $\mathrm{Prop}_U \coloneqq \sum_{A:U} \mathrm{isProp}(A)$ is the dominance, and where the $U$-small [[predicates]]/[[subsets]] are the $\mathrm{Prop}_U$-[[open sets]]

## Related entries

* [[first law of thought]]

* [[transport]]

* [[axiom of extensionality]]

* [[axiom of set truncation]]

* [[T0-space]], [[T1-space]]

## References

Leibniz's original paragraph (from an unpublished manuscript probably written after 1683) is reproduced in the Latin Original in

* {#Gerhard1890} K. Gerhard (ed.), Section XIX, [p. 228](https://archive.org/details/diephilosophisc00gerhgoog/page/228/mode/2up) in: *Die philosophischen Schriften von Gottfried Wilhelm Leibniz*, Siebenter Band, Weidmannsche Buchhandlung (1890) [[archive.org](https://archive.org/details/diephilosophisc00gerhgoog/page/n7/mode/2up)]

and in English translation in:

* {#Lewis18} [[Clarence I. Lewis]], Appendix (p. 373) of: *A Survey of Symbolic Logic*, University of California (1918) [[[LewisAppendix-LeibnizIndiscernibles.pdf:file]]]

and further discussed in:

* {#Cartwright71} [[Richard Cartwright]], *Identity and Substitutivity*, p. 119-133 of: Milton Munitz (ed.) *Identity and Individuation*, New York University Press (1971) [[[Cartwright-IdentityAndSubstitutivity.pdf:file]]]

* {#GriesSchneider98} David Gries, Fred Schneider, *Formalizations Of Substitution Of Equals For Equals* (1998) [[pdf](https://ecommons.cornell.edu/bitstream/handle/1813/7340/98-1686.pdf?sequence=1&isAllowed=y), [ecommons:1813/7340](https://ecommons.cornell.edu/handle/1813/7340)

Leibniz's original terminology "salva veritate" for the substitution principle is prominently used in:

* [[W. V. O. Quine]], §3 of *Two Dogmas of Empiricism* and *Three grades of modal involvement*, as reprinted in: Roger F. Gibson (ed.) *Quintessence -- Basic readings from the philosophy of W. V. Quine*, The Belknap Press of Harvard University Press (2004) [[ISBN:9780674027558](https://www.hup.harvard.edu/catalog.php?isbn=9780674027558)]

See also:

* Wikipedia, *[Identity of indiscernibles](http://en.wikipedia.org/wiki/Identity_of_indiscernibles)*

* Wikipedia, *[Salva veritate](https://en.wikipedia.org/wiki/Salva_veritate)*

* Standord Encyclopedia of Philosophy, *[The Identity of Indiscernibles](http://plato.stanford.edu/entries/identity-indiscernible/)*

Formalization in [[homotopy type theory]] is in

* [[Univalent Foundations Project]], section 1.12 of _[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]_ (2013)

* {#Favonia17} [[Favonia]], appendix B of _Higher-Dimensional Types in the Mechanization of Homotopy Theory_, PhD thesis 2017, ([pdf](https://www.cs.cmu.edu/~kuenbanh/files/thesis.pdf))

The understanding of "indiscernibility of identicals" as *[[transport]]* in [[homotopy type theory]] is made explicit in:

* {#LadymanPresnell15} [[James Ladyman]], [[Stuart Presnell]], §6.3 of: *Identity in Homotopy Type Theory, Part I: The Justification of Path Induction*, Philosophia Mathematica **23** 3 (2015) 386–406 [[doi:10.1093/philmat/nkv014](https://doi.org/10.1093/philmat/nkv014), [pdf](http://philsci-archive.pitt.edu/11079/1/Identity_in_HTT_public.pdf)]

* [[Benedikt Ahrens]], [[Paige Randall North]], §3.1 in: *Univalent foundations and the equivalence principle*, in: *Reflections on the Foundations of Mathematics*, Synthese Library **407** Springer (2019)  [[arXiv:2202.01892](https://arxiv.org/abs/2202.01892), [doi:10.1007/978-3-030-15655-8](https://doi.org/10.1007/978-3-030-15655-8)]

