I-space

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###Context###
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[homotopy theory]], by an *$\mathcal{I}$-space* [[Sagave & Schlichtkrull (2011)](#SagaveSchlichtkrull11)] one means a [[functor]] $\mathcal{I} \to \mathcal{S}$ from

1. the [[category]] $\mathcal{I}$ of [[finite sets]] (including the [[empty set]]) with [[injective maps]] between them;

1. a category such as [[TopologicalSpaces]] or [[SimplicialSets]] which carries a [[classical model structure on topological spaces|classical]] [[model category structure]]  presenting plain [[homotopy types]].

The [[functor category]] $\mathcal{S}^{\mathcal{I}}$ of $\mathcal{I}$-spaces then carries a [[model category]]-[[structure]] whose [[weak equivalences]] are those [[natural transformations]] which induce [[weak homotopy equivalences]] on [[homotopy colimits]] over $\mathcal{I}$; and this turns out to be  [[Quillen equivalence|Quillen equivalent]] to the [[classical model structure on topological spaces|classical model structure]] via the [[colimit]]-functor

\begin{tikzcd}
  \mathcal{S}^{\mathcal{I}}
  \ar[
    rr, shift left=10pt,
    "{ \mathrm{colim} }"
  ]
  \ar[
    from=rr, shift left=10pt,
    "{ \mathrm{const} }"
  ]
  \ar[rr, phantom, "{ \scalebox{.7}{$\simeq_{\mathrm{Qu}}$} }"]
  &&
  \mathcal{S}
\end{tikzcd}

As such, $\mathcal{I}$-spaces are just another presentation of [[homotopy theory]].

However, via [[Day convolution]] of the evident [[symmetric monoidal category]]-[[structure]] on $\mathcal{I}$ (whose [[tensor product]] is [[disjoint union]] of [[finite sets]]) the [[functor category]] $\mathcal{S}^{\mathcal{I}}$ of $\mathcal{I}$-spaces becomes itself [[symmetric monoidal category|symmetric monoidal]].

This monoidal structure on $\mathcal{I}$-spaces is more homotopy-truthful than the (cartesian) monoidal structure on plain spaces: strictly [[commutative monoids]] [[monoid object|internal to]] $\mathcal{I}$-spaces serve to model all [[E-infinity space|$E_\infty$-spaces]].

## Related concepts

* [[E-infinity space|$E_\infty$-space]]

* [[model structure on parameterized spectra]]

## References

* {#SagaveSchlichtkrull11} [[Steffen Sagave]], [[Christian Schlichtkrull]], §1.1 and §3 in: *Diagram spaces and symmetric spectra*, Advances in Mathematics **231** 3-4 (2012) 2116-2193 [[arXiv:1103.2764](https://arxiv.org/abs/1103.2764), [doi:10.1016/j.aim.2012.07.013](https://doi.org/10.1016/j.aim.2012.07.013)]

Discussion mostly in relation to [[symmetric spectra]]

* [[Stefan Schwede]], Section 3.4 of: *[[Symmetric spectra]]* (2012) [[pdf](http://www.math.uni-bonn.de/~schwede/SymSpec-v3.pdf)]

and [[parameterized spectra|paramterized]] [[symmetric spectra]]:

* {#HebestreitSagaveSchlichtkrull20} [[Fabian Hebestreit]], [[Steffen Sagave]], [[Christian Schlichtkrull]], *Multiplicative parametrized homotopy theory via symmetric spectra in retractive spaces*, Forum of Mathematics, Sigma **8** (2020) e16 [[arXiv:1904.01824](https://arxiv.org/abs/1904.01824), [doi:10.1017/fms.2020.11](https://doi.org/10.1017/fms.2020.11)]

[[!redirects I-spaces]]


I. A. Volodin

I. A. Volodin (И. А. Володин) is a Russian mathematician who in 1971 published an alternative approach to the higher algebraic [[K-theory]] of rings, which was later shown by [[Suslin]] to give equivalent $K$-groups to the theory using the [[Quillen plus construction]].

* [list of some papers](http://www.mathnet.ru/php/person.phtml?option_lang=eng&personid=23468)

category:people

[[!redirects Volodin]]
[[!redirects I. Volodin]]
[[!redirects I Volodin]]
[[!redirects I.A. Volodin]]
[[!redirects IA Volodin]]
[[!redirects I. A. Volodin]]
[[!redirects I A Volodin]]
[[!redirects Володин]]
[[!redirects И. Володин]]
[[!redirects И Володин]]
[[!redirects И.А. Володин]]
[[!redirects ИА Володин]]
[[!redirects И. А. Володин]]
[[!redirects И И Володин]]


[[!redirects I.10, Frobenius morphism and symmetric products]]

This entry is about a section of the text

* Michel [[Demazure, lectures on p-divisible groups]], [web](http://sites.google.com/site/mtnpdivisblegroupsworkshop/lecture-notes-on-p-divisible-groups)

We give here another characterization of the [[Frobenius morphism]] in terms of symmetric products.

Let $p$ be a prime number, let $k$ be a filed of [[characteristic]] $p$, let $V$ be a $k$-vector space, let $\otimes^p V$ denote the $p$-fold tensor power of $V$, let $TS^p V$ denote the subspace of symmetric tensors. Then we have the symmetrization operator

$$s_V:
\begin{cases}
\otimes^p V\to TS^p V
\\
a_1\otimes\cdots\otimes a_n\mapsto \Sigma_{\sigma\in S_p}a_{\sigma(1)}\otimes\cdots\otimes a_{\sigma(n)}
\end{cases}$$

end the linear map

$$\alpha_V:
\begin{cases}
TS^p V\to\otimes^p V
\\
a\otimes \lambda\mapsto\lambda(a\otimes\cdots\otimes a)
\end{cases}$$

then the map $V^{(p)}\stackrel{\alpha_V}{\to}TS^p V\to TS^p V/s(\otimes^p V)$ is bijective and we define $\lambda_V:TS^p V\to V^{(p)}$ by

$$\lambda_V\circ s=0$$

and

$$\lambda_V \circ \alpha_V= id$$

If $A$ is a $k$-ring we have that $TS^p A$ is a $k$-ring and $\lambda_A$ is a $k$-ring morphism.

If $X=Sp_k A$ is a ring spectrum we abbreviate $S^p X=S^p_k X:=Sp_k (TS^p A)$ and the following diagram is commutative.

$$\array{
X
&\stackrel{F_X}{\to}&
X^{(p)}
\\
\downarrow&&\downarrow
\\
X^p
&\stackrel{can}{\to}& S^p X
}$$


I8

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###Context###
#### Cohomology
+--{: .hide}
[[!include cohomology - contents]]
=--
#### String theory
+-- {: .hide}
[[!include string theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the [[string theory]]-literature "$I_8$" or "$X_8$" is standard notation for a certain [[characteristic class]] of [[manifolds]] (of their [[tangent bundles]]): It is a [[rational numbers|rational]] linear combination of the [[cup product|cup square]] of the [[first fractional Pontryagin class]]  with itself, and the [[second Pontryagin class]]:

\[
  \label{TheTerm}
  I_8
  \;\coloneqq\;
  \tfrac{1}{48}
  \Big(
    p_2 \;-\; \big( \tfrac{1}{2} p_1\big)^2
  \Big)
  \;\;
  \in
  H^8\big( -, \mathbb{Q}\big)
  \,.
\]

In general this is a [[cohomology class]] in [[rational cohomology]], though in applications it appears in further rational combination with other classes that in total yield a class in [[integral cohomology]].

The expression (eq:TheTerm) controls certain [[quantum anomaly cancellation]] in [[M-theory]] and [[type IIA string theory]] ([Vafa-Witten 95](#VafaWitten95), [Duff-Liu-Minasian 95 (3.10) with (3.14)](#DuffLiuMinasian95)). Since it was first obtain as a [[1-loop]]-contribution in [[perturbative quantum gravity|perturbative quantum]] [[supergravity]], it is often known as _the one-loop anomaly term_ or the _one-loop anomaly polynomial_ in [[M-theory]]/[[type IIA string theory]].

\linebreak

## Properties

### For $Spin(7)$- or $Sp(1)\cdot Sp(2)$-structure

If $X^{8}$ has

* [[Spin(7)-structure]] (hence in particular if it is a [[Calabi-Yau manifold]], which has $SU(4) = $ [[Spin(6)]]-structure)

or

* [[Sp(2).Sp(1)-structure]]

then

$$
  \tfrac{1}{2}\big( p_2 - \tfrac{1}{4}(p_1)^2  \big)
  \;=\;
  \chi_8
$$

is the [[Euler class]] (see [this Prop.](Spin7-manifold#CharacteristicClassesForSpinStructure) and [this Prop.](quaternion-Kähler+manifold#CharacteristicClassesForSpin5Spin3Structure), respectively), hence then

$$
  I_8 = \tfrac{1}{24} \chi_8
  \,.
$$

### As a higher curvature correction to 11d Supergravity
 {#AsHigherCurvatureCorrectionTo11dSugra}

A systematic analysis of the possible [[supersymmetry|supersymmetric]] [[higher curvature corrections]] of [[D=11 supergravity]] makes the $I_8$ appear as the higher curvature correction at order $\ell^6$, where $\ell$ is the [[Planck length]] in 11d ([Souères-Tsimpis 17, Section 4](#SoueresTsimpis17)).

At this order, the [[equation of motion]] for the [[supergravity C-field]] [[flux]] $G_4$ and its dual $G_7$ is ([Souères-Tsimpis 17, (4.3)](#SoueresTsimpis17))

$$
  d G_7(\ell)
  \;=\;
  -\tfrac{1}{2}
  G_4(\ell) \wedge G_4(\ell)
  +
  \ell^6 I_8(R)
  \,,
$$

where the flux forms themselves appear in their higher order corrected form as [[power series]] in the [[Planck length]]

$$
  G_4(\ell)
  \;=\;
  G_4 + \ell^6 G_4^{(1)} + \cdots
$$

$$
  G_7(\ell)
  \;=\;
  G_7 + \ell^6 G_7^{(1)} + \cdots
$$

([Souères-Tsimpis 17, (4.4)](#SoueresTsimpis17))

Beware that this is not the lowest order [[higher curvature correction]]: there is already one at $\mathcal{O}(\ell^3)$, given by $\ell^3 G_4 \wedge \tfrac{1}{2}p_1(R)$ ([Souères-Tsimpis 17, Section 3.2](#SoueresTsimpis17)). Hence the full correction at $\mathcal{O}(\ell^6)$ should be the further modification of (eq:11dSugraBIAtOrder6) to (cf. [Tsimpis 2004, p. 8](D=11+supergravity#Tsimpis04)):

\[
  \label{11dSugraBIAtOrder6}
  d G_7(\ell)
  \;=\;
  -\tfrac{1}{2}
  G_4(\ell) \wedge
  \big(
    G_4(\ell)
    -
    2\ell^3 \tfrac{1}{2} p_1(R)
  \big)
  +
  \ell^6 I_8(R)
  \,.
\]

