injection

#Contents#
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## Definition

A [[function]] $f$ from $A$ to $B$ is __injective__ if $x = y$ whenever $f(x) = f(y)$. Equivalently, a function is injective if all its [[fibers]] are [[subsingletons]]: for all elements $b \in B$ and for all elements $x \in A$ and $y \in A$, if $f(x) = b$ and $f(y) = b$, then $x = y$. An injective function is also called __one-to-one__ or an __injection__; it is the same as a [[monomorphism]] in [[Set|the category of sets]].

A _[[bijection]]_ is a function that is both injective and [[surjection|surjective]].

In [[constructive mathematics]], a [[strongly extensional function]] between sets equipped with tight [[apartness relation|apartness relations]] is called __strongly injective__ if $f(x) \ne f(y)$ whenever $x \ne y$ (which implies that the function is injective).  This is the same as a [[regular monomorphism]] in the category of such sets and strongly extensional functions (while any merely injective function, if strongly extensional, is still a monomorphism).  Some authors use 'one-to-one' for an injective function as defined above and reserve 'injective' for the stronger notion.

## In other categories
Since an element $a$ in a set $A$ in the [[category of sets]] is just a [[global element]] $a:1\rightarrow A$, one could define injections in any category $\mathcal{C}$ with a [[terminal object]] $1$:

+-- {: .num_defn}
###### Definition
A morphism $f:A\rightarrow B$ in $\mathcal{C}$ is an __injection__ or a __one-to-one morphism__ if, given any two global elements $x, y:1\rightarrow A$, $x = y$ if $f \circ x = f \circ y$.
=--

+-- {: .num_remark}
###### Remark
The term __injective morphism__ is already used in category theory in a different context to mean a morphism with a [[right lifting property]].
=--

+-- {: .num_prop}
###### Proposition
In a category $\mathcal{C}$ with a [[terminal object]] $1$, every [[monomorphism]] is an injection.
=--

This follows from the definition of a monomorphism.

+-- {: .num_prop}
###### Proposition
In a category $\mathcal{C}$ with a [[terminal object]] $1$, every global element $e:1\rightarrow A$ is an injection.
=--

+-- {: .proof}
###### Proof
By definition of [[terminal object]] $1$, the unique global element $i:1\rightarrow 1$ is the identity morphism of the terminal object. Thus for every global element $e:1\rightarrow A$, for any two global elements $x, y:1\rightarrow 1$, $x = y$ is always true, making $e:1\rightarrow A$ an injection.
=--

If the category has a [[strict initial object]] $\emptyset$, then every morphism $f:\emptyset\rightarrow B$ is vacuously an injection, since there are no global elements $x:1\rightarrow\emptyset$.

## Related concepts

* [[surjection]]

* [[monomorphism]]

* [[embedding]]

* [[injective-on-objects functor]]

* [[monomorphism in an (infinity,1)-category]]

* [[monotone function]]

* [[equivariant function]]

* [[decidable injection]]

[[!redirects injective function]]
[[!redirects strongly injective function]]
[[!redirects injective functions]]
[[!redirects strongly injective functions]]
[[!redirects injections]]
[[!redirects one-to-one function]]
[[!redirects one-to-one functions]]
[[!redirects injective]]

[[!redirects injective map]]
[[!redirects injective maps]]

[[!redirects inclusion]]
[[!redirects inclusions]]

[[!redirects injectivity]]


injection set

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## Definition

Given sets $A$ and $B$, the **injection set** $\mathrm{Inj}(A, B)$ is the set of [[injection]] between the set $A$ and $B$. In the [[foundations]] of mathematics, the existence of such a set may be taken to follow from the existence of [[power sets]], from the axiom of [[subset collection]], from the existence of [[function sets]], or as an axiom (the __axiom of injection sets__) in its own right.

## Properties

If the set theory has [[function sets]], then the injection set between $A$ and $B$ is a [[subset]] of the function set between $A$ and $B$: $\mathrm{Inj}(A, B) \subseteq B^A$.

The [[universal property]] of the bijection set states that given a function $X \to \mathrm{Inj}(A, B)$ from a set $X$ to the injection set between sets $A$ and $B$, there exists a injection $A \times X \hookrightarrow B \times X$ in the [[slice category]] $\mathrm{Set}/X$.

## Related concepts

* [[embedding type]]

* [[object of monomorphisms]]

* [[function set]], [[bijection set]]

[[!redirects injection set]]
[[!redirects injection sets]]
[[!redirects set of injections]]
[[!redirects sets of injections]]

[[!redirects axiom of injection sets]]


injective envelope

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#Contents#
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## Idea

In a [[concrete category]], an **injective hull** of an object $A$ is an [[extension]] $A \stackrel{m}{\longrightarrow} B$ of $A$ such that $B$ is [[injective object|injective]] and $m$ is an [[essential embedding]]. It is the dual concept to [[projective cover]].

Beware that, in general, there is no way of making the assignment of the injective hull to an object into a [[functor]] such that there is a [[natural transformation]] from the [[identity functor]] to that functor.

## Examples

* In [[Vect]] every object $A$ has an injective hull, $A \stackrel{id_A}{\longrightarrow} A$.  In other words, every vector space is already an [[injective object]].
* In [[Pos]] every object has an injective hull, its [[MacNeille completion]].
* In [[Ab]] every object has an injective hull. The embedding $\mathbb{Z} \hookrightarrow \mathbb{Q}$ is an example.
* In the category [[Field]] of [[fields]] and algebraic field extensions, every object has an injective hull, its [[algebraic closure]].
* In the category of metric spaces and [[short map]]s, the injective hull is a standard construction also known as the [[tight span]] (see [Wikipedia](http://en.wikipedia.org/wiki/Tight_span)).

## Generalization

Given a class $\mathcal{E}$ of objects in a category, an **$\mathcal{E}$-hull**
(or $\mathcal{E}$-envelope) of an object $A$ is a map $h\colon A\longrightarrow E$
such that the following two conditions hold:

1. Any map $k\colon A\longrightarrow E'$ to an object in $\mathcal{E}$ factors
through $h$ via some map $f: E\longrightarrow E'$.

2. Whenever a map $f\colon E\longrightarrow E$ satisfies $f\circ h = h$ then it
must be an automorphism.

On the other hand, given a class $\mathcal{H}$ of *morphisms* in a category, an **$\mathcal{H}$-injective hull** of an object $A$ is a map $h:A\to E$ in $\mathcal{H}$ such that:

1. $E$ is a $\mathcal{H}$-injective object and

2. $h$ is $\mathcal{H}$-essential, i.e. if $k\circ h \in \mathcal{H}$ then $k\in\mathcal{H}$.

## Related concepts

* [[projective object]], [[projective presentation]], [[projective cover]], [[projective resolution]]

  * [[projective module]]

* [[injective object]], [[injective presentation]], **injective envelope**, [[injective resolution]]

  * [[injective module]]

* [[free object]], [[free resolution]]

  * [[free module]]

* flat object, [[flat resolution]]

  * [[flat module]]

## References

Discussion in [[homological algebra]]:

* {#HiltonStammbach71} [[Peter Hilton]],  [[Urs Stammbach]], Section I.9 in: _A course in homological algebra_, Springer-Verlag, New York, 1971, Graduate Texts in Mathematics, Vol. 4 ([doi:10.1007/978-1-4419-8566-8](https://link.springer.com/book/10.1007/978-1-4419-8566-8), [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/hilton-stammbach.pdf))

Discussion in general [[concrete categories]]:

* [[Jiri Adamek|Jiří Adámek]], [[Horst Herrlich]], [[George Strecker]],  Definition 9.16 (page 156) in: [[The Joy of Cats]] (2004)

See also:

* Adámek, Herrlich, Rosický, Tholen; 2002; [Injective hulls are not natural](http://www.iti.cs.tu-bs.de/~adamek/injective.hulls.AHRT.ps)
* Jinzhog Xu; 1996; Flat Covers of Modules, SLNM 1634.
* [[Walter Tholen]], "Essential weak factorization systems".  *Contributions to General Algebra* 13 (2001) 321-333.  [ps](http://www.math.yorku.ca/~tholen/newessential.ps).

On injective hulls of [[partial order|partially ordered]] [[monoids]]:

* [[Joachim Lambek]], [[Michael Barr]], [[John Kennison]], [[Robert Raphael]], *Injective hulls of partially ordered monoids*, Theory Appl. Categories **26** 13 (2012) 338–348 [[tac:16-13](http://www.tac.mta.ca/tac/volumes/26/13/26-13abs.html)]

[[!redirects injective envelopes]]

[[!redirects injective hull]]
[[!redirects injective hulls]]


injective module

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## Definition

For $R$ a [[ring]], let $R$[[Mod]] be the category of $R$-[[modules]].

An **injective module** over $R$ is an [[injective object]] in $R Mod$.

This is the dual notion of a _[[projective module]]_.

## Properties

### Equivalent characterizations

Let $R$ be a [[commutative ring]] and  $C = R Mod$ the category of $R$-[[modules]]. We discuss [[injective modules]] over $R$ (see there for more).

+-- {: .num_theorem #BaerTheorem}
###### Theorem
**([[Baer's criterion]])**

If the [[axiom of choice]] holds,
then a [[module]] $Q \in R Mod$ is an [[injective module]] precisely if for $I$ any left $R$-[[ideal]] regarded as an $R$-module, any [[homomorphism]] $g : I \to Q$ in $C$ can be extended to all of $R$ along the inclusion $I \hookrightarrow R$.

=--

+-- {: .proof}
###### Sketch of proof

Let $i \colon M \hookrightarrow N$ be a [[monomorphism]] in $R Mod$, and let $f \colon M \to Q$ be a map. We must extend $f$ to a map $h \colon N \to Q$. Consider the [[poset]] whose elements are pairs $(M', f')$ where $M'$ is an intermediate [[submodule]] between $M$ and $N$ and $f' \colon M' \to Q$ is an extension of $f$, ordered by $(M', f') \leq (M'', f'')$ if $M''$ contains $M'$ and $f''$ extends $f'$. By an application of [[Zorn's lemma]], this poset has a [[maximal element]], say $(M', f')$. Suppose $M'$ is not all of $N$, and let $x \in N$ be an element not in $M'$; we show that $f'$ extends to a map $M'' = \langle x \rangle + M' \to Q$, a [[contradiction]].

The set $\{r \in R: r x \in M'\}$ is an ideal $I$ of $R$, and we have a module [[homomorphism]] $g \colon I \to Q$ defined by $g(r) = f'(r x)$. By [[hypothesis]], we may extend $g$ to a module map $k \colon R \to Q$. Writing a general element of $M''$ as $r x + y$ where $y \in M'$, it may be shown that

$$f''(r x + y) = k(r) + g(y)$$

is well-defined and extends $f'$, as desired.

=--

+-- {: .num_cor}
###### Corollary
{#DirectSumInjectives}

Assume that the [[axiom of choice]] holds.

Let $R$ be a [[Noetherian ring]], and let $\{Q_j\}_{j \in J}$ be a collection of [[injective modules]] over $R$. Then the [[direct sum]] $Q = \bigoplus_{j \in J} Q_j$ is also injective.

=--

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

By [Baer's criterion](#Baer), it suffices to show that for any [[ideal]] $I$ of $R$, a module [[homomorphism]] $f \colon I \to Q$ extends to a map $R \to Q$. Since $R$ is Noetherian, $I$ is [[finitely generated module|finitely generated]] as an $R$-module, say by elements $x_1, \ldots, x_n$. Let $p_j \colon Q \to Q_j$ be the [[projection]], and put $f_j = p_j \circ f$. Then for each $x_i$, $f_j(x_i)$ is nonzero for only finitely many summands. Taking all of these summands together over all $i$, we see that $f$ factors through

$$\prod_{j \in J'} Q_j = \bigoplus_{j \in J'} Q_j \hookrightarrow Q$$

for some finite $J' \subset J$. But a [[product]] of injectives is injective, hence $f$ extends to a map $R \to \prod_{j \in J'} Q_j$, which completes the proof.