See also:

* [Ladyman & Presnell (2016)](https://academic.oup.com/philmat/article/doi/10.1093/philmat/nkw023/2593919/Identity-in-Homotopy-Type-Theory-Part-II-The)

[[!redirects identity of indiscernibles]]

[[!redirects Leibniz's identity of indiscernibles]]
[[!redirects type theoretic identity of indiscernibles]]
[[!redirects type-theoretic identity of indiscernibles]]

[[!redirects Leibniz's Law]]
[[!redirects Leibniz's law]]

[[!redirects salva veritate]]


identity polynomial

A [[polynomial]] over a [[ring]] with [[unit]] is the **identity polynomial** if it is an order zero polynomial with value the unit of that ring.


identity system

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#Contents#
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## Idea

In [[dependent type theory]], an **identity system** is a [[correspondence]] on a type $A$

$$a:A, b:A \vdash R(a, b)$$

which behaves like a [[weak identity type]] on $A$.

Identity systems are very useful in characterizing the [[extensionality]] principle of [[types]] such as [[function types]], [[dependent function types]], and [[Tarski universes]].

## Definition

In [[dependent type theory]], an __identity system__ on a [[type]] $A$ is a [[correspondence]] on $A$, $a:A, b:A \vdash R(a, b)$, with a [[dependent function]]
$$a:A \vdash r_0(a):R(a, a)$$
such that for any [[dependent type|family of types]]
$$a:A, b:A, p:R(a, b) \vdash C(a, b, p)$$
there exists a dependent function
$$t:\prod_{c:A} C(c, c, r_0(c)), a:A, b:A, p:R(a, b) \vdash J(t, a, b, p):C(a, b, p)$$
and a dependent function
$$t:\prod_{c:A} C(c, c, r_0(c)), a:A \vdash \beta(t, a):J(t, a, a, r_0(a)) =_{C(a, a, r_0(a))} t(a)$$

## Extensionality principles

* [[product extensionality]] states that the correspondence
$$x:A \times B, y:A \times B \vdash (\pi_1(x) =_A \pi_1(y)) \times (\pi_2(x) =_B \pi_2(y))$$
is an identity system.

* dependent pair extensionality states that the correspondence
$$y:\sum_{x:A} B(x), z:\sum_{x:A} B(x) \vdash \sum_{p:\pi_1(y) =_A \pi_1(z)} \pi_2(y) =_{x:A.B(x)}^p \pi_2(z)$$
is an identity system.

* [[function extensionality]] states that the correspondence
$$f:A \to B, g:A \to B \vdash \prod_{x:A} \mathrm{ev}(f, x) =_B \mathrm{ev}(g, x)$$
is an identity system.

* dependent function extensionality states that the correspondence
$$f:\prod_{x:A} B(x), g:\prod_{x:A} B(x) \vdash \prod_{x:A} \mathrm{ev}(f, x) =_{B(x)} \mathrm{ev}(g, x)$$
is an identity system.

* [[univalence]] or [[universe extensionality]] states that the correspondence
$$A:U, B:U \vdash T(A) \simeq T(B)$$
is an identity system.

## See also ##

* [[identity type]]

* [[fundamental theorem of identity types]]

* [[one-to-one correspondence]]

* [[extensionality]]

## References ##

* [[Univalent Foundations Project]], Def. 5.8.3 in: *[[Homotopy Type Theory -- Univalent Foundations of Mathematics]]* (2013) [[web](http://homotopytypetheory.org/book/), [pdf](http://hottheory.files.wordpress.com/2013/03/hott-online-323-g28e4374.pdf)]

* {#Rijke22} [[Egbert Rijke]], Def. 11.2.1 in: *[[Introduction to Homotopy Type Theory]]*, Cambridge Studies in Advanced Mathematics, Cambridge University Press (2023) [[arXiv:2212.11082](https://arxiv.org/abs/2212.11082)]