### Inflow to M5-brane anomalies
 {#InflowToM5BraneAnomaly}

Consider an 11-dimensional [[spin structure|spin]]-[[manifold]] $X^{(11)}$ and a 2-parameter family of 6-dimensional [[submanifolds]] $Q_{M5} \hookrightarrow X^{(11)}$. When regarded as a family of [[worldvolumes]] of an [[M5-brane]], the family of [[normal bundles]] $N_X Q_{M5}$ of this inclusion carries a [[characteristic class]]

\[
  \label{IM5}
  I^{M5}
  \;\coloneqq\;
  I^{M5}_{\psi}
  +
  I_{C}
  \;\in\;
  H^8(F \times Q_{M5},\mathbb{Z})
\]

where

1. the first summand is the class of the [[chiral anomaly]] of [[chiral fermions]] on $Q_{M5}$ ([Witten 96, (5.1)](#Witten96)),

1. the second term the class of the [[quantum anomaly]] of a [[self-dual higher gauge field]] ([Witten 96, (5.4)](#Witten96))

Moreover, there is the restriction of the $I_8$-term (eq:TheTerm) to $Q_{M5}$, hence to the [[tangent bundle]] of $X^{11}$ to $Q_{M5}$ (the "anomaly inflow" from the [[bulk spacetime]] to the M5-brane)

\[
  \label{I8AnomalyInflow}
  I_8\vert_{M5}
  \;\coloneqq\;
   I_8
   \big(
     T_{Q_{M5}} X
   \big)
  \;\in\;
  H^8(F \times Q_{M5},\mathbb{Z})
  \,.
\]

The sum of these cohomology classes, evaluated on the [[fundamental class]] of $Q_{M5}$ is proportional to the [[second Pontryagin class]] of the [[normal bundle]]

\[
  \label{SumOfM5AndInflowAnomalyIsp2}
  I^{M5}
  \;+\;
  I_8\vert_{M5}
  \;=\;
  \tfrac{1}{24} p_2(N_{Q_{M5}})
\]

([Witten 96 (5.7)](#Witten96))

This result used to be "somewhat puzzling" ([Witten 96, p. 35](#Witten96)) since consisteny of the [[M5-brane]] in [[M-theory]] should require its total [[quantum anomaly]] to vanish. But $p_2(N_{Q_{M5}})$ does not in general vanish, and the right conditions to require under which it does vanish were "not clear" ([Witten 96, p. 37](#Witten96)).

(For more details on computations involved this and the following arguments, see also [Bilal-Metzger 03](#BilalMetzger03)).

A resolution was proposed in [Freed-Harvey-Minasian-Moore 98](#FreedHarveyMinasianMoore98), see also [Bah-Bonetti-Minasian-Nardoni 18 (5)](#BahBonettiMinasianNardoni18), [BBMN 19 (2.9) and appendix A.4, A.5](#BBMN19). By this proposal, the anomaly inflow from the bulk would not be just $I_8$, as in (eq:I8AnomalyInflow) but would be all of the following [[fiber integration]]

\[
  \label{FiberIntegration}
  \array{
    \pi_\ast
    \Big(
      -
      \tfrac{1}{6}
      G_4  G_4  G_4
      +
      G_4  I_8
    \Big)
    & =
    - \tfrac{1}{24} p_2 + \tfrac{1}{2}(G^{M5}_4)^2 + I_8
  }
\]

Here  we used [this Prop](Spin5#FiberIntegrationOfCupPowersOfChiOver4Sphere) to find that

$$
  \pi_\ast\big(
    \chi^3
  \big)
  \;=\;
  2 p_2
$$

which would cancel against the first term $\tfrac{1}{24} p_2$ in (eq:FiberIntegration). Hence with this proposal, the remaining M5-brane anomaly (eq:SumOfM5AndInflowAnomalyIsp2) would be canceled -- except for a remaining term $\tfrac{1}{2}(G^{M5}_4)^2$ which is ignored by fiat.

\linebreak

## Related concepts

* [[Pontryagin classes]], [[Stiefel-Whitney classes]], [[Wu classes]], [[Euler class]]

* [[C-field tadpole cancellation]]

## References

### General

The term showed  in [[string theory]]/[[M-theory]] [[anomaly cancellation]] in

* {#VafaWitten95} [[Cumrun Vafa]], [[Edward Witten]], _A One-Loop Test Of String Duality_, Nucl.Phys.B447:261-270, 1995 ([arXiv:hep-th/9505053](https://arxiv.org/abs/hep-th/9505053))

* {#DuffLiuMinasian95} [[Mike Duff]], [[James Liu]], [[Ruben Minasian]], _Eleven Dimensional Origin of String/String Duality: A One Loop Test_, Nucl.Phys. B452 (1995) 261-282 ([arXiv:hep-th/9506126](https://arxiv.org/abs/hep-th/9506126))

* {#Witten96} [[Edward Witten]], _Five-Brane Effective Action In M-Theory_, J.Geom.Phys.22:103-133, 1997 ([arXiv:hep-th/9610234](https://arxiv.org/abs/hep-th/9610234))

For further discussion see

* [[Hisham Sati]], _[[Geometric and topological structures related to M-branes]]_ ,

  part I, Proc. Symp. Pure Math. 81 (2010), 181-236 ([arXiv:1001.5020](http://arXiv.org/abs/1001.5020)),

  part _II: Twisted $String$ and $String^c$ structures_, J. Australian Math. Soc. 90 (2011), 93-108 ([arXiv:1007.5419](http://arxiv.org/abs/1007.5419));

  part _III: Twisted higher structures_, Int. J. Geom. Meth. Mod. Phys. 8 (2011), 1097-1116 ([arXiv:1008.1755](http://arxiv.org/abs/1008.1755))

### From higher curvature corrections to 11d Supergravity

Derivation from classification of [[higher curvature corrections]] to [[D=11 supergravity]]:

* [[Paul Howe]], [[Dimitrios Tsimpis]], Around (56) in: _On higher-order corrections in M theory_, JHEP 0309 (2003) 038 ([arXiv:hep-th/0305129](https://arxiv.org/abs/hep-th/0305129))

* {#SoueresTsimpis17} [[Bertrand Souères]], [[Dimitrios Tsimpis]], Section 4 of: _The action principle and the supersymmetrisation of Chern-Simons terms in eleven-dimensional supergravity_, Phys. Rev. D 95, 026013 (2017) ([arXiv:1612.02021](https://arxiv.org/abs/1612.02021))

### In relation to the quantum anomaly of the M5-brane

In relation to the [[quantum anomaly]] of the [[M5-brane]]:

The original computation of the total M5-brane anomaly due to

* [Witten 96](#Witten96)

left a remnant term of $\tfrac{1}{24} p_2$. It was argued in

* {#FreedHarveyMinasianMoore98} [[Dan Freed]], [[Jeff Harvey]], [[Ruben Minasian]], [[Greg Moore]], _Gravitational Anomaly Cancellation for M-Theory Fivebranes_, Adv.Theor.Math.Phys.2:601-618, 1998 ([arXiv:hep-th/9803205](https://arxiv.org/abs/hep-th/9803205))

* {#HarveyMinasianMoore98b} [[Jeff Harvey]], [[Ruben Minasian]], [[Greg Moore]], _Non-abelian Tensor-multiplet Anomalies_, JHEP9809:004, 1998 ([arXiv:hep-th/9808060](https://arxiv.org/abs/hep-th/9808060))

* {#BilalMetzger03} [[Adel Bilal]], Steffen Metzger, _Anomaly cancellation in M-theory: a critical review_, Nucl.Phys. B675 (2003) 416-446 ([arXiv:hep-th/0307152](https://arxiv.org/abs/hep-th/0307152))

that this term disappears (cancels) when properly taking into account the singularity of the [[supergravity C-field]] at the locus of the [[black brane|black]] [[M5-brane]].

This formulation via an anomaly 12-form is (re-)derived also in

* {#BahBonettiMinasianNardoni18} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], [[Emily Nardoni]], _Class $\mathcal{S}$ Anomalies from M-theory Inflow_, Phys. Rev. D 99, 086020 (2019) ([arXiv:1812.04016](https://arxiv.org/abs/1812.04016))

* {#BBMN19} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], [[Emily Nardoni]], _Anomaly Inflow for M5-branes on Punctured Riemann Surfaces_ ([arXiv:1904.07250](https://arxiv.org/abs/1904.07250))

[[!redirects one-loop anomaly polynomial I8]]


Iain Gordon

__Iain Grant Gordon__ is an algebraist at Edinburgh.

* [webpage](https://www.maths.ed.ac.uk/~igordon), [wikipedia](https://en.wikipedia.org/wiki/Iain_Gordon)

### related $n$Lab entires

* [[affine Lie algebra]], [[symplectic reflection algebra]], [[double Hecke algebra]]

### Papers

* Iain G. Gordon, _Symplectic reflection algebras_, in: Trends in Representation Theory of Algebras and Related Topics (Andrzej Skowroński, editor)
* K. Brown, I. Gordon, _Poisson orders, symplectic reflection algebras and representation theory_, J. Reine Angew. Math. __559__ (2003) 193--216

category: people

[[!redirects Iain G. Gordon]]
[[!redirects I. G. Gordon]]
[[!redirects I. Gordon]]


Iain Moffat

* [webpage](http://www.personal.rhul.ac.uk/uxah/001/)

## Selected writings

On [[Vassiliev invariants|Vassiliev]] [[knot invariants]]:

* [[David Jackson]], [[Iain Moffat]], _An Introduction to Quantum and Vassiliev Knot Invariants_, Springer 2019 ([doi:10.1007/978-3-030-05213-3](https://link.springer.com/book/10.1007/978-3-030-05213-3))

category: people


Iakovos Androulidakis

* [webpage](http://scholar.uoa.gr/iandroul)

category: people


Ian Affleck

* [webpage](https://www.phas.ubc.ca/~iaffleck/)

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

## Selected writings

On the [[Schwinger effect]]:

* {#AffleckManton82} [[Ian Affleck]], [[Nicholas Manton]], _Monopole pair production in a magnetic field_, Nuclear Physics B Volume 194, Issue 1, (1982) Pages 38-64 (doi:10.1016/0550-3213(82)90511-9

* {#AffleckAlvarezManton82} [[Ian Affleck]], [[Orlando Alvarez]], [[Nicholas Manton]], _Pair production at strong coupling in weak external fields_, Nuclear Physics B Volume 197, Issue 3 (1982) Pages 509-519 (doi:10.1016/0550-3213(82)90455-2)

  > (via [[worldline formalism]])

On [[flag manifold sigma-models]]:

* [[Ian Affleck]], [[Dmitri Bykov]], [[Kyle Wamer]], *Flag manifold sigma models: spin chains and integrable theories*,  Phys. Rept. **953** (2022) 1-93 [[arXiv:2101.11638](https://arxiv.org/abs/2101.11638), [doi:10.1016/j.physrep.2021.09.004](https://doi.org/10.1016/j.physrep.2021.09.004)]

category: people


Ian Agol

* [webpage](http://math.berkeley.edu/~ianagol/)

## related $n$Lab entries

* [[virtually fibered conjecture]]

category: people


Ian Coley

[My website](http://iancoley.org)

####Interests####
* [Derivators](https://ncatlab.org/nlab/show/derivator)
* [K-theory](https://ncatlab.org/nlab/show/K-theory)

####Papers####
* [Stabilization of derivators revisited](http://arxiv.org/abs/1802.04343)


Ian Durham

I'm a mathematical physicist interested in foundational issues, particularly in relation to quantum theory.  I'm a "big picture" thinker which is why categories appeal to me.  I'm relatively new to category theory itself but I'm slowly getting there.