=--

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

Conversely, $R$ is a [[Noetherian ring]] if [[direct sums]] of injective $R$-[[modules]] are injective.

=--

This is due to Bass and Papp. See ([Lam, Theorem 3.46](#Lam)).

### Preservation of injectives

+-- {: .num_lemma #RightAdjointsOfExactFunctorsPreserveInjectives}
###### Lemma

Given a pair of [[additive functor|additive]] [[adjoint functors]]

$$
  (L \dashv R)
  \;\colon\;
  \mathcal{B}
    \stackrel{\overset{L}{\longleftarrow}}{\underset{R}{\longrightarrow}}
  \mathcal{A}
$$

between [[abelian categories]] such that the [[left adjoint]] $L$ is an [[exact functor]], then the [[right adjoint]] preserves injective objects.

=--

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

Observe that an object in an abelian category is injective precisely if the [[hom-functor]] into it sends [[monomorphisms]] to [[epimorphisms]] ([prop.](injective+object#EquivalentCharacterizationOfInjectivesInAbelianCategories)), and that $L$ preserves monomorphisms by assumption of exactness. With this the statement follows via adjunction isomorphism

$$
  Hom_{\mathcal{A}}(-,R(I))\simeq Hom_{\mathcal{B}}(L(-),I)
  \,.
$$

=--

### Existence of enough injectives
 {#ExistenceOfEnoughInjectives}

We discuss that in the presence of the [[axiom of choice]] at least, the category $R$[[Mod]] has _[[injective object|enough injectives]]_ in that every module is a [[submodule]] of an injective one. We first consider this for $R = \mathbb{Z}$. We do assume prop. \ref{InjectiveAbelianGroupIsDivisibleGroup}, which may be proven using [[Baer's criterion]].

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

Assuming the [[axiom of choice]],
the [[category]] $\mathbb{Z}$[[Mod]] $\simeq$ [[Ab]] has [[injective object|enough injectives]].

=--

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

By prop. \ref{InjectiveAbelianGroupIsDivisibleGroup} an [[abelian group]]
is an injective $\mathbb{Z}$-module precisely if it is a [[divisible group]]. So we need to show that every [[abelian group]] is a [[subgroup]] of a [[divisible group]].

To start with, notice that the group $\mathbb{Q}$ of [[rational numbers]] is divisible and hence the canonical embedding $\mathbb{Z} \hookrightarrow \mathbb{Q}$ shows that the additive group of [[integers]] embeds into an injective $\mathbb{Z}$-module.

Now by the discussion at _[[projective module]]_ every [[abelian group]] $A$ receives an [[epimorphism]] $(\oplus_{s \in S} \mathbb{Z}) \to A$ from a [[free group|free]] abelian group, hence is the [[quotient group]] of a [[direct sum]] of copies of $\mathbb{Z}$. Accordingly it embeds into a quotient $\tilde A$ of a direct sum of copies of $\mathbb{Q}$.

$$
  \array{
    ker &\stackrel{=}{\to}& ker
    \\
    \downarrow && \downarrow
    \\
    (\oplus_{s \in S} \mathbb{Z}) &\hookrightarrow& (\oplus_{s \in S} \mathbb{Q})
    \\
    \downarrow && \downarrow
    \\
    A &\hookrightarrow& \tilde A
  }
$$

Here $\tilde A$ is divisible because the [[direct sum]] of divisible groups is again divisible, and also the [[quotient group]] of a divisible groups is again divisble. So this exhibits an embedding of any $A$ into a divisible abelian group, hence into an injective $\mathbb{Z}$-module.

=--

+-- {: .num_lemma #TransferOfEnoughInjectivesAlongAdjunctions}
###### Lemma

Given a pair of [[additive functor|additive]] [[adjoint functors]]

$$
  (L \dashv R)
  \;\colon\;
  \mathcal{B}
    \stackrel{\overset{L}{\longleftarrow}}{\underset{R}{\longrightarrow}}
  \mathcal{A}
$$

between [[abelian categories]] such that the [[left adjoint]] $L$ is

1. an [[exact functor]],

1. a [[faithful functor]].

Then if $\mathcal{B}$ has enough injectives, also $\mathcal{A}$ has enough injectives.

=--

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

Consider $A \in \mathcal{A}$. By the assumption that $\mathcal{B}$ has enough injectives, there is an injective object $I \in \mathcal{B}$ and a monomorphism $i \colon L(A) \hookrightarrow I$. The [[adjunct]] of this is a morphism

$$
  \tilde i \colon A \longrightarrow R(I)
$$

and so it is sufficient to show that

1. $R(I)$ is injective in $\mathcal{A}$;

1. $\tilde i$ is a monomorphism.

The first point is the statement of lemma \ref{RightAdjointsOfExactFunctorsPreserveInjectives}.

For the second point, consider the [[kernel]] of $\tilde i$ as part of the [[exact sequence]]

$$
  ker(\tilde i)\longrightarrow A \stackrel{\tilde i}{\longrightarrow} R(I)
  \,.
$$

By the assumption that $L$ is an [[exact functor]], the image of this sequence under $L$ is still exact

$$
  L(ker(\tilde i)) \longrightarrow L(A) \stackrel{L(\tilde i)}{\longrightarrow} L(R(I))
  \,.
$$

Now observe that $L(\tilde i)$ is a monomorphism: this is because its composite $L(A) \stackrel{L(\tilde i)}{\longrightarrow} L(R(I)) \stackrel{\epsilon}{\longrightarrow} I$ with the [[adjunction unit]] is (by the formula for [[adjuncts]]) the original morphism $i$, which by construction is a monomorphism. Therefore the exactness of the above sequence means that $L(ker(\tilde i)) \to L(A)$ is the [[zero morphism]]; and by the assumption that $L$ is a [[faithful functor]] this means that already $ker(\tilde i) \to A$ is zero, hence that $ker(\tilde i) = 0$, hence that $\tilde i$ is a monomorphism.

=--

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

As soon as the category [[Ab]] of [[abelian groups]] has enough injectives, so does the [[abelian category]] $R$[[Mod]] of [[modules]] over some [[ring]] $R$.

In particular if the [[axiom of choice]] holds, then $R Mod$ has enough injectives.

=--

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

Observe that the [[forgetful functor]] $U\colon R Mod \to AbGp$ has both a [[left adjoint]] $R_!$ ([[extension of scalars]] from $\mathbb{Z}$ to $R$) and a right adjoint $R_*$ ([[coextension of scalars]]).  Since it has a left adjoint, it is [[exact functor|exact]]. Thus the statement follows via lemma \ref{TransferOfEnoughInjectivesAlongAdjunctions} from prop. \ref{AbHasEnoughInjectives}.

=--

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

For $R = k$ a [[field]], hence $R$[[Mod]] = $k$[[Vect]], every object is both injective as well as [[projective object|projective]].

=--

## Examples

### Injective $\mathbb{Z}$-modules / abelian groups

Let $C = \mathbb{Z} Mod \simeq $ [[Ab]] be the abelian category of [[abelian groups]].

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

An [[abelian group]] $A$ is injective as a $\mathbb{Z}$-module precisely if it is a [[divisible group]], in that for all [[integers]] $n \in \mathbb{N}$ we have $n G = G$.

=--

Using Baer's criterion, prop. \ref{BaerTheorem}.

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

By prop. \ref{InjectiveAbelianGroupIsDivisibleGroup} the following [[abelian groups]] are injective in [[Ab]].

The group of [[rational numbers]] $\mathbb{Q}$ is injective in [[Ab]], as is the additive group of [[real numbers]] $\mathbb{R}$ and generally that underlying any [[field]]. The additive group underlying any [[vector space]] is injective. The [[quotient]] of any injective group by any other group is injective.

=--

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

**Not** injective in [[Ab]] is for instance the [[cyclic group]] $\mathbb{Z}/n\mathbb{Z}$ for $n \gt 1$.

=--

## Related concepts

* [[projective object]], [[projective presentation]], [[projective cover]], [[projective resolution]]

  * [[projective module]]

* [[injective object]], [[injective presentation]], [[injective envelope]], [[injective resolution]]

  * **injective module**

* [[free object]], [[free resolution]]

  * [[free module]]

* flat object, [[flat resolution]]

  * [[flat module]]

## References

The notion of injective module was introduced in

* {#Baer}R. Baer, _Abelian groups that are direct summands of every containing abelian group_ , Bulletin AMS **46** no. 10 (1940) pp.800-806. ([projecteuclid](http://projecteuclid.org/euclid.bams/1183503234))

(The dual notion of [[projective modules]] was considered explicitly only much later.)

A general discussion can be found in

* [[Masaki Kashiwara]], [[Pierre Schapira]], _[[Categories and Sheaves]]_

The general notion of [[injective objects]] is in section 9.5, the case of injective complexes in section 14.1.

[Baer's criterion](#Baer) is discussed in many texts, for example

* N. Jacobsen, _Basic Algebra II_, W.H. Freeman and Company, 1980.

See also

* T.-Y. Lam, _Lectures on modules and rings_, Graduate Texts in Mathematics 189, Springer Verlag (1999).
{#Lam}

Section 4.2 of

* [[Peter May]], _Notes on Tor and Ext_ ([pdf](http://www.math.uchicago.edu/~may/MISC/TorExt.pdf))
 {#May}

For [[abelian sheaves]] over the [[etale site]]:

* [[James Milne]], section 8 of _[[Lectures on Étale Cohomology]]_

A study of injective modules in higher algebra:

* [[Liran Shaul]], Injective DG-modules over non-positive DG-rings [arxiv](https://arxiv.org/abs/1709.01479)

[[!redirects injective modules]]


injective morphism in a dagger 2-poset

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## Contents ##
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## Idea ##

A morphism similar to how injective relations are defined in [[Rel]]

## Definition ##

A morphism $f:hom_A(a,b)$ of a [[dagger 2-poset]] $A$ is an **injective morphism** if $f^\dagger \circ f \leq 1_a$.

## See also ##

* [[onto morphism in a dagger 2-poset]]
* [[functional morphism in a dagger 2-poset]]
* [[entire morphism in a dagger 2-poset]]


injective object

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## Definition

There is a very general notion of _injective objects_ in a [[category]] $C$, and then there is a sequence of more concrete notions as $C$ is equipped with more [[stuff, structure, property|structure and property]], in particular for $C$ an [[abelian category]] [[additive and abelian categories|or a relative]] thereof.

The concept of [[resolutions]] by injective objects -- [[injective resolutions]] -- is crucial notably in the discussion of [[derived functors]] (in the context of [[abelian categories]]: [[derived functors in homological algebra]]).

Being injective is a *property* of an object; the corresponding *structure* is called an [[algebraic injective]].

### General definition

Let $C$ be a [[category]] and $J \subset Mor(C)$ a [[class]] of [[morphisms]] in $C$.

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

Frequently $J$ is the class of all [[monomorphisms]] or a related class.

This is notably the case for  $C$ is a [[category of chain complexes]] equipped with the injective [[model structure on chain complexes]] and $J$ is its class of [[cofibrations]].

=--

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

An object $I$ in $C$ is **$J$-injective** if all diagrams of the form

$$
  \array{
    X &\to& I
    \\
    {}^{\mathllap{j \in J}}\downarrow
    \\
    Z
  }
$$

admit an [[extension]]

$$
  \array{
    X &\to& I
    \\
    {}^{\mathllap{j \in J}}\downarrow & \nearrow_{\mathrlap{\exists}}
    \\
    Z
  }
  \,.
$$

If $J$ is the class of all [[monomorphisms]], we speak merely of an **injective object**.