I'm an Associate Professor and Chair of the Department of Physics at [Saint Anselm College](http://www.anselm.edu).  I live on the coast of Maine with my wife, two pre-teen kids, dog, and a tankful of guppies.

I've had a blog, [Quantum Moxie](http://quantummoxie.wordpress.com/), for about four years that is a dumping ground for things that don't otherwise fit inside my brain.

I'm really interested in collaborating with people which is what attracted me to this site.  Specifically, I'd be interested in working with anyone who is interested in exploring further extensions of **categorical [[quantum channels]]** as well as anyone interested in the application of category theory to **quantum communication on closed time-like curves** (colloquially known as wormholes).  I've written a paper on the subject (not including the category theory part) that I'm in the process of trying to publish and think there's some more that could be said about the topic using category theory.  If you're interested, please contact me.

You can e-mail me at idurham [at] anselm [dot] edu.

I also have some project pages up at [Quantum Collaborations](http://quantumcollaborations.org).  If you join and are interested in collaborating, please let me know.

category: people


Ian G. Macdonald

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

## Selected writings

On [[commutative algebra]]:

* [[Michael Atiyah]], [[Ian G. Macdonald]], _Introduction to commutative algebra_ (1969, 1994) [[pdf](http://math.univ-lyon1.fr/~mathieu/CoursM2-2020/AMD-ComAlg.pdf), [ISBN:9780201407518](https://www.routledge.com/Introduction-To-Commutative-Algebra/Atiyah/p/book/9780201407518)]

On [[affine Lie algebras]] in relation to [[modular forms]]:

* [[Ian G. MacDonald]], *Affine Lie algebras and modular forms*, Séminaire Bourbaki: vol. 1980/81, exposés 561-578, Séminaire Bourbaki, no. 23 (1981), Exposé no. 577 [[numdam:SB_1980-1981__23__258_0](http://www.numdam.org/book-part/SB_1980-1981__23__258_0/)]

On [[hypergeometric functions]]:

* [[Ian G. Macdonald]], _Hypergeometric functions I_, 1987 ([arxiv/1309.4568](http://arxiv.org/abs/1309.4568))

Introducing [[Macdonald polynomials]]:

* [[Ian G. Macdonald]],  *A New Class of Symmetric Functions*, Publ. I.R.M.A. **372** (S-20) (1988) 131-171 [[web](https://www.mat.univie.ac.at/~slc/opapers/s20macdonald.html)]

* [[Ian G. Macdonald]], _Symmetric functions and Hall polynomials_, Oxford Math. Monographs, 2nd enlarged ed. (1995) [[pdf](https://math.berkeley.edu/~corteel/MATH249/macdonald.pdf)]

On [[symmetric functions]] and [[Hall polynomials]]:

* {#Macdonald95} [[Ian G. Macdonald]], _Symmetric functions and Hall polynomials_, Oxford Math. Monographs, 2nd enlarged ed. 1995 ([ISBN:9780198739128](https://global.oup.com/academic/product/symmetric-functions-and-hall-polynomials-9780198739128?cc=ae&lang=en&))

category: people

[[!redirects Ian Macdonald]]

[[!redirects Ian G. MacDonald]]
[[!redirects Ian MacDonald]]


Ian Grojnowski

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

## Selected writings

On [[equivariant elliptic cohomology]] and [[loop groups]]:

* {#Grojnowski94} [[Ian Grojnowski]], _Delocalised equivariant elliptic cohomology_ (1994), in _Elliptic cohomology_, volume 342 of London Math. Soc. Lecture Note Ser., pages 114–121. Cambridge Univ. Press, Cambridge, 2007 ([pdf](http://hopf.math.purdue.edu/Grojnowski/deloc.pdf))

On [[instantons]] and [[Hilbert scheme of points]]:

* [[Ian Grojnowski]], _Instantons and affine algebras I: The Hilbert scheme and vertex operators_ ([arXiv:alg-geom/9506020](https://arxiv.org/abs/alg-geom/9506020))

category: people


Ian Hambleton

* [personal page](https://ms.mcmaster.ca/ian/ian.html)

## Selected writings

On [[free group actions on n-spheres]] and [[spherical space forms]]:

* [[Ian Hambleton]], _Topological spherical space forms_, Handbook of Group Actions (Vol. II), ALM 32 (2014), 151-172. International Press, Beijing-Boston ([arXiv:1412.8187](https://arxiv.org/abs/1412.8187), [ISBN:9781571463012](https://www.intlpress.com/site/pub/pages/books/items/00000439/index.php))

category: people


Ian MacDonald Anderson

[[!redirects Ian Anderson]]

* [institute page](http://www.math.usu.edu/people/ian-anderson)

* [Mathematics Genealogy page](https://www.genealogy.math.ndsu.nodak.edu/id.php?id=22869)

* [publications](https://works.bepress.com/ian_anderson/)

## Writings

* _The variational bicomplex_, Utah State University 1989 ([pdf]([[AndersonVariationalBicomplex.pdf:file]]))

  on [[variational calculus]] via the [[variational bicomplex]] and [[Euler-Lagrange complex]]

## related $n$Lab entries

* [[variational calculus]]

* [[variational bicomplex]]

* [[Euler-Lagrange complex]]

* [[jet bundle]]

category: people


Ian Orton

[Homepage](https://www.cl.cam.ac.uk/~rio22/)

## Publications

* _Axioms for Modelling Cubical Type Theory in a Topos_, [[Ian Orton]] and [[Andrew M. Pitts]], [pdf](http://drops.dagstuhl.de/opus/volltexte/2016/6564/pdf/LIPIcs-CSL-2016-24.pdf) [Agda code](http://www.cl.cam.ac.uk/~rio22/agda/cubical-topos/root.html)

category: people


Ian Porteous

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

## selected writings

* {#Porteous95} [[Ian Porteous]], _Clifford Algebras and the Classical Groups_, Cambridge Studies in Advanced Mathematics, Cambridge University Press (1995)

  (on [[Clifford algebra]], [[spin groups]] and the classical [[Lie groups]])

## related $n$Lab entries

* [[Spin(5)]]

* [[quaternionic Hopf fibration]]

category: people


Ian Putnam

* [personal page](https://www.math.uvic.ca/faculty/putnam/)

* [Institute page](https://www.uvic.ca/science/math-statistics/people/home/faculty/putnam_ian.php)

## Selected writings

On [[C-star algebras|$C^\ast$-algebras]]:

* [[Ian Putnam]], *Lecture notes on $C^\ast$-algebras* (2019) [[pdf](https://www.math.uvic.ca/faculty/putnam/ln/C*-algebras.pdf), [[Putnam-CStarAlgebras.pdf:file]]]

category: people


Ian Stark

* [homepage](https://homepages.inf.ed.ac.uk/stark/)

## Selected writings

On a form of [[bunched logic|bunched]] [[dependent linear type theory]]:

* {#SchöppStark04} [[Ulrich Schöpp]], [[Ian Stark]], *A Dependent Type Theory with Names and Binding*, in *Computer Science Logic. CSL 2004*, Lecture Notes in Computer Science **3210** (2004) 235–249 [[doi:10.1007/978-3-540-30124-0_20](https://doi.org/10.1007/978-3-540-30124-0_20), [web](https://homepages.inf.ed.ac.uk/stark/names+binding.html)]

category: people


Ib Madsen

* [website](http://www.math.ku.dk/~imadsen/)

## Selected writings

On [[infinite loop spaces]]:

* [[Ib Madsen]], [[Victor Snaith]], [[Jørgen Tornehave]], *Infinite loop maps in geometric topology*, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 81, Issue 3, (1977)([doi:10.1017/S0305004100053482](https://doi.org/10.1017/S0305004100053482))

On [[free group actions on n-spheres]] and their [[spherical space forms]]:

* {#MadsenThomasWall76} [[Ib Madsen]], [[Charles B. Thomas]], [[C. T. C. Wall]], *The topological spherical space form problem II: existence of free actions*, Topology Volume 15, Issue 4, 1976, Pages 375-382 (doi:10.1016/0040-9383(76)90031-8)

* {#MadsenThomasWall83} [[Ib Madsen]], [[Charles B. Thomas]], [[C. T. C. Wall]], *Topological spherical space form problem III: Dimensional bounds and smoothing*,  Pacific J. Math. 106(1): 135-143 (1983) ([pjm:1102721110](https://projecteuclid.org/journals/pacific-journal-of-mathematics/volume-106/issue-1/Topological-spherical-space-form-problem-III-Dimensional-bounds-and-smoothing/pjm/1102721110.full))

* {#Madsen78} [[Ib Madsen]], *Smooth spherical space forms*, Geometric Applications of Homotopy Theory I. Springer 1978. 303-352 ([doi:10.1007/BFb0069224](https://link.springer.com/book/10.1007/BFb0069224),  [pdf](https://link.springer.com/content/pdf/10.1007/BFb0069243.pdf))

## Related $n$Lab entries

* [[cobordism categories]]

category: people


Ibrahima Bah

* [institute page](https://physics-astronomy.jhu.edu/directory/ibrahima-bah/)

## Selected writings

On [[M5-branes]] and their [[quantum anomaly]]-cancellation:

* {#BahBonettiMinasianNardoni18} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], [[Emily Nardoni]], _Class $\mathcal{S}$ Anomalies from M-theory Inflow_, Phys. Rev. D 99, 086020 (2019) ([arXiv:1812.04016](https://arxiv.org/abs/1812.04016))

* {#BBMN19} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], [[Emily Nardoni]], _Anomaly Inflow for M5-branes on Punctured Riemann Surfaces_,  J. High Energ. Phys. 2019, 123 (2019)
([arXiv:1904.07250](https://arxiv.org/abs/1904.07250))

* {#BBMN19b} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], [[Emily Nardoni]], _Anomalies of QFTs from M-theory and Holography_, J. High Energ. Phys. 2020, 125 (2020) ([arXiv:1910.04166](https://arxiv.org/abs/1910.04166))

* {#BahBonettiMinasian20} [[Ibrahima Bah]], [[Federico Bonetti]], [[Ruben Minasian]], _Discrete and higher-form symmetries in SCFTs from wrapped M5-branes_, J. High Energ. Phys. 2021, 196 (2021) ([arXiv:2007.15003](https://arxiv.org/abs/2007.15003))

On sesolution of [[scalar field]]-dressed [[Schwarzschild black holes]] in [[D=11 supergravity]]:

* [[Ibrahima Bah]], Pierre Heidmann, *Geometric Resolution of Schwarzschild Horizon* [[arXiv:2303.10186](https://arxiv.org/abs/2303.10186)]

category: people


Ichiro Satake

* [Wikipedia entry](https://en.wikipedia.org/wiki/Ichir%C5%8D_Satake)

## Selected writings

Introducing the concept of [[orbifolds]]:

* {#Satake} [[Ichiro Satake]], _On a generalisation of the notion of manifold_, Proc. Nat. Acad. Sci. U.S.A. 42 (1956), 359–363 ([doi:10.1073/pnas.42.6.359]( https://doi.org/10.1073/pnas.42.6.359))

and applying it to the [[Gauss-Bonnet theorem]]:

* [[Ichiro Satake]], _The Gauss–Bonnet theorem for $V$-manifolds_, J. Math. Soc. Japan 9 (1957), 464–492 ([euclid:1261153826](https://projecteuclid.org/euclid.jmsj/1261153826))

category: people


Ichiro Shimada

* [webpage](http://www.math.sci.hiroshima-u.ac.jp/shimada/index.html)

## Selected writings

On [[elliptically fibered K3-surfaces]]:

* {#Shimada00} [[Ichiro Shimada]], _On elliptic K3 surfaces_, Michigan Math. J. Volume 47, Issue 3 (2000), 423-446 ([euclid:mmj/1030132587](https://projecteuclid.org/euclid.mmj/1030132587))

category: people


Ichiro Yokota

* [MathGenealogy page](https://www.mathgenealogy.org/id.php?id=238773)

## Selected writings

On the [[Pontrjagin rings]] of the [[classical Lie groups]]:

* [[Ichiro Yokota]], *On the homology of classical Lie groups*, J. Inst. Polytech. Osaka City Univ. Ser. A **8** 2 (1957) 93-120 [[euclid:ojm/1353054814](https://projecteuclid.org/journals/osaka-journal-of-mathematics/volume-8/issue-2/On-the-homology-of-classical-Lie-groups/ojm/1353054814.full)]

On [[exceptional Lie groups]]:

* [[Ichiro Yokota]], _Exceptional Lie groups_ [[arXiv:0902.0431](http://arxiv.org/abs/0902.0431)]

category: people


ICM

__ICM__ is the abbreviation for the __International Congress of Mathematicians__, which takes place once per 4 years. It is organized by
International Mathematical Union ([[IMU]]).

The proceedings of ICM-s up to 2014 are free online at

* [ICM archive](http://www.mathunion.org/ICM)

The proceedings of ICM 2018 are available here:

* [ICM 2018](https://impa.br/icm2018/)

[[!redirects International Congress of Mathematicians]]


icon

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### 2-Category theory
+--{: .hide}
[[!include 2-category theory - contents]]
=--
=--
=--

# Icons
* table of contents
{: toc}

## Definition

### For 2-categories

Let $C,D$ be [[2-categories]] and $F,G:C\to D$ be [[functors]].  An **icon** $\alpha:F\to G$ consists of the following:

* the assertion that $F$ and $G$ agree on objects.
* for each morphism $u:x\to y$ in $C$, a 2-cell $\alpha_u:F(u) \to G(u)$ in $D$ (note that this only makes sense because $F$ and $G$ agree on objects, so that $F(u)$ and $G(u)$ are [[parallel morphism|parallel]].
* for each 2-cell $\mu:u\to v$ in $C$, we have $\alpha_v . F(\mu) = G(\mu).\alpha_u$.
* for each object $x$ of $C$, $\alpha_{1_x}$ is an identity (modulo the unit constraints of $F$ and $G$, if they are not strict functors).
* for each [[composable pair]] $x\overset{u}{\to} y \overset{v}{\to} z$ in $C$, we have $\alpha_v {*} \alpha_u = \alpha_{v u}$ (modulo the composition constraints of $F$ and $G$, if they are not strict functors).

If $D$ is a [[strict 2-category]] (or at least strictly unital), then an icon is identical to an [[oplax natural transformation]] whose 1-cell components are identities.  In general, there is a bijection between icons and such oplax natural transformations, obtained by pre- and post-composing with the unit constraints of $D$.  The name "icon" derives from this correspondence: it is an Identity Component Oplax Natural-transformation.

## Applications

Icons have technical importance in the theory of 2-categories.  For instance, there is no 2-category (or even 3-category) of 2-categories, functors, and lax or oplax transformations (even with modifications), but there is a 2-category of 2-categories, functors, and icons.  (In fact, this 2-category is the 2-category of algebras for a certain [[2-monad]].)

Additionally, if [[monoidal categories]] are regarded as one-object 2-categories, then monoidal functors can be identified with 2-functors, and monoidal transformations can be identified with icons.

Icons are also used to construct [[distributors]] in the context of enriched bicategories.

## Relation to pseudo double categories

A bicategory can be viewed as a [[pseudo double category]] whose tight-cells are trivial. An icon is then precisely a transformation of oplax functors of pseudo double categories. See [Paré](#Pare13) for details.

## References

An early observation that restricting to icons allows one to form a bicategory of bicategories and lax functors appears in:

* [[Aurelio Carboni]], and [[Robert Rosebrugh]]. _Lax monads: Indexed monoidal monads_, Journal of Pure and Applied Algebra 76.1 (1991): 13-32.

The terminology "ICON" was introduced in:

* {#Lack10} [[Stephen Lack]], _Icons_. Appl. Categ. Structures. Volume 18, Issue 3 (2010), pp 289–307. [arxiv:0711.4657](http://arxiv.org/abs/0711.4657)

* {#GarnerShulman16} [[Richard Garner]], [[Mike Shulman]], Section 3.8 of _Enriched categories as a free cocompletion_, Adv. Math. 289 (2016), 1--94

* {#Pare13} [[Robert Paré]]. _Composition of modules for lax functors_. Theory and Applications of Categories 27.16 (2013): 393-444.

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], Section 4.6 of: *2-Dimensional Categories*, Oxford University Press 2021 ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378))

[[!redirects icon]]
[[!redirects icons]]


icosahedral group

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Group Theory
+-- {: .hide}
[[!include group theory - contents]]
=--
#### Exceptional structures
+-- {: .hide}
[[!include exceptional structures -- contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

The _icosahedral group_ is the [[group]] of [[symmetries]] of an [[icosahedron]].

As a symmetry group of one of the [[Platonic solids]], the icosahedral group participates in the _ADE pattern_:

[[!include ADE -- table]]

More in detail, there are variants of the icosahedral group corresponding to the stages of the [Whitehead tower of O(3)](Whitehead+tower#OfTheOrthogonalGroup):

* the full icosahedral group is the [[subgroup]] of [[orthogonal group|O(3)]]

  $$
    I_h \hookrightarrow O(3)
  $$

  which is the [[stabilizer subgroup|stabilizer]] of the standard embedding of the [[icosahedron]] into [[Cartesian space]] $\mathbb{R}^3$;

* the _rotational icosahedral group_ $I \hookrightarrow SO(3)$ is the restriction to orientation-preserving symmetries, hence to [[special orthogonal group|SO(3)]]; this is [[isomorphism|isomorphic]] to the [[alternating group]] $A_5$

* finally the _binary icosahedral group_ is the [[double cover]] (see at _[[covering of alternating group]]_), hence the lift of $I$ to [[spin group|Spin(3)]]$\simeq$ [[special unitary group|SU(2)]];

* next there is a [[string 2-group]] lift $\mathcal{I} \hookrightarrow String_{SU(2)}$ of the icosahedral group ([Epa 10](#Epa10), [Epa-Ganter 16](#EpaGanter16))

$$
  \array{
     String_{2I} &\hookrightarrow& String_{SU(2)}
     \\
     \downarrow && \downarrow
     \\
     2 I &\hookrightarrow & Spin(3) \simeq SU(2)
     \\
     \downarrow && \downarrow
     \\
     I \simeq A_5 &\hookrightarrow&  SO(3)
     \\
     \downarrow && \downarrow
     \\
     I_h \simeq A_5\times \mathbb{Z}/2 &\hookrightarrow & O(3)
  }
$$

## Definition
 {#Definition}

Regard the [[icosahedron]], determined uniquely up to [[isometry]] on $\mathbb{R}^3$ as a regular convex [[polyhedron]] with $20$ faces, as a [[metric space|metric]] subspace $S$ of $\mathbb{R}^3$. Then the *icosahedral group* may be defined as the group of [[isometries]] of $S$.

(...)

The elements of the _binary icosahedral group_ form the [[vertices]] of the [[120-cell]].

More to be added.

## Properties

### General properties

The [[subgroup]] of [[orientation]]-preserving symmetries of the [[icosahedron]] is the [[alternating group]] $A_5$ whose [[order of a group|order]] is 60. The full icosahedral group is [[isomorphism|isomorphic]] to the [[Cartesian product]] $A_5 \times \mathbb{Z}_2$ (with the [[group of order 2]]).

Hence the [[order of a group|order]] of the full icosahedral group is $ 60 \times 2 = 120 $, as is that of the binary icosahedral group $2 I$.

+-- {: .num_prop #IsomorphismToSL25}
###### Proposition

There is an [[exceptional isomorphism]]

$$
  I \;\simeq\; PSL_2(\mathbb{F}_5)
$$

of the [[icosahedral group]] with the [[projective special linear group]] over the [[prime field]] $\mathbb{F}_5$.

and, covering this,

$$
  2I \;\simeq\; SL_2(\mathbb{F}_5)
$$

of the [[binary icosahedral group]] with the [[special linear group]] over $\mathbb{F}_5$.

=--

+-- {: .num_prop #2IIsPerfect}
###### Proposition

The [[binary icosahedral group]] $2I$ is a [[perfect group]]: its [[abelianization]] is the [[trivial group]].

In fact, up to [[isomorphism]], the [[binary icosahedral group]] is the unique [[finite group]] of [[order of a group|order]] 120 which is a [[perfect group]].

=--

+-- {: .num_prop #QuaternionSubgroup}
###### Proposition
**([[quaternion group]] inside [[binary icosahedral group]])**

The [[binary icosahedral group]] contains the [[quaternion group]] of [[order]] 8, hence the [[binary dihedral group]] of [[order of a group|order]] 8, as a [[subgroup]] (not [[normal subgroup|normal]]):

$$
  2 D_4 =Q_8 \subset 2 I
  \,.
$$

In fact the only [[finite subgroups of SU(2)]] which contain $2 D_4 =Q_8$ as a proper subgroup are the exceptional ones, hence the [[binary tetrahedral group]], the [[binary octahedral group]] and the [[binary icosahedral group]].

=--

See [this Prop](quaternion+group#InclusionInLargerFininteSubgroupsOfSU2) at _[[quaternion group]]_.

+-- {: .num_prop #NormalSubgroupsOf2I}
###### Proposition
**([[normal subgroups]] of [[binary icosahedral group]]

The only proper [[normal subgroup]] of the [[binary icosahedral group]] is its [[center]] $Z(2I) \simeq \mathbb{Z}/2$.

=--

### Character table

[[!include character table of 2I]]

### Quotient spaces

The [[coset space]] $SU(2)/2I$ is the [[Poincaré homology sphere]].