=--

\begin{remark}\label{InjectiveObjectsAsInjectiveMorphisms}
If $C$ has a [[terminal object]] $\ast$, then the $J$-injective objects $I$ according to def. \ref{InjectiveObjects} are those for which the unique morphism $I \stackrel{\exists!}{\to} \ast$ is a $J$-[[injective morphism]].
\end{remark}

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

Ones says that a category $C$ has **enough injectives** if every object admits a [[monomorphism]] into an injective object.

=--

(The [[formal duality|dual]] notion is that of a *[[projective object]]*.)

Assuming the [[axiom of choice]], we have the following easy result:

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

Any [[small set|small]] [[Cartesian product]] of injective objects is injective.

=--

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

If $C$ has a [[terminal object]] $*$ then these extensions are equivalently [[lifts]]

$$
  \array{
    X &\longrightarrow& I
    \\
    \mathllap{{}^{j \in J}}
    \big\downarrow
    &
    \nearrow_{\mathrlap{\exists}}
    &
    \big\downarrow
    \\
    Z
    &\longrightarrow&
    *
  }
$$

and hence the $J$-injective objects are precisely those that have the [[right lifting property]] against the class $J$.

=--

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

If $C$ is a [[locally small category]] then $I$ is $J$-injective precisely if the [[hom-functor]]

$$
  Hom_C(-,I)
  \,\colon\,
  C^{op} \longrightarrow Set
$$

takes morphisms in $J$ to [[epimorphism]]s in [[Set]].

=--

### In abelian categories

The term _injective object_ is used most frequently in the context that $C$ is an [[abelian category]].

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

For $C$ an [[abelian category]] the class $J$ of monomorphisms is the same as the class of morphisms $f : A \to B$ such that $0 \to A \stackrel{f}{\to} B$ is [[exact sequence|exact]].

=--

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

By definition of [[abelian category]] every monomorphism $A \hookrightarrow B$ is a [[kernel]], hence a [[pullback]] of the form

$$
  \array{
    A &\to& 0
    \\
    \downarrow && \downarrow
    \\
    B &\to& C
  }
$$

for $0$ the (algebraic) [[zero object]]. By the pasting law for pullbacks we find that also the left square in

$$
  \array{
    0 &\to& A &\to& 0
    \\
    \downarrow && \downarrow && \downarrow
    \\
    0 &\to& B &\to& C
  }
$$

is a pullback, hence $0 \to A \to B$ is exact.

=--

+-- {: .num_cor #EquivalentCharacterizationOfInjectivesInAbelianCategories}
###### Corollary

An [[object]] $I$ of an abelian category $C$ is then **injective** if it satisfies the following equivalent conditions:

* the [[hom-functor]] $Hom_C(-, I) : C^{op} \to \mathscr{Ab}$ is [[exact functor|exact]] where $\mathscr{Ab}$ is the category of abelian groups;

* for all [[morphism]]s $f : X \to Y$ such that $0 \to X \to Y$ is [[exact sequence|exact]] and for all $k : X \to I$, there exists $h : Y \to I$ such that $ h\circ f = k$.

$$
  \array{
    0 &\to& X &\stackrel{f}{\to}& Y
    \\
    && \downarrow^{\mathrlap{k}} & \swarrow_{\mathrlap{\exists h}}
    \\
    &&
    I
  }
  \,.
$$

=--

By the [[formal dual]] of [this prop.](projective+module#NProjectiveIFFHomNExact).

### In chain complexes

See [[homotopically injective object]] for a relevant generalization to categories of chain complexes, and its relationship to ordinary injectivity.

## Examples

### Injective modules
 {#InjectiveModules}

Let $R$ be a [[commutative ring]] and  $C = R Mod$ the category of $R$-[[modules]]. We discuss [[injective modules]] over $R$ (see there for more).

The following criterion says that for identifying [[injective modules]] it is sufficient to test the [[right lifting property]] which characterizes injective objects by def. \ref{InjectiveObjects}, only on those [[monomorphisms]] which include an [[ideal]] into the base ring $R$.

+-- {: .num_theorem #BaerTheorem}
###### Proposition
**([[Baer's criterion]])**

If the [[axiom of choice]] holds,
then a [[module]] $Q \in R Mod$ is an [[injective module]] precisely if for $I$ any left $R$-[[ideal]] regarded as an $R$-module, any [[homomorphism]] $g : I \to Q$ in $C$ can be extended to all of $R$ along the inclusion $I \hookrightarrow R$.

=--

This is due to ([Baer](#Baer)).

+-- {: .proof}
###### Sketch of proof

Let $i \colon M \hookrightarrow N$ be a [[monomorphism]] in $R Mod$, and let $f \colon M \to Q$ be a map. We must extend $f$ to a map $h \colon N \to Q$. Consider the [[poset]] whose elements are pairs $(M', f')$ where $M'$ is an intermediate [[submodule]] between $M$ and $N$ and $f' \colon M' \to Q$ is an extension of $f$, ordered by $(M', f') \leq (M'', f'')$ if $M''$ contains $M'$ and $f''$ extends $f'$. By an application of [[Zorn's lemma]], this poset has a [[maximal element]], say $(M', f')$. Suppose $M'$ is not all of $N$, and let $x \in N$ be an element not in $M'$; we show that $f'$ extends to a map $M'' = \langle x \rangle + M' \to Q$, a [[contradiction]].

The set $\{r \in R: r x \in M'\}$ is an ideal $I$ of $R$, and we have a module [[homomorphism]] $g \colon I \to Q$ defined by $g(r) = f'(r x)$. By [[hypothesis]], we may extend $g$ to a module map $k \colon R \to Q$. Writing a general element of $M''$ as $r x + y$ where $y \in M'$, it may be shown that

$$f''(r x + y) = k(r) + f'(y)$$

is well-defined and extends $f'$, as desired.
=--

+-- {: .num_corollary #DirectSumInjectives}
###### Corollary
(Assume that the [[axiom of choice]] holds.) Let $R$ be a [[Noetherian ring]], and let $\{Q_j\}_{j \in J}$ be a collection of [[injective modules]] over $R$. Then the [[direct sum]] $Q = \bigoplus_{j \in J} Q_j$ is also injective.
=--

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

By Baer's criterion, theorem \ref{BaerTheorem}, it suffices to show that for any [[ideal]] $I$ of $R$, a module [[homomorphism]] $f \colon I \to Q$ extends to a map $R \to Q$. Since $R$ is Noetherian, $I$ is [[finitely generated module|finitely generated]] as an $R$-module, say by elements $x_1, \ldots, x_n$. Let $p_j \colon Q \to Q_j$ be the [[projection]], and put $f_j = p_j \circ f$. Then for each $x_i$, $f_j(x_i)$ is nonzero for only finitely many summands. Taking all of these summands together over all $i$, we see that $f$ factors through

$$\prod_{j \in J'} Q_j = \bigoplus_{j \in J'} Q_j \hookrightarrow Q$$

for some finite $J' \subset J$. But a [[product]] of injectives is injective, hence $f$ extends to a map $R \to \prod_{j \in J'} Q_j$, which completes the proof.

=--

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

Conversely, $R$ is a [[Noetherian ring]] if [[direct sums]] of injective $R$-[[modules]] are injective.

=--

This is due to Bass and Papp. See ([Lam, Theorem 3.46](#Lam)).

### Injective abelian groups

Let $C = \mathbb{Z} Mod \simeq $ [[Ab]]  be the [[abelian category]] of [[abelian groups]].

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

If the [[axiom of choice]] holds,
then an [[abelian group]] $A$ is an injective object in [[Ab]] precisely if it is a [[divisible group]], in that for all [[integers]] $n \in \mathbb{N}$ we have $n G = G$.

=--

This follows for instance using [[Baer's criterion]], prop. \ref{BaerTheorem}.

An explicit proof is spelled out at
[Planet math -- abelian group is divisible if and only if it is an injective object](https://planetmath.org/abeliangroupisdivisibleifandonlyifitisaninjectiveobject)

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

By prop. \ref{InjectiveAbelianGroupIsDivisibleGroup} the following [[abelian groups]] are injective in [[Ab]].

The group of [[rational numbers]] $\mathbb{Q}$ is injective in [[Ab]], as is the additive group of [[real numbers]] $\mathbb{R}$ and generally that underlying any [[field]] of characteristic zero. The additive group underlying any [[vector space]] is injective. The [[quotient]] of any injective group by any other group is injective.

=--

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

**Not** injective in [[Ab]] is for instance the [[cyclic group]] $\mathbb{Z}/n\mathbb{Z}$ for $n \gt 1$.

=--

### In toposes

In any [[topos]], the [[subobject classifier]] $\Omega$ is an injective object, as is any [[powering|power]] of $\Omega$ ([[Mac Lane-Moerdijk]], IV.10).

Also one can define various notions of *internally* injective objects. These turn out to be equivalent:

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

In any [[elementary topos]] $\mathcal{E}$ with a [[natural numbers object]], the following statements about an object $I \in \mathcal{E}$ are equivalent.

1. The functor $[-, I] : \mathcal{E}^op \to \mathcal{E}$ maps monomorphisms in $\mathcal{E}$ to epimorphisms.
2. The functor $[-, I] : \mathcal{E}^op \to \mathcal{E}$ maps monomorphisms in $\mathcal{E}$ to morphisms for which any global element of the target locally (after [[change of base]] along an epimorphism) possesses a preimage.
3. For any morphism $p : A \to 1$ in $\mathcal{E}$, the object $p^*I$ has property 1. as an object of $\mathcal{E}/A$.
4. For any morphism $p : A \to 1$ in $\mathcal{E}$, the object $p^*I$ has property 2. as an object of $\mathcal{E}/A$.
5. The interpretation of the statement "$I$ is an injective object" using the [[stack semantics]] of $\mathcal{E}$ holds.

=--

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

The implications "1. $\Rightarrow$ 2.", "3 $\Rightarrow$ 4.", "3. $\Rightarrow$ 1.", and "4. $\Rightarrow$ 2." are trivial.

The equivalence "3. $\Leftrightarrow$ 5." follows directly from the interpretation rules of the stack semantics.

The implication "2. $\Rightarrow$ 4." employs the [[base change#GeometricMorphism|extra left-adjoint]] $p_! : \mathcal{E}/A \to \mathcal{E}$ to $p^* : \mathcal{E} \to \mathcal{E}/A$, as in the usual proof that injective sheaves remain injective when restricted to smaller open subsets: We have that $p_* \circ [-, p^*I]_{\mathcal{E}/A} \cong [-, I]_{\mathcal{E}} \circ p_!$, the functor $p_!$ preserves monomorphisms, and one can check that $p_*$ reflects the property that global elements locally possess preimages. Details are in ([Harting, Theorem 1.1](#Harting)).

The implication "4. $\Rightarrow$ 3." follows by performing an extra change of base, since any non-global element becomes a global element after a suitable change of base.

=--

Somewhat surprisingly, and in stark contrast with the situation for [[internally projective objects]], internal injectivity coincides with external injecticity.

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

Let $\mathcal{E}$ be the topos of [[sheaves]] over a [[locale]]. Then an object $I \in \mathcal{E}$ is internally injective (in any of the senses given by Proposition \ref{EquivalenceOfInternalNotionsOfInjectivity}) if and only if $I$ is injective as in Definition \ref{InjectiveObjects}.

=--

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

Let $I$ be an externally injective object. Then $I$ satisfies condition 2. of Proposition \ref{EquivalenceOfInternalNotionsOfInjectivity}, even without having to pass to a cover.

Conversely, let $I$ be an internally injective object. Let $m : X \to Y$ be a monomorphism and let $k : X \to I$ be an arbitrary morphism. We want to show that there exists an extension $Y \to I$ of $k$ along $m$. To this end, consider the sheaf
$$ F \coloneqq \{ k' : \mathcal{H}om(Y,I) | k' \circ m = k \}. $$
One can check that $F$ is [[flabby sheaf|flabby]] (this is particularly easy using the [[internal language]], details will be added later) and therefore has a global section.