### Group cohomology

For a little bit about the [[group cohomology]] (or at least the homology) of the binary icosahedral group $SL_2(\mathbb{F}_5)$, see *[Tomoda & Zvengrowski 08, Section 4.3](#TomodaZvengrowski08)*, *[Epa & Ganter 16, p. 12](#EpaGanter16)* _[Groupprops](#GrouppropsGroupCohomology)_

## Related concepts

* [[Platonic 2-group]]

## References

Origin:

* {#Klein1884} [[Felix Klein]], chapter I.8 of: _Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade_, 1884, translated as _Lectures on the Icosahedron and the Resolution of Equations of Degree Five_ by George Morrice 1888, [online version](https://archive.org/details/cu31924059413439)

See also:

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

* {#Groupprops} [[Groupprops]], _[Alternating group:A5](http://groupprops.subwiki.org/wiki/Alternating_group:A5)_

* [[Philip Boalch]], _The fifty-two icosahedral solutions to Painlevé VI_, J. Reine Angew. Math. 596 (2006) 183--214 ([arXiv:0406281](http://arxiv.org/abs/math/0406281))

Discussion of the [[group cohomology]]:

* {#TomodaZvengrowski08} [[Satoshi Tomoda]], [[Peter Zvengrowski]], Section 4.3 of: *Remarks on the cohomology of finite fundamental groups of 3-manifolds*, Geom. Topol. Monogr. 14 (2008) 519-556 ([arXiv:0904.1876](https://arxiv.org/abs/0904.1876))

* {#GrouppropsGroupCohomology} _[Group cohomology of algebrating group:A5](http://groupprops.subwiki.org/wiki/Group_cohomology_of_alternating_group:A5)_

Discussion of [[Platonic 2-group]]-extensions:

* {#Epa10} [[Narthana Epa]], _Platonic 2-groups_, 2010 ([pdf](http://www.ms.unimelb.edu.au/documents/thesis/Epa-Platonic2-Groups.pdf))

* {#EpaGanter16} [[Narthana Epa]], [[Nora Ganter]], _Platonic and alternating 2-groups_, ([arXiv:1605.09192](http://arxiv.org/abs/1605.09192))

[[!redirects rotational icosahedral group]]
[[!redirects binary icosahedral group]]
[[!redirects stringy icosahedral group]]

[[!redirects 2I]]


icosahedron

#Contents#
* table of contents
{:toc}

## Idea

 (image source)

 (image source)

The _regular icosahedron_ is the unique [[regular polyhedron|regular]] [[convex set|convex]] [[polyhedron]] with 20 ([[triangle|triangular]]) faces. One of the [[Platonic solids]].

Its [[symmetry]] [[group]] is called the [[icosahedral group]].

$\,$

$\,$

$\,$

$\,$

$\,$

[[!include ADE -- table]]

## References

* {#Klein1884} [[Felix Klein]], chater I.8 of _Vorlesungen über das Ikosaeder und die Auflösung der Gleichungen vom fünften Grade_, 1884, translated as _Lectures on the Icosahedron and the Resolution of Equations of Degree Five_ by George Morrice 1888, [online version](https://archive.org/details/cu31924059413439)

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


ideal

# Contents
* table of contents
{: toc}

## Definitions

Ideals show up both in [[ring]] theory and in [[lattice]] theory.  We recall both of these below and look at some slight generalizations.

### In rings (and other rigs)

A __left ideal__ in a [[ring]] (or even [[rig]]) $R$ is a [[subset]] $I$ of (the underlying set of) $R$ such that:

*  $0 \in I$;
*  $x + y \in I$ whenever $x, y \in I$;
*  $x y \in I$ whenever $y \in I$, regardless of whether $x \in I$.

A __right ideal__ in $R$ is a subset $I$ such that:

*  $0 \in I$;
*  $x + y \in I$ whenever $x, y \in I$;
*  $x y \in I$ whenever $x \in I$.

A __two-sided ideal__ in $R$ is a subset $I$ that is both a left and right ideal; that is:

*  $0 \in I$;
*  $x + y \in I$ whenever $x \in I$ and $y \in I$;
*  $x y \in I$ whenever $x \in I$ or $y \in I$.

This generalises to:

*  $x_1 + \cdots + x_n \in I$ whenever $x_k \in I$ for every $k$;
*  $x_1 \cdots x_n \in I$ whenever $x_k \in I$ for some $k$.

Notice that all three kinds of ideal are equivalent for a commutative ring.

+-- {: .num_remark}
###### Remarks
* A left ideal in a ring $R$ may be equivalently defined as an $R$-submodule of $R$, viewing the latter as a left $R$-[[module]].
* A right ideal in a ring $R$ may be equivalently defined as an $R$-submodule of $R$, viewing the latter as a right $R$-[[module]].
* A two-sided ideal in a ring $R$ may be equivalently defined as a sub-bimodule of $R$, viewing the latter as an $R$-[[bimodule]].
* The collection of all two-sided ideals in a ring $R$ form a [[concrete category]], see [[category of two-sided ideals in a ring]]
* The preceding remarks apply to rigs as well.
* Considering the category of rings as a Barr-exact category, there is a natural bijection between [[congruence|congruence relations]] on a ring $R$ (internal to the category of rings) and two-sided ideals of $R$; this associates to each ideal $I$ the relation $\sim_I$ where $x \sim_I y$ means $x - y \in I$. This observation does not apply to the category of rigs.
=--

This definition also makes sense for [[nonassociative rings]] (or rigs), with the left/right/two-sided coincidence in the commutative case.

In the case of a [[nonunital ring]] (or rig), if $R$ is being thought of as a [[module]] over some other ring $K$, then it won\'t immediately follow that $I$ is a [[submodule]] of $R$ over $K$, and so one usually includes that requirement as well:

*  $a x \in I$ whenever $a \in K$ and $x \in I$

in the case of left-modules, and similarly in the case of right modules or [[bimodules]].  (Technically, this should be distinguished in the terminology, say by calling $I$ an __ideal of $R$ over $K$__ or the like.)  In particular, a (real or complex) [[Lie ideal]] of a [[Lie algebra]] $R$ is an ideal of $R$ over the real or complex field $K$.

### In lattices (and other prosets)

An __ideal__ in a [[lattice]] (or even [[preorder|proset]]) $L$ is a [[subset]] $I$ of (the underlying set of) $L$ such that:

*  There is an element of $I$ (so that $I$ is [[inhabited set|inhabited]]);
*  if $x, y \in I$, then $x, y \leq z$ for some $z \in I$;
*  if $x \in I$ and $y \leq x$, then $y \in I$ too.

We can make this look more algebraic if $L$ is a (bounded) join-[[semilattice]]:

*  $\bot \in I$;
*  $x \vee y \in I$ if $x, y \in I$;
*  $y \in I$ whenever $x \vee y \in I$.

If $L$ is indeed a lattice, then we can make this look just like the ring version:

*  $\bot \in I$;
*  $x \vee y \in I$ whenever $x, y \in I$;
*  $x \wedge y \in I$ whenever $x \in I$.

The concept of ideal is dual to that of [[filter]].  A subset of $L$ that satisfies the first two of the three axioms for an ideal in a proset is precisely a [[direction|directed subset]] of $L$; notice that this is weaker than being a sub-join-semilattice even if $L$ is a lattice.

### In both at once

There are some common situations where these two kinds of ideal might seem to clash but fortunately do not:

* A [[distributive lattice]] is both a lattice and a commutative rig; the two concepts of ideal are the same, as can be seen by comparing the definition for rigs to the last definition for lattices.

* A [[Boolean algebra]] is a both a distributive lattice and a [[Boolean ring]]; again, the two concepts of ideal are the same (partly because the multiplication operators are the same, although there is still some checking to do regarding closure under addition).

On the other hand, every poset is a poset in an [[opposite poset|opposite]] way, and this does *not* give the same concept of ideal; an ideal in one is a [[filter]] in the opposite one.  We are lucky that the convention for interpreting a Boolean ring as a lattice goes in the correct direction, or the two notions of ideal in a Boolean algebra would not match; or perhaps it is not a matter of luck, but the convention for which way to define ideals in a lattice was chosen precisely to match the conventions for Boolean algebras!

### In monoids

There is a notion of ideal in a [[monoid]] (or even [[semigroup]]), or more generally in a [[monoid object]] in any [[monoidal category]] $C$, which generalises the notion of ideal in a ri(n)g or in a (semi)lattice.  That is, if $C$ is [[Ab]], then a monoid in $C$ is a [[ring]]; if $C$ is [[Ab Mon]], then a monoid in $C$ is a [[rig]]; and a [[semilattice]] is a commutative idempotent monoid in [[Set]].  A left (right) ideal in a monoid (or semigroup) is a subset closed under multiplications with arbitrary elements in the semigroup from the left (right). See [[ideal in a monoid]].

This generalizes all of the above notions of ideal *except* for ideals in prosets that are not (possibly unbounded) join-semilattices.

### In categories

More generally still, passing from monoids to their many-object version and from prosets to their many-morphism version, a right ideal or _[[sieve]]_ in a [[category]] is a subcategory closed under precomposition with morphisms from the entire category, and a cosieve or left ideal is the dual notion closed under postcompositions with morphisms in the entire category.  See [[sieve]].

### In additive categories

In ringoids (small additive categories) and additive categories in general, a left/right/2-sided ideal is just the ideal (sieve/cosieve) in the sense of categories which is also closed under group operations in hom groups.

## Kinds of ideals

Ideals form [[complete lattices]] where arbitrary meets are given by set-theoretic intersection. In other words, ideals form a [[Moore collection]] of subsets of $R$ if $R$ is a rig, or of $L$ if $L$ is a lattice. This implies we have an ideal __generated__ by any [[subset]]: the intersection of all ideals containing the subset.  A subset $S$ that generates a given ideal $I$ may be called a __[[subbase]]__ of $I$; then $S$ is a __[[base]]__ if every element of $I$ is a multiple (in a rig) or a predecessor (in an order) of some element of $S$.  (In particular, every [[singleton subset]] is a base of its generated ideal.)  See also [[filter base]] and dualize for more about bases and subbases of ideals in lattices and other posets.

Certain kinds of ideals are often characterized by the roles they play in ideal lattices, or in terms of the Moore [[closure operator]]. Some examples follow.

The top element of an ideal lattice is called the *improper ideal*. That is to say, an ideal $I$ is the improper ideal if $x \in I$ for every $x$ (which follows if $1 \in I$ for the case of rigs, or $\top \in I$ for the case of bounded lattices). An ideal $I$ is _proper_ if it is not the improper ideal: if there exists an element $x$ such that $x \notin I$.  So in a [[rig]], $I$ is proper iff $1 \notin I$; in a (bounded) lattice, $I$ is proper iff $\top \notin I$.

An ideal is a __[[maximal ideal]]__ if it is maximal among *proper* ideals.  A maximal ideal in a rig (including in a distributive lattice, but not in every lattice) is necessarily [[prime ideal|prime]]; conversely, a prime ideal in a [[Boolean algebra]] is necessarily maximal.

An ideal $I$ is a __[[principal ideal]]__ if it is generated by a singleton. This means there exists an element $x \in I$ such that $y$ is a multiple of $x$ (in a rig) or $y \leq x$ (in an ordered set) whenever $y \in I$; we say that $I$ is __generated__ by $x$. Thus every element $x$ generates a unique principal ideal, the set of all left/right/two-sided multiples of $x$ ($a x$, $x b$, or $a x b$ if we are talking about left/right/two-sided ideals in a rig) or the [[downset]] of $x$ (in an an order). Clearly, every ideal $I$ is a join over all the principal ideals $P_x$ generated by the elements $x$ of $I$.

As discussed at [[ideals in a monoid]], there is for two-sided ideals an operation of ideal multiplication, making the ideal lattice a [[quantale]] (cf. [[Day convolution]]). Namely, if $I, J$ are ideals, then their product $I J$ is the ideal generated by all products $x y$ with $x \in I, y \in J$ in the case of rigs. Similarly, in the case of lattices, we could define $I J$ to be the ideal generated by all meets $x \wedge y$ -- but in this case the result is the same as $I \cap J$. In any case, we say that a proper ideal $P$ is __[[prime ideal|prime]]__ if for any ideals $I, J$, the condition $I J \subseteq P$ implies $I \subseteq P$ or $J \subseteq P$. In [[order theory]], the term 'prime ideal' is usually used for a _[[strongly irreducible ideal]]_, since the two are equivalent for prosets.

An (left, right, or two-sided) ideal $I$ is __[[irreducible ideal|irreducible]]__ if it is proper and, whenever it is written as the [[intersection]] of two ideals (of the same kind) $I = J \cap K$, then at least one of $J$ and $K$ equals $I$.  If we replace equality here by containment, then we get something more analogous to the definition of prime ideal, called a __[[strongly irreducible ideal]]__.  (Indeed, for [[prosets]], prime ideals and strongly irreducible ideals are the same.)  In general, prime ideals are strongly irreducible, and strongly irreducible ideals are irreducible, but the converses fail.  An ideal is __[[completely irreducible ideal|completely irreducible]]__ if whenever $I$ is an intersection of any (possibly infinite) collection of ideals, $I$ is equal to at least one of them.  (This is _not_ analogous to [[completely prime ideals]].)

An ideal $I$ is a __nil ideal__ if for every $x\in I$, there is an $n\in \mathbb{N}$ with $x^n=0$. That is, every element of the ideal is nilpotent. If on the other hand, there is an $n\in \mathbb{N}$, such that for every $x\in I$, $x^n=0$, the ideal is called __nilpotent__.

+-- {: .num_prop}
###### Proposition
A maximal ideal $M$ is prime.
=--

+-- {: .proof}
###### Proof
Because the ideal lattice is a [[quantale]], multiplication of ideals distributes over ideal joins. Suppose $I J \subseteq M$ for two ideals $I, J$. If neither is contained in $M$, then $I \vee M = \top = J \vee M$ (the improper ideal) since $M$ is maximal. Then

$$\top = \top \cdot \top = (I \vee M) \cdot (J \vee M) = I J \vee M J \vee I M \vee M M$$

where all four summands are contained in $M$ ($I J \subseteq M$ by supposition, and the other containments hold since $M$ is an ideal). Thus their join is contained in $M$, so we have proved $\top \subseteq M$, contradiction.
=--

That every ideal is contained in a prime ideal is a [[prime ideal theorem]]; that every ideal is contained in a maximal ideal is a [[maximal ideal theorem]].

## Related concepts

* [[differential ideal]]

* [[germ-determined ideal]]

* [[ideal in a monoid]]

* [[filter of a ring]]

* [[modular ideal]]

* [[ideal predicate]]

[[!redirects ideal]]
[[!redirects ideals]]

[[!redirects ideal of a ring]]
[[!redirects ideals of a ring]]
[[!redirects ideals of rings]]
[[!redirects ideal in a ring]]
[[!redirects ideals in a ring]]
[[!redirects ideals in rings]]
[[!redirects ideal of a semiring]]
[[!redirects ideals of a semiring]]
[[!redirects ideals of semirings]]
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[[!redirects ideals in a semiring]]
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[[!redirects ideal of a rig]]
[[!redirects ideals of a rig]]
[[!redirects ideals of rigs]]
[[!redirects ideal in a rig]]
[[!redirects ideals in a rig]]
[[!redirects ideals in rigs]]

[[!redirects ideal of a lattice]]
[[!redirects ideals of a lattice]]
[[!redirects ideals of lattices]]
[[!redirects ideal in a lattice]]
[[!redirects ideals in a lattice]]
[[!redirects ideals in lattices]]
[[!redirects ideal of a semilattice]]
[[!redirects ideals of a semilattice]]
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[[!redirects left ideal]]
[[!redirects left ideals]]
[[!redirects right ideal]]
[[!redirects right ideals]]
[[!redirects two-sided ideal]]
[[!redirects two-sided ideals]]

[[!redirects proper ideal]]
[[!redirects proper ideals]]

[[!redirects generated ideal]]
[[!redirects generated ideals]]
[[!redirects ideal generated]]
[[!redirects ideals generated]]
[[!redirects ideal base]]
[[!redirects ideal bases]]
[[!redirects base of an ideal]]
[[!redirects bases of an ideal]]
[[!redirects bases of ideals]]
[[!redirects ideal subbase]]
[[!redirects ideal subbases]]
[[!redirects subbase of an ideal]]
[[!redirects subbases of an ideal]]
[[!redirects subbases of ideals]]


ideal class group

##Definition

The ideal class group $C_K$ of a number field $K$ is defined as the group of fractional ideals of $\mathcal{O}_K$ modded out by the group of principal ideals in $\mathcal{O}_K$ (both groups freely generated as $K$-modules). An equivalent definition is that $C_K$ := $Pic(Spec \text{ } \mathcal{O}_K)$.

This was created to measure how far $\mathcal{O}_K$ is from being a PID.

More generally, for $R$ a Dedekind domain, we define $C_R$ to be the group of fractional ideals of $R$ modded out by the group of principal ideals (freely generated as $Frac(R)$-modules). In other words, $C_R$ := $Pic(Spec \text{ } R)$.

## Properties

### Relation to the idele class group

A much larger variant of the ideal class group is the [[idele class group]]...

### Relation to the pole of the zeta function

The [[Dedekind zeta function]] $\zeta_K$ of the [[number field]] $K$ has a [[simple pole]] at $s = 1$. The _[[class number formula]]_ says that its [[residue]] there is proportional the the product of the [[regulator of a number field|regulator]] with the [[class number]] of $K$

$$
  \underset{s\to 1}{\lim} (s-1) \zeta_K(s)
  \propto
  ClassNumber_K \cdot Regulator_K
  \,.
$$

## References

* Wikipedia, _[Ideal class group](http://en.wikipedia.org/wiki/Ideal_class_group)_


ideal completion

The ideal completion of a [[category]] $C$ generalises the notion of [[ideal]] completion of a [[partially ordered set]]. It is formed by taking the category of [[ideals]] of $C$. It is the completion of $C$ under [[filtered colimit]]s of [[monomorphisms]], thus producing an [[M-complete category]].

See page 24 of [An Outline of Algebraic Set Theory](http://www.phil.cmu.edu/projects/ast/Papers/awodey_outline.pdf).

"The actual definition requires either some care in specifying choices of monomorphisms, as is done in [4], or a sheaf-theoretic approach as in [5]."

* [4] S. Awodey, C. Butz, A. Simpson and T. Streicher, Relating first-order set theories, toposes and categories of classes. In preparation, 2007. Preliminary version available [here](http://www.phil.cmu.edu/projects/ast/).

* [5] S. Awodey and H. Forssell, [Algebraic models of intuitionistic theories of sets and classes](http://www.tac.mta.ca/tac/volumes/15/5/15-05abs.html), Theory and Applications of Categories 15(1): 147-163, 2005.

This is part of [[algebraic set theory]].

* Steve Awodey, Henrik Forssell and Michael A. Warren,
_Algebraic models of sets and classes in categories of ideals_ [PDF](http://www.phil.cmu.edu/projects/ast/Papers/afw_06.pdf)

## Related pages

* [[M-complete category]]


ideal in a monoid

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Monoid theory
+-- {: .hide}
[[!include monoid theory - contents]]
=--
=--
=--

# Contents
* table of contents
{: toc}

## Definition

Given a [[monoid]] (or [[semigroup]]) $S$, a __left ideal__ in $S$ is [[subset]] $A$ of $S$ such that $S A$ is contained in $A$.  Similarly, a __right ideal__ is a subset $A$ such that $A S \subseteq A$.  Finally, a __two-sided ideal__, or simply __ideal__, in $S$ is a subset $A$ that is both a left ideal and a right ideal.

Given a [[monoidal category]] $(C, \otimes, I)$ and a [[monoid object]] (or semigroup object) $S$ of $C$, we can [[internalisation|internalise]] the above.  For instance, if $m: S \otimes S \to S$ is the binary multiplication and $\mu = m \circ (m \otimes 1_S) = m \circ (1_S \otimes m): S \otimes S \otimes S \to S$ the ternary multiplication, a two-sided ideal is a [[subobject]] $A$ of $S$, i.e., a mono $i: A \to S$ in $C$, such that the composite

$$S \otimes A \otimes S \stackrel{1_S \otimes i \otimes 1_S}{\to} S \otimes S \otimes S \stackrel{\mu}{\longrightarrow} S$$

factors through $i: A \to S$. Clearly $i: A \to S$ is not necessarily a [[submonoid]], inasmuch as the monoid unit $e: I \to S$ need not factor through $i: A \to S$.

In particular for $C = $ [[Ab]], a monoid in $C$ is a [[ring]] and the corresponding notion of _[[ideal in a ring]]_ is the most common notion of ideal.

See [[ideal]] for ideals in more well known contexts: commutative idempotent monoids ([[semilattices]]) and monoids in [[Ab]] ([[rings]]).

## Properties and constructions

An ideal $A$ (on either side) must be a [[subsemigroup]] of $S$, but it is a [[submonoid]] iff $1 \in A$, in which case $A = S$.

### Ideals forming a quantale

(Two-sided) ideals of a monoid $A$ are frequently the elements of a [[quantale]] whose multiplication is called taking the *product of ideals*. In the classical case of ideals over a [[ring]] $R$, the product $I J$ of ideals $I, J \subseteq R$ is the smallest ideal containing all products $i j: i \in I, j \in J$; the sup-lattice of such ideals ordered by inclusion is a [[residuated lattice]], in that there are also division operations where

$$K/J \coloneqq \{r \in R: r J \subseteq K\}; \qquad I\backslash K = \{r \in R: I r \subseteq K\}$$

satisfying the expected [[adjunction|adjointness]] relations: $I \subseteq K/J$ iff $I J \subseteq K$ iff $J \subseteq I\backslash K$.

A reasonably general context might be as follows.

Let $\mathbf{C}$ be a [[well-powered category|well-powered]] [[regular category|regular]] [[cosmos]] ('cosmos' in the sense of complete cocomplete symmetric monoidal closed category). Just using the fact that $\mathbf{C}$ is a cosmos, we may construct a [[monoidal bicategory]] $Mod(\mathbf{C})$ whose objects are monoids $S$ in $\mathbf{C}$, whose 1-morphisms $S \to T$ are left-$S$ right-$T$ [[modules]], and whose 2-morphisms are bimodule homomorphisms.

For each monoid $S$, there is a subbicategory of $Mod(\mathbf{C})$ whose only object is $S$; this is a  complete and cocomplete [[biclosed monoidal category]] $Mod_S$ whose objects are bimodules, i.e., 1-morphisms $S \to S$ in $Mod(\mathbf{C})$, and whose morphisms are bimodule homomorphisms. The unit of the monoidal product is $S$ with its standard $S$-bimodule structure, and hence the [[slice category|slice]] $Mod_S/S$ (see also [[semicartesian monoidal category]]) forms another complete and cocomplete biclosed monoidal category.

An ideal of $S$ is just a subobject of $S$ in $Mod_S$. Under the assumption that $\mathbf{C}$ is well-powered, the category of subobjects $Sub(S) \hookrightarrow Mod_S/S$ is a (small) sup-lattice. Under the regularity assumption on $\mathbf{C}$, the subcategory $Sub(S) \hookrightarrow Mod_S/S$ is [[reflective subcategory|reflective]], and by applying the reflector to the monoidal product on $Mod_S/S$, we obtain a product on $Sub(S)$ which preserves arbitrary joins in each variable, hence a quantale. The unit of the quantale is the top element, namely $S$ considered as an ideal.