=--

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

The analogs of Proposition \ref{EquivalenceOfInternalNotionsOfInjectivity} and Proposition \ref{EquivalenceOfInternalAndExternalInjectivity} for abelian group objects instead of unstructured objects hold as well, with mostly the same proofs. Condition 1. then refers to the functor $[-, X] : Ab(\mathcal{E})^op \to Ab(\mathcal{E})$.

=--
### In topological spaces

In the category [[Top]] of all topological spaces, the injective objects are precisely the [[inhabited set|inhabited]] [[indiscrete spaces]].

In the category of $T_0$ spaces (see [[separation axiom]]), the injective objects are the terminal spaces.

In the above two cases, this refers to injectivity with respect to monomorphisms.

In the category of $T_0$ spaces, the injective objects with respect to homeomorphic embeddings are precisely those given by [[Scott topologies]] on [[continuous lattices]]; as [[locales]] these are [[local compactum|locally compact]] and [[spatial locale|spatial]]. (Such spaces are usually called, perhaps confusingly, *injective spaces*.)

In the category of all spaces, the injectives with respect to homeomorphic embeddings (i.e., [[regular monomorphisms]]) are the spaces whose $T_0$ reflections are continuous lattices under the Scott topology.

### In Boolean algebras

Injective objects in the category of [[Boolean algebras]] are precisely [[complete Boolean algebras]]. This is the dual form of a theorem of Gleason, saying that the [[projective objects]] in the category of [[Stone spaces]] are the [[extremally disconnected topological space|extremally disconnected]] ones (the closure of every open set is again open).

## Properties

### Preservation of injective objects

+-- {: .num_lemma #RightAdjointsOfExactFunctorsPreserveInjectives}
###### Lemma

Given a pair of [[additive functor|additive]] [[adjoint functors]]

$$
  (L \dashv R)
  \;\colon\;
  \mathcal{B}
    \stackrel{\overset{L}{\longleftarrow}}{\underset{R}{\longrightarrow}}
  \mathcal{A}
$$

between [[abelian categories]] such that the [[left adjoint]] $L$ is a [[left exact functor]] (thus automatically exact), then the [[right adjoint]] preserves injective objects.

=--

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

Observe that an object is injective precisely if the [[hom-functor]] into it sends [[monomorphisms]] to [[epimorphisms]], and that $L$ preserves monomorphisms by assumption of (left-)exactness. With this the statement follows via adjunction isomorphism

$$
  Hom_{\mathcal{A}}(-,R(I))\simeq Hom_{\mathcal{B}}(L(-),I)
  \,.
$$

=--

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

Additivity of the left adjoint follows from the remaining assumptions, since [[additive functor#SufficientConditions|exact functors preserve biproducts]].

=--

The preceding lemma has the following variant:

+-- {: .num_prop #Adjuncts_Injectives}
###### Proposition
Let $\mathcal{C}$, $\mathcal{D}$ be categories and $L\dashv R:\mathcal{D}\to\mathcal{C}$ be an adjunction. If $L$ maps monos to monos, then $R$ maps injectives to injectives.
=--

+-- {: .proof}
###### Proof
Let $I\in\mathcal{D}$ be injective. Consider the following diagram in $\mathcal{C}$:

$$
\array{
A & \overset{m}{\rightarrowtail} & B
\\
f\downarrow &    &
\\
R(I) &   &
}
$$

Let $\theta: Hom_\mathcal{C}(X,R(Y))\overset{\simeq}{\rightarrow}Hom_\mathcal{D}(L(X),Y)$ the natural bijection given by the adjunction. Consider now the following diagram in $\mathcal{D}$ where the assumptions ensure that $L(m)$ is mono:

$$
\array{
L(A) & \overset{L(m)}{\rightarrowtail} & L(B)
\\
\theta(f)\downarrow &    &
\\
I &   &
}
$$
Since $I$ is injective, there there exists a filler $\theta(g):L(B)\to I$ which by the adjunction must come from a uniquely determined $g:B\to R(I)$. But the naturality of the bijection with respect to composition says that

\[
\frac{L(A)\overset{L(m)}{\rightarrowtail} L(B)\overset{\theta(g)}{\rightarrow}I}{A\overset{m}{\rightarrowtail} B\overset{g}{\rightarrow}R(I)}
\]

correspond to each other under the bijection whence $\theta(g\circ m)=\theta(g)\circ L(m)$ but from the commutativity of the second diagram we have $\theta(g)\circ L(m)=\theta(f)=\theta(g\circ m)$. Since $\theta$ is a bijection it follows that $f=g\circ m$ which proves that $R(I)$ is injective.

=--

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

The proof transposes the proof of the dual statement 10.2. in  ([Hilton-Stammbach 1971](#HiltonStammbach71), p. 82): In situation $L\dashv R$, if $R$ maps epis to epis then $L$ maps projectives to projectives.

=--

+-- {: .num_remark #Exponential_injectives}
###### Remark
Let $X\in\mathcal{C}$ such that ${}_-\times:\mathcal{C}\to\mathcal{C}$ exists and has a right adjoint ${}_-\times X\dashv (_-)^X$. Since it is easy to check that ${}_-\times X$ preserves monos it follows that $(_-)^X$ preserves injectives.

In particular, for toposes this implies that all power objects $\Omega^X$ are injective since the injectivity of $\Omega$ follows more or less straightforwardly from its classifying properties.
=--

### Existence of enough injectives
 {#ExistenceOfEnoughInjectives}

We discuss a list of classes of categories that have _enough injectives_
according to def. \ref{EnoughInjectives}.

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

Every [[topos]] has enough injectives.

=--

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

Every [[power object]] can be shown to be injective (cf. the above [remark](#Exponentials_injectives)), and every object embeds into its power object by the "singletons" map.

=--

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

Assuming some form of the [[axiom of choice]], the category of [[abelian groups]] has enough injectives.

=--

Full AC is much more than required, however; [[small violations of choice]] suffices.

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

By prop. \ref{InjectiveAbelianGroupIsDivisibleGroup} an [[abelian group]]
is an injective $\mathbb{Z}$-module precisely if it is a [[divisible group]]. So we need to show that every [[abelian group]] is a [[subgroup]] of a [[divisible group]].

To start with, notice that the group $\mathbb{Q}$ of [[rational numbers]] is divisible and hence the canonical embedding $\mathbb{Z} \hookrightarrow \mathbb{Q}$ shows that the additive group of [[integers]] embeds into an injective $\mathbb{Z}$-module.

Now by the discussion at _[[projective module]]_ every [[abelian group]] $A$ receives an [[epimorphism]] $(\oplus_{s \in S} \mathbb{Z}) \to A$ from a [[free group|free]] abelian group, hence is the [[quotient group]] of a [[direct sum]] of copies of $\mathbb{Z}$. Accordingly it embeds into a quotient $\tilde A$ of a direct sum of copies of $\mathbb{Q}$.

$$
  \array{
    ker &\stackrel{=}{\to}& ker
    \\
    \downarrow && \downarrow
    \\
    (\oplus_{s \in S} \mathbb{Z}) &\hookrightarrow& (\oplus_{s \in S} \mathbb{Q})
    \\
    \downarrow && \downarrow
    \\
    A &\hookrightarrow& \tilde A
  }
$$

Here $\tilde A$ is divisible because the [[direct sum]] of divisible groups is again divisible, and also the [[quotient group]] of a divisible groups is again divisble. So this exhibits an embedding of any $A$ into a divisible abelian group, hence into an injective $\mathbb{Z}$-module.

=--

+-- {: .num_lemma #TransferOfEnoughInjectivesAlongAdjunctions}
###### Lemma

Given a pair of [[additive functor|additive]] [[adjoint functors]]

$$
  (L \dashv R)
  \;\colon\;
  \mathcal{B}
    \stackrel{\overset{L}{\longleftarrow}}{\underset{R}{\longrightarrow}}
  \mathcal{A}
$$

between [[abelian categories]] such that the [[left adjoint]] $L$ is

1. an [[exact functor]],

1. a [[faithful functor]].

Then if $\mathcal{B}$ has enough injectives, also $\mathcal{A}$ has enough injectives.

=--

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

Consider $A \in \mathcal{A}$. By the assumption that $\mathcal{B}$ has enough injectives, there is an injective object $I \in \mathcal{B}$ and a monomorphism $i \colon L(A) \hookrightarrow I$. The [[adjunct]] of this is a morphism

$$
  \tilde i \colon A \longrightarrow R(I)
$$

and so it is sufficient to show that

1. $R(I)$ is injective in $\mathcal{A}$;

1. $\tilde i$ is a monomorphism.

The first point is the statement of lemma \ref{RightAdjointsOfExactFunctorsPreserveInjectives}.

For the second point, consider the [[kernel]] of $\tilde i$ as part of the [[exact sequence]]

$$
  ker(\tilde i)\longrightarrow A \stackrel{\tilde i}{\longrightarrow} R(I)
  \,.
$$

By the assumption that $L$ is an [[exact functor]], the image of this sequence under $L$ is still exact

$$
  L(ker(\tilde i)) \longrightarrow L(A) \stackrel{L(\tilde i)}{\longrightarrow} L(R(I))
  \,.
$$

Now observe that $L(\tilde i)$ is a monomorphism: this is because its composite $L(A) \stackrel{L(\tilde i)}{\longrightarrow} L(R(I)) \stackrel{\epsilon}{\longrightarrow} I$ with the [[adjunction unit]] is (by the formula for [[adjuncts]]) the original morphism $i$, which by construction is a monomorphism. Therefore the exactness of the above sequence means that $L(ker(\tilde i)) \to L(A)$ is the [[zero morphism]]; and by the assumption that $L$ is a [[faithful functor]] this means that already $ker(\tilde i) \to A$ is zero, hence that $ker(\tilde i) = 0$, hence that $\tilde i$ is a monomorphism.

=--

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

As soon as the category [[Ab]] of [[abelian groups]] has enough injectives, so does the [[abelian category]] $R$[[Mod]] of [[modules]] over some [[ring]] $R$.

In particular if the [[axiom of choice]] holds, then $R Mod$ has enough injectives.

=--

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

Observe that the [[forgetful functor]] $U\colon R Mod \to AbGp$ has both a [[left adjoint]] $R_!$ ([[extension of scalars]] from $\mathbb{Z}$ to $R$) and a right adjoint $R_*$ ([[coextension of scalars]]).  Since it has a left adjoint, it is [[exact functor|exact]]. Thus the statement follows via lemma \ref{TransferOfEnoughInjectivesAlongAdjunctions} from prop. \ref{AbHasEnoughInjectives}.

=--

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

For $R = k$ a [[field]], hence $R$[[Mod]] = $k$[[Vect]], every object is both injective as well as [[projective object|projective]].

=--

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

The category of [[abelian sheaves]] $Ab(Sh(C))$ on any [[small site]] $C$, hence the category of abelian groups in the [[sheaf topos]] over $C$, has enough injectives.

=--

A proof of can be found in [[Peter Johnstone]]'s book *Topos Theory*, p261.

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

This is in stark contrast to the situation for [[projective objects]], which generally do not exist in categories of sheaves.

=--

+-- {: .num_cor}
###### Corollary

The category of sheaves of modules over any [[sheaf of rings]] on any [[small site]] also enough injectives.

=--

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

Combining prop. \ref{AbelianSheavesHaveEnoughProjectives} with prop. \ref{AbHasEnoughInjectives} (which relativizes to any topos).

=--

This slick proof of this important fact was pointed out by [[Colin McLarty]] in an email to the categories list dated 10 Oct 2010.