## Related concepts

* [[sieve]]

* [[principal ideal of a monoid]]

* [[ideal of a ring]]

[[!redirects ideal of a monoid]]
[[!redirects ideals of a monoid]]
[[!redirects ideals of monoids]]
[[!redirects ideal in a monoid]]
[[!redirects ideals in a monoid]]
[[!redirects ideals in monoids]]

[[!redirects ideal of a semigroup]]
[[!redirects ideals of a semigroup]]
[[!redirects ideals of semigroups]]
[[!redirects ideal in a semigroup]]
[[!redirects ideals in a semigroup]]
[[!redirects ideal in semigroups]]
[[!redirects ideals in semigroups]]


ideal predicate

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

\tableofcontents

## Idea

In [[material set theory]], an ideal of a [[commutative ring]] $R$ is a subset $I \subseteq R$ which satisfies

* $0 \in I$
* for all $x \in R$ and $y \in R$, $x \in I$ and $y \in I$ implies that $x + y \in I$
* for all $x \in R$ and $y \in R$, $x \cdot y \in I$ implies that $x \in I$ or $y \in I$

In [[dependently sorted set theory]], where membership $x \in S$ is not a [[relation]], the above statement that $x \in I$ for every element $x \in R$ in material set theory is equivalently a predicate in the logic $x \in R \vdash P_I(x)$. The given ideal $I$ is then defined by [[restricted separation]] as the set $I \coloneqq \{x \in R \vert P_I(x)\}$, which in structural set theory automatically comes with an [[injection]]
$$i:\{x \in R \vert P_I(x)\} \hookrightarrow R$$
such that
$$\exists y \in \{x \in R \vert P_I(x)\}.x = i(y) \iff P(x)$$
Hence, the notion of ideal predicate, a formulation of the notion of ideal as a predicate rather than a subset.

## Definition

Given a commutative ring $R$, an **ideal predicate** on $R$ is a predicate $x \in R \vdash P_I(x)$ which satisfies

* $P_I(0)$
* for all $x \in R$ and $y \in R$, $P_I(x)$ and $P_I(y)$ implies that $P_I(x + y)$
* for all $x \in R$ and $y \in R$, $P_I(x \cdot y)$ implies that $P_I(x)$ or $P_I(y)$

The **[[ideal]]** $I$ is then defined by [[restricted separation]] as
$I \coloneqq \{x \in R \vert P_I(x)\}$

## Examples

The various definitions of ideals translate over from material set theory to ideal predicates in dependently sorted structural set theory by replacing $x \in I$ with $P_I(x)$ throughout the definition:

* A **[[proper ideal]] predicate** on a commutative ring $R$ is an ideal predicate $P_I$ where $P_I(1)$ is false.

* A **[[maximal ideal]] predicate** on a commutative ring $R$ is an ideal predicate $P_I$ where for all $x \in R$, $P_I(x)$ being false implies that $x$ is an invertible element.
$$\forall x \in R.(\neg P_I(x)) \implies \exists y \in R.x \cdot y = 1$$

* A **[[prime ideal]] predicate** on a commutative ring $R$ is a proper ideal predicate $P_I$ where for all $x \in R$ and $y \in R$, $P_I(x \cdot y)$ implies that $P_I(x)$ or $P_I(y)$.
$$\forall x \in R.\forall y \in R.P_I(x \cdot y) \implies P_I(x) \vee P_I(y)$$

* A **[[principal ideal]] predicate** on a commutative ring $R$ generated by an element $a \in R$ is an ideal predicate $P_I$ where for all $x \in R$, $P_I(a \cdot x)$.

## See also

* [[ideal]]
* [[predicate]]
* [[anti-ideal predicate]]
* [[restricted separation]]

[[!redirects ideal predicate]]
[[!redirects ideal predicates]]


idealism

+-- {: .rightHandSide}
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### Context
#### Philosophy
+-- {: .hide}
[[!include philosophy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

According to [[Georg Hegel]]:

> [WdL §316](Science+of+Logic#316) Anmerkung 2. Der Satz, daß das Endliche ideell ist, macht den Idealismus aus. Der Idealismus der Philosophie besteht in nichts anderem, als darin, das Endliche nicht als ein wahrhaft Seyendes anzuerkennen. Jede Philosophie ist wesentlich Idealismus, oder hat denselben wenigstens zu ihrem Princip, und die Frage ist dann nur, inwiefern dasselbe wirklich durchgeführt ist. Die Philosophie ist es so sehr als die Religion; denn die Religion anerkennt die Endlichkeit ebenso wenig als ein wahrhaftes Seyn, als ein Letztes, Absolutes, oder als ein Nicht-Gesetztes, Unerschaffenes, Ewiges. Der Gegensatz von idealistischer und realistischer Philosophie ist daher ohne Bedeutung. Eine Philosophie, welche dem endlichen Daseyn als solchem wahrhaftes, letztes, absolutes Seyn zuschriebe, verdiente den Namen Philosophie nicht; Principien älterer oder neuerer Philosophien, das Wasser, oder die Materie oder die Atome sind Gedanken, Allgemeine, Ideelle, nicht Dinge, wie sie sich unmittelbar vorfinden, d. h. in sinnlicher Einzelnheit, selbst jenes thaletische Wasser nicht; denn, obgleich auch das empirische Wasser, ist es außerdem zugleich das Ansich oder Wesen aller anderen Dinge; und diese sind nicht selbstständige, in sich gegründete, sondern aus einem Anderen, dem Wasser, gesetzte, d. i. ideelle. Indem vorhin das Princip, das Allgemeine, das Ideelle genannt worden, wie noch mehr der Begriff, die Idee, der Geist, Ideelles zu nennen ist, und dann wiederum die einzelnen sinnlichen Dinge als ideell im Princip, im Begriffe, noch mehr im Geiste, als aufgehoben sind, so ist dabei auf dieselbe Doppelseite vorläufig aufmerksam zu machen, die bei dem Unendlichen sich gezeigt hat, nämlich daß das eine Mal das Ideelle das Konkrete, Wahrhaftseyende ist, das andere Mal aber ebenso sehr seine Momente das Ideelle, in ihm Aufgehobene sind, in der That aber nur das Eine konkrete Ganze ist, von dem die Momente untrennbar sind.

## Related entries

* [[doctrine of ideas]]

* idealism

  * [[objective idealism]], [[objective logic]]

  * [[subjective idealism]]

  * [[absolute idealism]]

* [[universal exceptionalism]]

* [[empiricism]]

* [[structuralism]]

## References

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

* Wikipedia, _[German idealism](http://en.wikipedia.org/wiki/German_idealism)_

* Encyclopedia Britannica, _[Idealism](http://www.britannica.com/EBchecked/topic/281802/idealism)_

* {#Ewing74} [[Alfred Ewing]], _Idealism: A Critical Survey_, Barnes & Noble (1974) ([PhiPapers](http://philpapers.org/rec/EWIIAC), [Review on JSTOR](http://www.jstor.org/stable/2249864))

[[!redirects German idealism]]


idempotent

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

# Idempotents
* table of contents
{: toc}

## Idea

The notion of an _idempotent_ [[morphism]] in a [[category]] generalizes the notion of _[[projector]]_ in the context of [[linear algebra]]: it is an [[endomorphism]] $e \colon X \to X$ of some [[object]] $X$ that "squares to itself" in that the [[composition]] of $e$ with itself is again $e$:

$$
  e \circ e = e
  \,.
$$

Accordingly, given any idempotent $e \colon X \to X$ it is of interest to ask what [[subobject]] $A \stackrel{i}{\hookrightarrow} X$ of $X$ it is the projector onto, in that there is a [[projection]] $X \stackrel{p}{\to} A$ such that the idempotent is the composite of this projection followed by including $A$ back into $X$:

$$
  e \colon X \stackrel{p}{\to} A \stackrel{i}{\hookrightarrow} X
  \,.
$$

As opposed to the case of linear algebra, in general such a factorization into a projection onto a subobject $A$ need not actually exists for an idempotent $e$ in a generic category. If it exists, one says that $e$ is a _[[split idempotent]]_.

Accordingly, one is interested in those categories for which every idempotent is split. These are called _[[idempotent complete categories]]_ or _[[Cauchy complete categories]]_. If a category is not yet idempotent complete it can be completed to one that is: its _[[Karoubi envelope]]_ or _[[Cauchy completion]]_.

## Definition

An [[endomorphism]] $e\colon B \to B$ in a [[category]] is an **idempotent** if the [[composition]] with itself equals itself

$$
  e \circ  e = e
  \,.
$$

A **splitting** of an idempotent $e$ consists of morphisms $s\colon A \to B$ and $r\colon B \to A$ such that $r \circ s = 1_A$ and $s \circ r = e$.  In this case $A$ is a [[retract]] of $B$, and we call $e$ a [[split idempotent]].

Of course, we can simply consider the __idempotent elements__ of any [[monoid]].

## Properties

### The algebra of idempotents

Given an [[abelian monoid]] $R$, the idempotent elements form a [[submonoid]] $Idem(R)$.

Given a [[commutative ring]] $R$, the idempotent elements of $R$ form a [[Boolean algebra]] $Idem(R)$ with these operations:

*  $\top \coloneqq 1$,
*  $P \wedge Q \coloneqq P Q$,
*  $\bot \coloneqq 0$,
*  $P \vee Q \coloneqq P - P Q + Q$,
*  $\neg{P} \coloneqq 1 - P$.

This is important in [[measure theory]]; if $R$ is the ring $L^\infty(X,\mathcal{M},\mathcal{N})$ of [[essentially bounded function|essentially bounded]] [[real number|real]]-valued [[measurable functions]] on some [[measurable space]] $(X,\mathcal{M})$ modulo an ideal $\mathcal{N}$ of [[null sets]], then $Idem(R)$ is the Boolean algebra of [[characteristic functions]] of [[measurable sets]] modulo null sets, which is [[isomorphic]] to the Boolean algebra $\mathcal{M}/\mathcal{N}$ of measurable sets modulo null sets itself.

If $R$ is a commutative $*$-[[star-ring|ring]], then we may restrict to the [[self-adjoint element|self-adjoint]] idempotent elements to get the Boolean algebra $Proj(R)$.  In measure theory, if $R$ is the [[complex number|complex]]-valued version of $L^\infty(X,\mathcal{M},\mathcal{N})$, then $Proj(R)$ will still reconstruct $\mathcal{M}/\mathcal{N}$.  In [[operator algebra]] theory, the self-adjoint idempotent elements of an operator algebra are called [[projection operator]]s, which is the origin of the notation $Proj$.  (Sometimes one requires projection operators to be _proper_: to have norm $1$; the only projection operator that is not proper is $0$.)

The projection operators of a commutative $W^\star$-[[W-star-algebra|algebra]] give the link between operator algebra theory and measure theory; in fact, the [[categories]] of commutative $W^\star$-algebras and of [[localisable measurable spaces]] (or [[measurable locales]]) are [[dual equivalence|dual]], and $W^\star$-algebra theory in general may be thought of as noncommutative measure theory.  In noncommutative measure theory, the projection operators are still important, but they no longer form a Boolean algebra.

### The universal idempotent-split completion

Given a [[category]] $\mathcal{C}$ one may ask for the [[universal construction|universal]] category obtained from $\mathcal{C}$ subject to the constraint that all idempotents are turned into [[split idempotents]]. This is called the _[[Karoubi envelope]]_  of $\mathcal{C}$. More generally, in [[enriched category theory]] it is called the _[[Cauchy completion]]_ of $\mathcal{C}$.