### Injective resolutions
 {#InjectiveResolutions}

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

Let $\mathcal{A}$ be an [[abelian category]]. Then for every object
$X \in \mathcal{A}$ there is an [[injective resolution]], hence
a [[chain complex]]

$$
  J^\bullet = [J^0 \to \cdots \to J^n \to \cdots] \in Ch_(\mathcal{A})
$$

equipped with a a [[quasi-isomorphism]] of [[cochain complexes]]  $X \stackrel{\sim}{\to} J^\bullet$

$$
  \array{
    X &\to& 0 &\to& \cdots &\to& 0 &\to& \cdots
    \\
    \downarrow && \downarrow && && \downarrow
    \\
    J^0 &\to& J^1 &\to& \cdots &\to& J^n &\to& \cdots
  }
  \,.
$$

=--

## Related concepts

* [[projective object]], [[projective presentation]], [[projective cover]], [[projective resolution]]

  * [[projective module]]

* **injective object**, [[injective presentation]], [[injective envelope]], [[injective resolution]]

  * [[injective module]]

* [[algebraically injective object]]

* [[free object]], [[free resolution]]

  * [[free module]]

* flat object, [[flat resolution]]

  * [[flat module]]

* [[orthogonality]]

## References

The notion of [[injective modules]] was introduced in

* {#Baer}R. Baer, _Abelian groups that are direct summands of every containing abelian group_ , Bulletin AMS **46** no. 10 (1940) pp.800-806. ([projecteuclid](http://projecteuclid.org/euclid.bams/1183503234))

The dual notion of [[projective modules]] was considered explicitly only much later in

* {#CartanEilenberg56} [[Henri Cartan]], [[Samuel Eilenberg]], *[[Homological Algebra]]*, Princeton Univ. Press (1956), Princeton Mathematical Series **19** (1999) [[ISBN:9780691049915](https://press.princeton.edu/books/paperback/9780691049915/homological-algebra-pms-19-volume-19), [doi:10.1515/9781400883844](https://doi.org/10.1515/9781400883844), [pdf](http://www.math.stonybrook.edu/~mmovshev/BOOKS/homologicalalgeb033541mbp.pdf)]

Textbook accounts:

* {#HiltonStammbach71} [[Peter Hilton]],  [[Urs Stammbach]], p. 30 in: _A course in homological algebra_, Springer-Verlag, New York, 1971, Graduate Texts in Mathematics, Vol. 4 ([doi:10.1007/978-1-4419-8566-8](https://link.springer.com/book/10.1007/978-1-4419-8566-8), [pdf](https://web.math.rochester.edu/people/faculty/doug/otherpapers/hilton-stammbach.pdf))

(used, e.g., for the proof of [Lemma](#Adjuncts_Injectives));

* [[Masaki Kashiwara]], [[Pierre Schapira]], sections 9.5, 14.1 of:_[[Categories and Sheaves]]_

Using tools from the theory of [[accessible categories]], injective objects are discussed in

* [[Jiri Rosicky]], _Injectivity and accessible categories_ ([ps](http://www.math.muni.cz/~rosicky/papers/acc5.ps))

[Baer's criterion](#Baer) is discussed in many texts, for example

* N. Jacobsen, _Basic Algebra II_, W.H. Freeman and Company, 1980.

See also

*
{#Lam} T.-Y. Lam, _Lectures on modules and rings_, Graduate Texts in Mathematics 189, Springer Verlag (1999).

For injective objects in a [[topos]] see

* [[Francis Borceux]], _Handbook of Categorical Algebra vol. 3_ , Cambridge UP 1994. (section 5.6, pp.314-315)

* [[Saunders Mac Lane]], [[Ieke Moerdijk]], _Sheaves in geometry and Logic_ , Springer Heidelberg 1994. (section IV.10, pp.210-213)

* D. Higgs, _Injectivity in the topos of complete Heyting algebra valued sets_ , Can. J. Math. **36** (1984) pp.550-568. ([pdf](http://cms.math.ca/openaccess/cjm/v36/cjm1984v36.0550-0568.pdf))

* [[Peter Johnstone]], [[Fred Linton]], [[Robert Paré]], _Injective Objects in Topoi II: Connections with the axiom of choice_ , pp.207-216 in LNM **719** Springer Heidelberg 1979.

* T. Kenney, _Injective Power Objects and the Axiom of Choice_ ,  JPAA **215** (2011) pp.131–144.

* [[Fred Linton]], _Injective Objects in Topoi III: Stability under coproducts_ , Bull. Acad. Polon. Sci. Sér. Sci. Math. Astronom. Phys. **29** (1981) pp.341-347.

* [[Fred Linton]], [[Robert Paré]], _Injective Objects in Topoi I: Representing coalgebras as algebras_ , pp.196-206 in LNM **719** Springer Heidelberg 1979.

Discussion of injective objects ([[types]]) in [[homotopy type theory]]:

* {#Escardo19} [[Martín Escardó]], _Injectives types in univalent mathematics_ ([arXiv:1903.01211](https://arxiv.org/abs/1903.01211))

For a detailed discussion of internal notions of injectivity see

* {#Harting} Roswitha Harting, _Locally injective abelian groups in a topos_, Communications in Algebra 11 (4), 1983.

For injective toposes in the 2-category of [[bounded topos|bounded toposes]] see

* [[Peter Johnstone]], _Injective Toposes_ , pp.284-297 in LNM **871** Springer Heidelberg 1981.

* [[Peter Johnstone]], _Sketches of an Elephant vol. 2_ , Cambridge UP 2002. (section C4.3, pp.738-745)

[[!redirects injective objects]]

[[!redirects enough injectives]]

[[!redirects injective type]]
[[!redirects injective types]]


injective or projective morphism

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Factorization systems
+--{: .hide}
[[!include factorization systems - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Injective and projective morphisms are [[morphisms]] in a [[category]] satisfying some [[lifting property]]. Often they appear jointly to form [[weak factorization systems]].

## Definition

### Injective and projective

Given a [[category]] $\mathcal{C}$ and a [[class]] $J \subset Mor(\mathcal{C})$ of its [[morphisms]], then any morphism $f \in Mor(\mathcal{C})$ is called

* a **$J$-injective morphism** if it has the [[right lifting property]] against elements in $J$;

* a **$J$-projective morphism** if it has the [[left lifting property]] against elements in $J$.

If $\mathcal{C}$ has a [[terminal object]] $\ast$, then an object $X$ for which $X \stackrel{\exists!}{\to} \ast$ is a $J$-injective morphism is called a $J$-[[injective object]].

If $\mathcal{C}$ has an [[initial object]] $\emptyset$, then an object $X$ for which $\emptyset \stackrel{\exists!}{\to} X$ is a $J$-projective morphism is called a $J$-[[projective object]].

### Fibration and cofibration

Frequently one furthermore says:

* a **$J$-[[cofibration]]** is a morphism that has the [[left lifting property]] against all $J$-injective morphisms;

* a **$J$-[[fibration]]** is a morphism that has the [[right lifting property]] against all $J$-projective morphisms.

## Properties

### Closure properties
 {#ClosureProperties}

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

Let $\mathcal{C}$ be a [[category]] and let $K\subset Mor(\mathcal{C})$ be a [[class]] of [[morphisms]]. Write $K Proj$ and $K Inj$, respectively, for the sub-classes of $K$-[[projective morphisms]] and of $K$-[[injective morphisms]] (the "[[Quillen negation]]" of $K$). Then:

1. Both classes contain the class of [[isomorphism]] of $\mathcal{C}$.

1. Both classes are closed under [[composition]] in $\mathcal{C}$.

   $K Proj$ is also closed under [[transfinite composition]].

1. Both classes are closed under forming [[retracts]] in the [[arrow category]] $\mathcal{C}^{\Delta[1]}$.

1. $K Proj$ is closed under forming [[pushouts]] of morphisms in $\mathcal{C}$ ("[[cobase change]]").

   $K Inj$ is closed under forming [[pullback]] of morphisms in $\mathcal{C}$ ("[[base change]]").

1. $K Proj$ is closed under forming [[coproducts]] in $\mathcal{C}^{\Delta[1]}$.

   $K Inj$ is closed under forming [[products]] in $\mathcal{C}^{\Delta[1]}$.

=--

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

We go through each item in turn.

**containing isomorphisms**

Given a [[commuting square]]

$$
  \array{
    A &\overset{f}{\rightarrow}& X
    \\
    {}_{\mathllap{\in Iso}}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{p}}
    \\
    B &\underset{g}{\longrightarrow}& Y
  }
$$

with the left morphism an isomorphism, the a [[lift]] is given by using the [[inverse]] of this isomorphism ${}^{{f \circ i^{-1}}}\nearrow$. Hence in particular there is a lift when $p \in K$ and so $i \in K Proj$. The other case is [[formal dual|formally dual]].

**closure under composition**

Given a [[commuting square]] of the form

$$
  \array{
    A &\longrightarrow& X
    \\
    \downarrow && \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in K Inj}}
    \\
    {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in K Inj}}
    \\
    B &\longrightarrow& Y
  }
$$

consider its [[pasting]] decomposition as

$$
  \array{
    A &\longrightarrow& X
    \\
    \downarrow
    &\searrow& \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in K Inj}}
    \\
    {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in K Inj}}
    \\
    B &\longrightarrow& Y
  }
  \,.
$$

Now the bottom commuting square has a lift, by assumption. This yields another [[pasting]] decomposition

$$
  \array{
    A &\longrightarrow& X
    \\
    {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow
    && \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in K Inj}}
    \\
    \downarrow &\nearrow& \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in K Inj}}
    \\
    B &\longrightarrow& Y
  }
$$

and now the top commuting square has a lift by assumption. This is now equivalently a lift in the total diagram, showing that $p_1\circ p_1$ has the right lifting property against $K$ and is hence in $K Inj$. The case of composing two morphisms in $K Proj$  is [[formal dual|formally dual]]. From this the closure of $K Proj$ under [[transfinite composition]] follows since the latter is given by [[colimits]] of sequential composition and successive lifts against the underlying sequence as above constitutes a [[cocone]], whence the extension of the lift to the colimit follows by its [[universal property]].

**closure under retracts**

Let $j$ be the [[retract]] of an $i \in K Proj$, i.e. let there be a [[commuting diagram]] of the form.

$$
  \array{
    id_A \colon & A &\longrightarrow& C &\longrightarrow& A
    \\
    & \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{\mathrlap{\in K Proj}} && \downarrow^{\mathrlap{j}}
    \\
    id_B \colon & B &\longrightarrow& D &\longrightarrow& B
  }
  \,.
$$

Then for

$$
  \array{
     A &\longrightarrow& X
     \\
     {}^{\mathllap{j}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}}
     \\
     B &\longrightarrow& Y
  }
$$

a [[commuting square]], it is equivalent to its [[pasting]] composite with that retract diagram

$$
  \array{
    A &\longrightarrow& C &\longrightarrow& A &\longrightarrow& X
    \\
    \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{\mathrlap{\in K Proj}} && \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}}
    \\
    B &\longrightarrow& D &\longrightarrow& B &\longrightarrow & Y
  }
  \,.
$$

Now the pasting composite of the two squares on the right has a lift, by assumption,

$$
  \array{
    A &\longrightarrow& C &\longrightarrow& A &\longrightarrow& X
    \\
    \downarrow^{\mathrlap{j}}
      &&
    \downarrow^{\mathrlap{i}}_{\mathrlap{\in K Proj}}
      &&
    \nearrow
      &&
    \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}}
    \\
    B &\longrightarrow& D &\longrightarrow& B &\longrightarrow & Y
  }
  \,.
$$

By composition, this is also a lift in the total outer rectangle, hence in the original square. Hence $j$ has the left lifting property against all $p \in K$ and hence is in $K Proj$. The other case is [[formal duality|formally dual]].

**closure under pushout and pullback**

Let $p \in K Inj$ and and let

$$
  \array{
    Z \times_f X &\longrightarrow& X
    \\
    {}^{\mathllap{{f^* p}}}\downarrow && \downarrow^{\mathrlap{p}}
    \\
    Z &\stackrel{f}{\longrightarrow} & Y
  }
$$

be a [[pullback]] diagram in $\mathcal{C}$. We need to show that $f^* p$ has the [[right lifting property]] with respect to all $i \in K$. So let

$$
  \array{
     A &\longrightarrow& Z \times_f X
     \\
     {}^{\mathllap{i}}_{\mathllap{\in K}}\downarrow && \downarrow^{\mathrlap{\mathrlap{f^* p}}}
     \\
     B &\stackrel{g}{\longrightarrow}& Z
  }
$$

be a [[commuting square]]. We need to construct a diagonal lift of that square. To that end, first consider the [[pasting]] composite with the pullback square from above to obtain the commuting diagram

$$
  \array{
     A &\longrightarrow& Z \times_f X &\longrightarrow& X
     \\
     {}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{f^* p}}
     && \downarrow^{\mathrlap{p}}
     \\
     B &\stackrel{g}{\longrightarrow}& Z &\stackrel{f}{\longrightarrow}& Y
  }
  \,.
$$

By the right lifting property of $p$, there is a diagonal lift of the total outer diagram

$$
  \array{
    A &\longrightarrow& X
    \\
    \downarrow^{\mathrlap{i}} &{}^{\hat {(f g)}}\nearrow&  \downarrow^{\mathrlap{p}}
    \\
    B &\stackrel{f g}{\longrightarrow}& Y
  }
  \,.
$$

By the [[universal property]] of the [[pullback]] this gives rise to the lift $\hat g$ in

$$
  \array{
     && Z \times_f X &\longrightarrow& X
     \\
     &{}^{\hat g} \nearrow& \downarrow^{\mathrlap{f^* p}}
     && \downarrow^{\mathrlap{p}}
     \\
     B &\stackrel{g}{\longrightarrow}& Z &\stackrel{f}{\longrightarrow}& Y
  }
  \,.
$$

In order for $\hat g$ to qualify as the intended lift of the total diagram, it remains to show that

$$
  \array{
     A &\longrightarrow& Z \times_f X
     \\
     \downarrow^{\mathrlap{i}} & {}^{\hat g}\nearrow
     \\
     B
  }
$$

commutes. To do so we notice that we obtain two [[cones]] with tip $A$:

* one is given by the morphisms
  1. $A \to Z \times_f X \to X$
  2. $A \stackrel{i}{\to} B \stackrel{g}{\to} Z$

  with universal morphism into the pullback being

  * $A \to Z \times_f X$

* the other by
  1. $A \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X \to X$
  2. $A \stackrel{i}{\to} B \stackrel{g}{\to} Z$.

  with universal morphism into the pullback being

  * $A \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X$.

The commutativity of the diagrams that we have established so far shows that the first and second morphisms here equal each other, respectively. By the fact that the universal morphism into a pullback diagram is _unique_ this implies the required identity of morphisms.

The other case is [[formal dual|formally dual]].

**closure under (co-)products**

Let $\{(A_s \overset{i_s}{\to} B_s) \in K Proj\}_{s \in S}$ be a set of elements of $K Proj$. Since [[colimits]] in the [[presheaf category]] $\mathcal{C}^{\Delta[1]}$ are computed componentwise, their [[coproduct]] in this [[arrow category]] is the universal morphism out of the coproduct of objects $\underset{s \in S}{\coprod} A_s$ induced via its [[universal property]] by the set of morphisms $i_s$:

$$
  \underset{s \in S}{\sqcup} A_s
  \overset{(i_s)_{s\in S}}{\longrightarrow}
  \underset{s \in S}{\sqcup} B_s
  \,.
$$

Now let

$$
  \array{
    \underset{s \in S}{\sqcup} A_s &\longrightarrow& X
    \\
    {}^{\mathllap{(i_s)_{s\in S}}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}}
    \\
    \underset{s \in S}{\sqcup} B_s
    &\longrightarrow&
    Y
  }
$$

be a [[commuting square]]. This is in particular a [[cocone]] under the [[coproduct]] of objects, hence by the [[universal property]] of the coproduct, this is equivalent to a set of commuting diagrams

$$
  \left\{
  \;\;\;\;\;\;\;\;\;
  \array{
    A_s &\longrightarrow& X
    \\
    {}^{\mathllap{i_s}}_{\mathllap{\in K Proj}}\downarrow
      &&
    \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}}
    \\
    B_s
    &\longrightarrow&
    Y
  }
  \;\;\;\;
  \right\}_{s\in S}
  \,.
$$

By assumption, each of these has a lift $\ell_s$. The collection of these lifts

$$
  \left\{
  \;\;\;\;\;\;\;\;\;
  \array{
    A_s &\longrightarrow& X
    \\
    {}^{\mathllap{i_s}}_{\mathllap{\in Proj}}\downarrow
      &{}^{\ell_s}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}}
    \\
    B_s
    &\longrightarrow&
    Y
  }
  \;\;\;\;
  \right\}_{s\in S}
$$

is now itself a compatible [[cocone]], and so once more by the [[universal property]] of the coproduct, this is equivalent to a lift $(\ell_s)_{s\in S}$ in the original square

$$
  \array{
    \underset{s \in S}{\sqcup} A_s &\longrightarrow& X
    \\
    {}^{\mathllap{(i_s)_{s\in S}}}\downarrow
     &{}^{(\ell_s)_{s\in S}}\nearrow&
    \downarrow^{\mathrlap{f}}_{\mathrlap{\in K}}
    \\
    \underset{s \in S}{\sqcup} B_s
    &\longrightarrow&
    Y
  }
  \,.
$$

This shows that the coproduct of the $i_s$ has the left lifting property against all $f\in K$ and is hence in $K Proj$. The other case is [[formal dual|formally dual]].

=--

## Related concepts

* the [[small object argument]] is about factoring morphisms into $J$-cofibrations followed by $J$-injective morphisms.

[[!redirects injective or projective morphisms]]

[[!redirects injective morphism]]
[[!redirects injective morphisms]]

[[!redirects projective morphism]]
[[!redirects projective morphisms]]

[[!redirects injective and projective morphisms]]
[[!redirects projective and injective morphisms]]


injective presentation

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

#Contents#
* table of contents
{:toc}

## Idea

The [[duality|dual]] notion of _[[projective presentation]]_.

## Related concepts

* [[projective object]], [[projective presentation]], [[projective cover]], [[projective resolution]]

  * [[projective module]]

* [[injective object]], **injective presentation**, [[injective envelope]], [[injective resolution]]

  * [[injective module]]

* [[free object]], [[free resolution]]

  * [[free module]]

* flat object, [[flat resolution]]

  * [[flat module]]

[[!redirects injective presentations]]


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

#Contents#
* table of contents
{:toc}

## Statement

+-- {: .num_prop #InjectiveProperMapsAreEquivalentlyTheClosedEmbeddings}
###### Proposition
**(injective proper maps to locally compact spaces are equivalently the closed embeddings)**

Let

1. $X$ be a [[topological space]]

1. $Y$ a [[locally compact topological space]]

1. $f \colon X \to Y$ be a [[continuous function]].

Then the following are equivalent

1. $f$ is an [[injective function|injective]] [[proper map]],

1. $f$ is a closed [[embedding of topological spaces]].

=--

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

In one direction, if $f$ is an injective proper map,
then since [[proper maps to locally compact spaces are closed]], it follows that $f$ is also  [[closed map]]. The claim then follows since [[closed injections are embeddings]], and since the image of a closed map is closed.

Conversely, if $f$ is a closed embedding, we only need to show that the embedding map is proper. So for $C \subset Y$ a [[compact subspace]], we need to show that the [[pre-image]] $f^{-1}(C) \subset X$ is also compact. But since $f$ is an injection (being an embedding), that pre-image is just the intersection $f^{-1}(C) \simeq C \cap f(X)$. By the nature of the [[subspace topology]], this is compact if $C$ is.

=--

## Related statements

* [[closed injections are embeddings]]

* [[proper maps to locally compact spaces are closed]]


injective topos

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

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

## Idea

An **injective topos** is a topos-theoretic generalization of the notion of an [[injective object]] familiar from algebra, or in other words, a topos with good extension properties along [[subtopos|subtopos inclusions]].

## Definition

+-- {: .num_defn}
###### Definition
A [[Grothendieck topos]] $\mathcal{E}$ is called _injective_ (with respect to subtopos inclusions) in the [[2-category]] of Grothendieck toposes $GrTop$ if for every inclusion $i:\mathcal{F}\hookrightarrow\mathcal{G}$ and 1-cell $f:\mathcal{F}\to\mathcal{E}$ there exists a 1-cell $g:\mathcal{E}\to\mathcal{G}$ and a 2-isomorphism $g\circ i\cong f$ .
=--

## Properties

The main characterization of injective toposes is achieved by the following result due to Johnstone-Joyal ([1982](#JJ82)).

+-- {: .num_prop}
###### Theorem
There is an equivalence of 2-categories between the full sub-2-category of $GrTop$ with 0-cells the injective toposes and the 2-category of cocomplete ind-small [[continuous category|continuous categories]] with 1-cells the filtered colimit preserving functors. This equivalence sends an injective topos to its [[category of points]].

=--

Note, that this gives also a characterization of the continuous categories involved as category of points of injective toposes.

+-- {: .num_prop}
###### Proposition
The lex totally distributive categories with a small dense generator are precisely the injective Grothendieck toposes.

=--

This occurs as theorem 4.5 in Lucyshyn-Wright ([2011](#LW11)).

## Example

 * $Set$ is the only injective [[Boolean topos]] (cf [Johnstone 2002](#JT02), p.740).

## Related entries

* [[injective object]]

* [[continuous category]]

* [[totally distributive category]]

* [[exponentiable topos]]

* [[De Morgan topos]]

## References

* [[Peter Johnstone]], _Injective Toposes_ , pp.284-297 in LNM **871** Springer Heidelberg 1981.

* {#JT02}[[Peter Johnstone]], _Sketches of an Elephant vol. 2_ , Cambridge UP 2002. (section C4.3, pp.738-745)

* {#JJ82}[[Peter Johnstone]], [[André Joyal]], _Continuous categories and exponentiable toposes_ ,  JPAA **25** (1982) pp.255-296.

* {#LW11} [[Rory Lucyshyn-Wright]], _Totally distributive toposes_ , arXiv:1108.4032 (2011). ([abstract](https://arxiv.org/abs/1108.4032))


injective-on-objects functor

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

#Contents#
* table of contents
{:toc}

## Injective-on-objects functors

A [[functor]] between [[strict categories]] is called **injective-on-objects** if its object-[[function]] is [[injective function|injective]].

Injective-on-objects functors are exactly:

* the [[isocofibrations]] in [[Cat]],

* the [[cofibrations]] in the [[canonical model structure on categories]] (also in the [[canonical model structure on groupoids]]).

## Related pages

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

[[!redirects injective-on-objects functor]]
[[!redirects injective-on-objects functors]]
[[!redirects injective on objects functor]]
[[!redirects injective on objects functors]]


Inna Zakharevich

* [personal page](https://pi.math.cornell.edu/~zakh/)

* [Institute page](https://www.ias.edu/scholars/inna-zakharevich)

## Selected writings

On [[Pontrjagin rings]]:

* [[Inna Zakharevich]], pp. 35 of: *Stable homotopy*, Section 11 of: *[K-theory and characteristic classes](https://pi.math.cornell.edu/~zakh/6530/)*, lecture notes (2017) [[pdf](https://pi.math.cornell.edu/~zakh/6530/sec11.pdf), [[Zakharevich-StableHomotopy.pdf:file]]]

On the [[Dennis trace]] on [[algebraic K-theory]] and relating to [[bicategorical trace]]:

* {#CLMPZ20} [[Jonathan Campbell]], [[John Lind]], [[Cary Malkiewich]], [[Kate Ponto]], [[Inna Zakharevich]], _K-theory of endomorphisms, the TR-trace, and zeta functions_ [[arXiv:2005.04334](https://arxiv.org/abs/2005.04334)]

category: people


inner automorphism

An __inner automorphism__ $\phi:G\to G$ of a group $G$ is any [[automorphism]] $\phi_g$ of the form $h\mapsto g h g^{-1}$. The inner automorphisms form a [[subgroup]] $Inn(G)$, called the __inner automorphism group__ of $G$, of the entire [[automorphism group]] $Aut(G)$; it is the [[image]] of the natural map $G\to Aut(G)$ given by $g\mapsto\phi_g$. The [[center]] of a group $G$ is precisely the [[kernel]] of this natural map. Similarly, the monoidal center due to Drinfel'd and Majid, in the case when the monoidal category is Picard, is a $2$-category-theoretic kernel (an observation due to L. Breen).

Higher analogues of the inner automorphism group were studied by Roberts and Schreiber.

[[!redirects inner automorphisms]]
[[!redirects inner automorphism group]]


inner automorphism 2-group

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

## Idea

The _inner automorphism 2-group_ of a [[group]] is essentially the sub-[[2-group]] of the [[automorphism 2-group]] $AUT(G)$ of $G$ of those automorphisms that are connected by a [[transfor|transformation]] to the identity: this makes the automorphism necessarily an [[inner automorphism]].

In fact, more precisely the inner automorphism 2-group is the 2-group of these connecting transformations, i.e. it remembers the group element and the inner automorphism that it induces under conjugation.

## Definition

Let $G$ be a [[group]]. Write $\mathbf{B}G$ for its [[delooping]].

The **inner automorphism 2-group** $INN(G)$ of $G$ is the [[strict 2-group]]

* whose [[object]]s are diagrams

  $$
    \array{
      & \nearrow \searrow^{\mathrlap{Id}}
      \\
      \mathbf{B}G
      &\Downarrow^{\eta}&
      \mathbf{B}G
      \\
      & \searrow \nearrow_{\mathrlap{\alpha}}
    }
  $$

  in [[Grpd]].

* morphisms $\kappa : (\eta_1, \alpha_1) \to (\eta_2, \alpha_2)$ are commuting triangles of transformations

  $$
    \array{
       && Id
       \\
       & {}^{\mathllap{\eta_1}}\swArrow && \seArrow^{\mathrlap{\eta_2}}
       \\
       \alpha_1 &&\stackrel{\kappa}{\Rightarrow}&&
       \alpha_2
    }
  $$

Equivalently, this is the [[action groupoid]]

$$
  INN(G) = G//G \eqqcolon \mathbf{E}G
$$

of $G$ acting on itself.

Equivalently , this is the [[strict 2-group]] corresponding to the [[crossed module]]

$$
  [INN(G)] =
  \left(
    G \stackrel{Id}{\to} G
  \right)
$$

with action $G \to Aut(G)$ given by right multiplication in $G$.

This makes it evident that $INN(G)$ is contractible

$$
  \mathbf{B} INN(G) \stackrel{\simeq}{\to} {*}
  \,.
$$

In fact, we may think of $INN(G)$ as the [[generalized universal bundle|universal]] $G$-[[principal bundle]] in its incarnation in [[Grpd]] (as opposed to the more tradition incarnation in [[Top]], to which it is [[Quillen equivalence|Quillen equivalent]] by the [[homotopy hypothesis]] theorem).

To emphasize this we also write

$$
  \mathbf{E}G \coloneqq INN(G)
  \,.
$$

We have a natural sequence of groupoids

$$
  G \to \mathbf{E}G \to \mathbf{B}G
  \,.
$$

It is an old theorem by [[Graeme Segal]] that under [[nerve]] followed by [[geometric realization]] this maps to the sequence of [[topological space]]s

$$
  G \to \mathcal{E} G \to \mathcal{B}G
$$

that is the universal $G$-bundle over the [[classifying space]] $\mathcal{B}G$ in its incarnation in [[Top]].

The 2-group structure on $INN(G)$ is evident, and hence makes the fact evident that the universal $G$-bundle itself carries a group structure, which is compatibel with the group structure, in that the morphism $ G \to \mathbf{E}G$ [[delooping|deloops]] to a morphism

$$
  \mathbf{B}G \to \mathbf{B} \mathbf{E}G
  \,.
$$

This fact is useful in various applications in [[nonabelian cohomology]].


inner derivation Lie 2-algebra

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### $\infty$-Lie theory
+--{: .hide}
[[!include infinity-Lie theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For **inner derivation Lie 2-algebra** $inn(\mathfrak{g})$ of a [[Lie algebra]] $\mathfrak{g}$ is the ([[strict Lie 2-algebra|strict]]) [[Lie 2-algebra]] equivalently given as

* the [[dg-Lie algebra]] of contractions $\iota_x$ and inner derivations $\mathcal{L}_x = [d_{CE},x]$ acting on the [[Chevalley-Eilenberg algebra]] of $\mathfrak{g}$;

* the [[differential crossed module]] $(\mathfrak{g} \stackrel{Id}{\to} \mathfrak{g})$ with the action being the [[adjoint action]] of $\mathfrak{g}$ on itself;

* the dual of the [[Weil algebra]] $W(\mathfrak{g})$.

In the first formulation this may be identified with the [[dg-Lie algebra]] whose

* elements in degree -1 are the contractions $\iota_x : CE(\mathfrak{g}) \to CE(\mathfrak{g})$ with $x \in \mathfrak{g}$;

* elements in degree 0  are the inner derivations $\mathcal{L}_x = [d_{CE(\mathfrak{g})}, \iota_x] : CE(\mathfrak{g}) \to CE(\mathfrak{g})$;

* the differential $\partial : \mathfrak{g} \to \mathfrak{g}$ is given by the commutator $\partial = [d_{CE(\mathfrak{g})}, -]$;

* the bracket is the graded commutator bracket of [[derivation]]s:

  * $[\iota_x, \iota_y] = 0$

  * $[\mathcal{L}_x, \iota_y] = \iota_{[x,y]}$

  * $[\mathcal{L}_x, \mathcal{L}_y] = \mathcal{L}_{[x,y]}$.

So this is the full subalgebra of the [[automorphism ∞-Lie algebra]] of $CE(\mathfrak{g})$ on the inner derivations.

See Weil algebra as CE-algebra of inner derivations for more details.

## Properties

* The first formulation makes manifest that $inn(\mathfrak{g})$ is the structure that has historically been called [[Cartan calculus]].

* In the [[(∞,1)-category]] of [[∞-Lie algebra]]s $inn(\mathfrak{g})$ is equivalent to the point. See [[Weil algebra]] for details on this.

## References

The structure of $inn(\mathfrak{g})$ is of course in itself very simple and goes as such back at least to Cartan.

Its role as a [[Lie 2-algebra]] in the context of [[∞-Chern-Weil theory]] has been discussed in section 6 of

* [[Hisham Sati]], [[Urs Schreiber]], [[Jim Stasheff]], _$L_{\infty}$ algebra connections and applications to String- and Chern-Simons $n$-transport_ (ref)
{#SSSI}


inner fibration

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

#Contents#
* table of contents
{:toc}

## Idea

The notion of _inner fibration_ of [[simplicial set]]s is one of the notion of [[fibrations of quasi-categories]].

When the notion of [[(∞,1)-category]] is incarnated in terms of the notion of [[quasi-category]], an _inner fibration_ is a morphism of [[simplicial set]]s $C \to D$ such that each fiber is a quasi-category and such that over each morphism $f : d_1 \to d_2$ of $D$, $C$ may be thought of as the [[cograph of a profunctor|cograph of an (∞,1)-profunctor]] $C_{d_1} ⇸ C_{d_2}$.

So when $D = {*}$ is the point, an inner fibration $C \to {*}$ is precisely a [[quasi-category]] $C$.

And when $D = N(\Delta[1])$ is the [[nerve]] of the [[interval category]], an inner fibration $C \to \Delta[1]$ may be thought of as the [[cograph of a profunctor|cograph of an (∞,1)-profunctor]] $C ⇸ D$.

This $(\infty,1)$-profunctor comes from an ordinary [[(∞,1)-functor]] $F : C \to D$ precisely if the inner fibration $K \to \Delta[1]$ is even a [[coCartesian fibration]]. And it comes from a functor $G : D \to C$ precisely if the fibration is even a [[Cartesian fibration]]. This is the content of the [[(∞,1)-Grothendieck construction]].

And precisely if the inner fibration/cograph of an $(\infty,1)$-profunctor $K  \to \Delta[1] $ is both a Cartesian as well as a coCartesian fibration does it exhibit a pair of [[adjoint (∞,1)-functor]]s.

## Definition

A [[morphism]] of [[simplicial set]]s $F : X \to Y$ is an **inner fibration** or **[[inner Kan fibration]]** if its has the [[right lifting property]] with respect to all inner [[horn]] inclusions, i.e. if for all [[commuting diagram]]s

$$
  \array{
    \Lambda[n]_i &\to& X
    \\
    \downarrow && \downarrow^{\mathrlap{F}}
    \\
    \Delta[n] &\to& Y
  }
$$

for $0 \lt i \lt n$ there exists a lift

$$
  \array{
    \Lambda[n]_i &\to& X
    \\
    \downarrow &\nearrow& \downarrow^{\mathrlap{F}}
    \\
    \Delta[n] &\to& Y
  }
  \,.
$$

The morphisms with the [[left lifting property]] against all inner fibrations are called **inner anodyne**.

## Properties

### General properties

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

By the [[small object argument]] we have that every morphism $f : X \to Y$ of simplicial sets may be factored as

$$
  f : X \to Z \to Y
$$

with $X \to Z$ a left/right/inner anodyne cofibration and $Z \to Y$ accordingly a left/right/inner Kan fibration.

=--

## Related concepts

* [[Kan fibration]], [[anodyne morphism]]

* [[right/left Kan fibration]], [[right/left anodyne map]]

* **inner fibration**

* [[Cartesian fibration]]

* [[fibration]] in the [[model structure for quasi-categories]]

* [[fibration of (infinity,1)-operads]]

## References

Inner fibrations were introduced by [[Andre Joyal]]. A comprehensive account is in section 2.3 of

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

Their relation to [[cograph of a profunctor|cographs]]/correspondence is discussed in section 2.3.1 there.

[[!redirects inner fibrations]]

[[!redirects inner Kan fibration]]
[[!redirects inner Kan fibrations]]


inner local system

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+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Higher geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Given an [[orbifold]] $\mathcal{X}$, an *inner local system* ([Ruan 00, Def. 3.1.6](#Ruan00)) is a family of [[local systems]] (in fact [[flat connections]] with [[coefficients]] in [[circle group|U(1)]]) on loci with non-trivial [[automorphisms]] (i.e. on $H\subset G$-[[fixed loci]] in the case that $\mathcal{X}$ is presented as a [[global quotient orbifold]] $X \sslash G$).

These inner local systems make an appearance in the [[twisted equivariant Chern character]] of the [[orbifold K-theory]] of $\mathcal{X}$, which lands, fixed-locus wise (in fact: [[automorphism]]-wise), in the corresponding [[twisted de Rham cohomology]].

## References

The terminology "inner local system" is due to

* {#Ruan00} [[Yongbin Ruan]],  Def. 3.1.6 in: *Stringy Geometry and Topology of Orbifolds* $[$[arXiv:math/0011149](https://arxiv.org/abs/math/0011149)$]$

As a contribution to the twist in the [[twisted equivariant Chern character]] the concept appears (not under this name) in:

* {#FreedHopkinsTeleman07} [[Daniel Freed]], [[Michael Hopkins]], [[Constantin Teleman]], Def. 3.6 of: _Twisted equivariant K-theory with complex coefficients_, Journal of Topology, Volume 1, Issue 1, 2007 ([arXiv:math/0206257](https://arxiv.org/abs/math/0206257), [doi:10.1112/jtopol/jtm001](https://doi.org/10.1112/jtopol/jtm001))

* {#TuXu06} [[Jean-Louis Tu]], [[Ping Xu]], Prop. 3.9, Def. 3.10 in: *Chern character for twisted K-theory of orbifolds*, Advances in Mathematics Volume 207, Issue 2, 20 December 2006, Pages 455-483 $[$[arXiv:math/0505267](https://arxiv.org/abs/math/0505267), [doi:10.1016/j.aim.2005.12.001](https://doi.org/10.1016/j.aim.2005.12.001)$]$

> (On the other hand, this inner local system twist seems to be missing in the twisted equivariant Chern character of [Mathai &Stevenson 03](twisted+equivariant+Chern+character#MathaiStevenson03)?)

A possibly more transparent (?) derivation of this subtle twisting by inner local systems, for the simple special case of a global singularity, is offered in:

* [[Hisham Sati]], [[Urs Schreiber]], Section 3 of: *[[schreiber:Anyonic defect branes in TED-K-theory]]* $[$[arXiv:2203.11838](https://arxiv.org/abs/2203.11838), [exposition pdf](https://ncatlab.org/schreiber/files/DefectBranes-slides_220509f.pdf)$]$

[[!redirects inner local systems]]


inner product abelian group

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### Context
#### Algebra
+-- {: .hide}
[[!include higher algebra - contents]]
=--
#### Group theory
+-- {: .hide}
[[!include group theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

An **inner product abelian group** is an [[abelian group]] $G$ with a binary function $q: G \times G \to \mathbb{Z}$ called the *inner product* such that the following properties hold:

* for all $a \in G$, $ \langle 0, a \rangle = 0 $ and $ \langle a, 0 \rangle = 0 $;
* for all $a, b, c \in G$, $ \langle a + b, c \rangle = \langle a, c \rangle + \langle b, c \rangle $ and $ \langle a, b + c \rangle = \langle a, b \rangle + \langle a, c \rangle $
* for all $a, b \in G$ and $ \langle a, b \rangle = \overline{\langle b, a \rangle}$.

where $\overline{(-)}:\mathbb{Z} \to \mathbb{Z}$ is an [[involution]] on the integers.

Typically, the inner product is defined with another axiom

* for all $a, b \in G$ and $c \in \mathbb{Z}$, $ \langle c a, b \rangle = c \langle a, b \rangle $ and
$\langle a, \overline{c} b \rangle = \langle a, b \rangle \overline{c}$

but this is provable for any binary function which is left and right distributive over the abelian group operations.

## Properties

* Every inner product abelian group is a [[quadratic abelian group]] with $q(x) \coloneqq \langle x, x \rangle$.

## See also

* [[inner product]]

* [[abelian group]]

* [[ring]]

* [[quadratic abelian group]]

* [[Clifford ring]]

[[!redirects inner product abelian groups]]


inner product of multisets

# Contents #

* tic
{: toc}

## Definition ##

A [[multiset]] $\mathcal{X} = \langle X,\mu_X\rangle$ consists of a set $X$ and a function $\mu_X:U\to\mathbb{N}$, where $U$ is a universal set and $\mu_X(e) \gt 0$ if and only if $e\in X$.

We can add two multisets $\mathcal{X} = \langle X,\mu_X\rangle$ and $\mathcal{Y} = \langle Y,\mu_Y\rangle$ to get

$$\mathcal{X} + \mathcal{Y} = \langle X\cup Y,\mu_X+\mu_Y\rangle.$$

Note that we can write

$$k\mathcal{X} = \langle X,k\mu_X\rangle$$

for $k\in\mathbb{N}$.

We can also define an **inner product of multisets** via

$$\langle \mathcal{X},\mathcal{Y}\rangle = \sum_{e\in\mathcal{X}\cup \mathcal{Y}} \mu_X(e) \mu_Y(e).$$

Note that when $\mathcal{X}$ and $\mathcal{Y}$ are simply sets, $\mu_X$ and $\mu_Y$ are the characteristic functions and

$$\langle \mathcal{X},\mathcal{Y}\rangle = |X\cap Y|,$$

where $|\cdot|$ denotes the cardinality of the set.

Using this inner product, we can define the **angle between multisets** as

$$\cos\theta_{\mathcal{X},\mathcal{Y}} = \frac{\langle \mathcal{X},\mathcal{Y}\rangle}{\sqrt{\langle \mathcal{X},\mathcal{X}\rangle \langle \mathcal{Y},\mathcal{Y}\rangle}}.$$

In particular, when $\mathcal{X}=\mathcal{Y}$ we have

$$cos\theta_{{\mathcal{X}},{\mathcal{Y}}} = 1$$

and when $X\cap Y = \emptyset$ we have

$$cos\theta_{{\mathcal{X}},{\mathcal{Y}}} = 0.$$

When $\mathcal{X}$ and $\mathcal{Y}$ are simply sets, the angle between them is given by

$$cos\theta_{{\mathcal{X}},{\mathcal{Y}}} = \frac{|X\cap Y|}{\sqrt{|X||Y|}}.$$

With this notion of addition, the collection of multisets in $U$ becomes the $\mathbb{N}$-[[module]] (that is [[abelian monoid]]) $\mathbb{N}^U$; this inner product makes it an [[inner product space]] analogous to the Banach space $\mathbb{R}^n$.

## Machine Learning ##

The inner product of multisets is closely related to the "bag of words" kernel in machine learning (see n-Cafe).

## Related concepts

* [[multiset]]


inner product on vector bundles

[[!redirects inner product of vector bundles]]

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###Context###
#### Bundles#
+-- {: .hide}
[[!include bundles - contents]]
=--
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The concept of _inner product on vector bundles_ is the evident generalization of that of [[inner product]] on [[vector spaces]] to [[vector bundles]]: a [[fiber]]-wise inner product of vector spaces.

## Definition

+-- {: .num_defn #InnerProductOnTopologicalVectorBundles}
###### Definition
**(inner product on [[topological vector bundles]])**

Let

1. $k$ be a [[topological field]] (such as the [[real numbers]] or [[complex numbers]] with their [[Euclidean space|Euclidean]] [[metric topology]] ),

1. $X$ be a [[topological space]],

1. $E \to X$ a [[topological vector bundle]] over $X$ (over $\mathbb{R}$, say).

Then an _inner product_ on $E$ is

* a vector bundle [[homomorphism]]

  $$
    \langle -,-\rangle
    \;\colon\;
    E \otimes_X E
     \longrightarrow
    X \times \mathbb{R}
  $$

  from the [[tensor product of vector bundles]] of $E$ with itself to the trivial [[line bundle]]

such that

* for each point $x \in X$ the function

  $$
    \langle -,-\rangle|_x \colon E_x \otimes E_x \to \mathbb{R}
  $$

  is an [[inner product]] on the [[fiber]] [[vector space]], hence a positive-definite symmetric [[bilinear form]].

=--

## Properties

### Existence

\begin{prop}
\label{ExistenceOfInnerProductOfTopologicalVectorBundlesOverParacompactHausdorffSpaces}
Let $X$ be a [[paracompact Hausdorff space]]. Then on every [[topological vector bundle]] $E \to X$ on $X$ there exists an inner product of topological vector bundles (def. \ref{InnerProductOnTopologicalVectorBundles})
\end{prop}

(e.g. [Hatcher, prop. 1.2](#Hatcher))

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

Let $\{U_i \subset X\}_{i \in I}$ be an [[open cover]] of $X$ over which $E \to X$ admits a [[local trivialization]]

$$
  \left\{
    \phi_i \;\colon\;
    U_i \times \mathbb{R}^n
      \overset{\simeq}{\longrightarrow}
    E|_{U_i}
  \right\}_{i \in I}
  \,.
$$

Since [[paracompact Hausdorff spaces equivalently admit subordinate partitions of unity]] we may choose a [[partition of unity]]

$$
  \{f_i \;\colon\; X \to [0,1]\}_{i \in I}
  \,.
$$

Write

$$
  \langle -,-\rangle_{\mathbb{R}^n}
  \;\colon\;
  \mathbb{R}^n
    \otimes
  \mathbb{R}^n
    \longrightarrow
  \mathbb{R}
$$

for the standard inner product on $\mathbb{R}^n$.

By the [[compact support]] of $f_i$ inside $U_i \subset X$, the functions

$$
  \array{
    \langle -,-\rangle_i
    &\colon&
    E|_{U_i} \otimes_X E|_{U_i}
      &\overset{ \phi_i^{-1} \otimes_{U_i} \phi_i^{-1} }{\longrightarrow}&
    (U_i \times \mathbb{R}^n ) \otimes_{U_i} (U_i \times \mathbb{R}^n)
      &\overset{\simeq}{\to}&
    U_i \times (\mathbb{R}^n \otimes \mathbb{R}^n)
      &\overset{}{\longrightarrow}&
    U_i \times \mathbb{R}
    \\
    && && &&
    ((x,i),(v_1,v_2))
      & \overset{\phantom{AAA}}{\mapsto} &
    ((x,i), f_i(x) \cdot \langle v_1, v_2\rangle_{\mathbb{R}^n} )
  }
  \,.
$$

extend by zero to continuous functions on all of $E \otimes_X E$, which we denote by the same symbol $\langle -,-\rangle_i \colon E \otimes_X E \to X \times \mathbb{R}$.

We then claim that the sum

$$
  \langle -,-\rangle
    \;\colon\;
  \underset{i \in I}{\sum} \langle -,-\rangle_i
  \;\colon\;
  E \otimes_X E
  \longrightarrow
  X \times \mathbb{R}
$$

is an inner product as required.  Notice that the sum is well defined by [[locally finite cover|local finiteness]] of the supports of the partition functions $f_i$. Hence it is pointwise a finite sum of positive definite symmetic bilinear forms on $E_x$, and as such itself pointwise a positive definite symmetric bilinear form.

=--

## Related concepts

* [[topological Hermitian K-theory]]

* [[vector bundle]]

  * [[real vector bundle]]

  * [[complex vector bundle]]

    * [[holomorphic vector bundle]], [[pseudoholomorphic vector bundle]]

  * [[universal vector bundle]]

  * [[dual vector bundle]]

  * [[module bundle]]

  * [[direct sum of vector bundles]]

  * [[short exact sequence of vector bundles]]

  * [[connection on a vector bundle]]

  * [[flat vector bundle]]

  * [[real vector bundle]], [[complex vector bundle]]

  * [[super vector bundle]]

## References

* [[Glenys Luke]], [[Alexandr S. Mishchenko]], *Vector bundles and their applications*, Math. and its Appl. **447** Kluwer (1998) [[doi:10.1007/978-1-4757-6923-4](https://doi.org/10.1007/978-1-4757-6923-4), [MR99m:55019](http://www.ams.org/mathscinet-getitem?mr=99m:55019)]

Discussion with an eye towards [[topological K-theory]] is in

* [[Max Karoubi]], _K-theory. An introduction_, Grundlehren der Mathematischen Wissenschaften __226__, Springer 1978. xviii+308 pp.

* {#Hatcher} [[Allen Hatcher]], section 1.1 of: _Vector bundles and K-Theory_ ([webpage](https://pi.math.cornell.edu/~hatcher/VBKT/VBpage.html), [pdf](https://pi.math.cornell.edu/~hatcher/VBKT/VB.pdf))

[[!redirects inner products on vector bundles]]

[[!redirects inner products of vector bundles]]