## Related concepts

* [[retract]], [[section]]

* [[Cauchy complete category]]

* [[idempotent complete (infinity,1)-category]]

* [[idempotent semiring]]

## References

* [[Saunders MacLane]], §I.5 of: *[[Categories for the Working Mathematician]]*, Graduate Texts in Mathematics **5** Springer (1971, second ed. 1997) [[doi:10.1007/978-1-4757-4721-8](https://link.springer.com/book/10.1007/978-1-4757-4721-8)]

Formalization in [[homotopy type theory]]:

* {#Shulman14} [[Mike Shulman]], _[Splitting idempotents](http://homotopytypetheory.org/2014/12/08/splitting-idempotents/)_

[[!redirects idempotent]]
[[!redirects idempotents]]
[[!redirects idempotent element]]
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[[!redirects idempotent operator]]
[[!redirects idempotent operators]]
[[!redirects idempotent morphism]]
[[!redirects idempotent morphisms]]


idempotent (infinity,1)-monad

#Contents#
* table of contents
{:toc}

## Idea

An [[(∞,1)-monad]] such that applying to its unit its unit yields an [[equivalence]]. Exhibiting a [[reflective sub-(∞,1)-category]].

The [[(∞,1)-category]] refinement of a _[[idempotent monad]]_.

## Related concepts

* [[modal operator]]

* [[Bousfield-Friedlander theorem]]

* [[modal type theory]]

[[!redirects idempotent (infinity,1)-monads]]

[[!redirects idempotent (∞,1)-monad]]
[[!redirects idempotent (∞,1)-monads]]

[[!redirects idempotent (infinity,1)-comonad]]
[[!redirects idempotent (infinity,1)-comonads]]

[[!redirects idempotent (∞,1)-comonad]]
[[!redirects idempotent (∞,1)-comonads]]

[[!redirects idempotent (infinity,1)-co-monad]]
[[!redirects idempotent (infinity,1)-co-monads]]

[[!redirects idempotent (∞,1)-co-monad]]
[[!redirects idempotent (∞,1)-co-monads]]


idempotent adjunction

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###Context###
#### Idempotents
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[[!include idempotents - contents]]
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#### Category theory
+--{: .hide}
[[!include category theory - contents]]
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=--

# Idempotent adjunctions
* table of contents
{: toc}

## Definition
 {#Definition}

\begin{prop}\label{EquivalentConditionsForIdempotency}
**(equivalent conditions for idempotency)**
\linebreak
Let $F \,\colon\, C \rightleftarrows D \,\colon\, G$ be an [[adjunction]] with [[unit of an adjunction|unit]] $\eta$ and [[counit of an adjunction|counit]] $\varepsilon$.  Then the following conditions on their [[whiskering]] are equivalent:

1. $F \eta$ is a [[natural isomorphism]].

1. $\varepsilon F$ is a natural isomorphism.

1. $G \varepsilon F$ is a natural isomorphism — i.e. the [[monad]] induced by the adjunction is an [[idempotent monad]].

1. $G F \eta = \eta G F$.

1. $G F \eta G = \eta G F G$.

1. $G\varepsilon$ is a natural isomorphism.

1. $\eta G$ is a natural isomorphism.

1. $F \eta G$ is a natural isomorphism — i.e. the [[comonad]] induced by the adjunction is an [[idempotent comonad]].

1. $F G \varepsilon = \varepsilon F G$.

1. $F G \varepsilon F = \varepsilon F G F$.

1. {#IdempotentAdjunctionFactoredThroughFixedPoints} The adjunction factors through its [[fixed point of an adjunction|fixed points]] as

   $$
     C
     \underoverset
       {\underset{G_1}{\hookleftarrow}}
       {\overset{F_1}{\longrightarrow}}
       {\;\;\; \bot \;\;\;}
     E
     \underoverset
       {\underset{G_2}{\longleftarrow}}
       {\overset{F_2}{\hookrightarrow}}
       {\;\;\; \bot \;\;\;}
     D
     \,,
   $$

   where $F_2$ and $G_1$ are [[fully faithful functor|fully faithful]], i.e. $F_1\dashv G_1$ is a [[reflective subcategory|reflection]] and $F_2 \dashv G_2$ is a [[coreflective subcategory|coreflection]].

\end{prop}
\begin{definition}\label{IdempotentAdjunction}
**(idempotent adjunction)**
\linebreak
When the equivalent conditions from Prop. \ref{EquivalentConditionsForIdempotency} hold, the adjunction is said to be *idempotent*.
\end{definition}

An original reference for the equivalence of all but the last of these conditions is [MacDonald & Stone 1982, Prop. 2.8](#MacDonaldStone82); a textbook account is in [Grandis 2021, Thm. 3.8.2](#Grandis21). The full statement including the (co)reflective factorization through the fixed points is made explicit in the proof of [Grandis 2021, Thm. 3.8.8](#Grandis21), which also makes explicit that:

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

For an idempotent adjunction as in def. \ref{IdempotentAdjunction}, the functors $F$ and $G$ restrict to an [[equivalence of categories]] between the [[full images]] of $F$ and of $G$ (which are, respectively, a [[coreflective subcategory]] of $D$ and a [[reflective subcategory]] of $C$, both equivalent to the $E$ in the last item above).  In other words, for an idempotent adjunction, the category of [[fixed point of an adjunction|fixed points]] has a particularly simple form.

=--

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

If an idempotent adjunction is [[monadic adjunction|monadic]], then (up to equivalence) it consists of the inclusion and reflection of a [[reflective subcategory]] (i.e. the algebras for an [[idempotent monad]]).  Dually, if it is [[comonadic adjunction|comonadic]], it consists of the inclusion and coreflection of a [[coreflective subcategory]].  Thus, the primary interest in isolating the notion of *idempotent adjunction* is when considering adjunctions which are neither monadic nor comonadic.

=--

## Examples

### General

\begin{example}
Any adjunction between [[posets]] is idempotent.  This is a central fact in the theory of [[Galois connections]].  Thus, in a sense, *non-idempotent* adjunctions are an important new idea arising by the "groupoidal" form of [[vertical categorification]].
\end{example}

\begin{example}
More generally, an adjunction in which the [[full image]] of either functor is a [[poset]] must be idempotent.  This follows from conditions 4, 5, 9, and 10 in Prop. \ref{EquivalentConditionsForIdempotency}.  This fact arises when constructing [[generalized kernels]].
\end{example}

\begin{example}
The [[material-structural adjunction]] between [[material set theory|material set theories]] and [[structural set theory|structural set theories]] is idempotent.  The [[fixed point of an adjunction|fixed categories]] consist of the models satisfying appropriate versions of the [[axiom of foundation]] or anti-foundation.
\end{example}

\begin{example}\label{CoCommaAdjunction}
**((co)comma construction)**
\linebreak
The [[comma category]] construction forms part of an adjunction
$$
  cocomma
  \;\colon\;
  Span(X,Y)
  \; \rightleftarrows \;
  Cospan(X,Y)
  \;\colon\;
  comma
$$
between [[spans]] and [[cospans]] of categories whose feet are given by categories $X$ and $Y$ ([Shulman 2016](#Shulman16)).
This adjunction is idempotent and factors into the [[reflection]] into [[discrete]] [[two-sided fibrations]] in the category $Span(X,Y)$ and the [[coreflection]] from [[codiscrete cofibrations]] in $Cospan(X,Y)$.
\end{example}

### Involving topological spaces

\begin{example}
The "[[frame of opens]]" and "space of points" functors between [[topological spaces]] and [[locales]] form an idempotent adjunction.  The resulting [[equivalence of categories]] is between [[sober spaces]] (which are reflective in [[Top]]) and [[spatial locales]] (which are coreflective in [[Loc]]).
\end{example}

\begin{example}
For any [[topological space]] $X$, there is an idempotent adjunction between the category $[O(X)^{\op}, Set]$ of [[presheaves]] on $X$ and the [[slice category]] $Top_{/X}$ of [[TopologicalSpaces]] over $X$ (the right adjoint gives the presheaf of [[sections]] of a space over $X$).  The resulting equivalence of categories is between [[sheaves]] in the modern sense of presheaves satisfying [[descent]], and sheaves in the original sense as [[étalé spaces]].  See [this blog post](http://golem.ph.utexas.edu/category/2010/02/sheaves_do_not_belong_to_algeb.html).
\end{example}

### Between topological and diffeological spaces
 {#BetweenTopologicalAndDiffeologicalSpaces}

[[!include adjunction between topological spaces and diffeological spaces]]

## Related concepts

* A [[categorification]] of the notion of idempotent adjunction is that of [[lax-idempotent 2-adjunction]].

* In an [[adjoint triple]], one of the two [[adjoint pairs]] is idempotent if and only if the other one is; see [there](adjoint+triple#Idempotent) for more.

## References

The characterization of idempotent adjunctions is proven/due to:

* {#MacDonaldStone82} [[John L. MacDonald]], [[Arthur L. Stone]], Prop. 2.8 in: *The tower and regular decomposition*, Cahiers de Topologie et Géométrie Différentielle Catégoriques, Tome 23 (1982) no. 2, pp. 197-213 ([numdam:CTGDC_1982__23_2_197_0](http://www.numdam.org/item/?id=CTGDC_1982__23_2_197_0))

Textbook accounts:

* {#Grandis21} [[Marco Grandis]], Section 3.8 in: _Category Theory and Applications: A Textbook for Beginners_, World Scientific 2021 ([doi:10.1142/12253](https://doi.org/10.1142/12253))

See also:

* Alessandro Ardizzoni, José Gómez-Torrecillas, Claudia Menini, Prop. 1.3 in:  *Monadic Decompositions and Classical Lie Theory*,  Appl Categor Struct 23, 93–105 (2015)  ([arXiv:1203.2881](https://arxiv.org/abs/1203.2881), [doi:10.1007/s10485-013-9326-7](https://doi.org/10.1007/s10485-013-9326-7))

The [example of D-topological spaces](#BetweenTopologicalAndDiffeologicalSpaces) inside [[topological spaces|topological]]/[[diffeological spaces]] is due to:

* {#SYH10} [[Kazuhisa Shimakawa]], K. Yoshida, [[Tadayuki Haraguchi]], Section 3 of: _Homology and cohomology via enriched bifunctors_, Kyushu Journal of Mathematics, 2018 Volume 72 Issue 2 Pages 239-252 ([arXiv:1010.3336](https://arxiv.org/abs/1010.3336), [doi:10.2206/kyushujm.72.239]( https://doi.org/10.2206/kyushujm.72.239))

The example \ref{CoCommaAdjunction} of the (co)comma adjunction is also mentioned in:

* {#Shulman16} [[Mike Shulman]], reply [MO:a/247311/381](https://mathoverflow.net/a/247311/381) (2016) to: "An 'explicit' description of cocomma categories?"

[[!redirects idempotent adjunctions]]


idempotent complete (infinity,1)-category

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###Context###
#### Idempotents
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[[!include idempotents - contents]]
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#### $(\infty,1)$-Category theory
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[[!include quasi-category theory contents]]
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#Contents#
* table of contents
{:toc}

## Idea

An ordinary [[category]] is _[[idempotent complete category|idempotent complete]]_, aka _[[Karoubi envelope|Karoubi complete]]_ or _[[Cauchy completion|Cauchy complete]]_ , if every [[idempotent]] splits.  Since the splitting of an idempotent is a limit *or* colimit of that idempotent, any category with [[finitely complete category|all finite limits]] or [[finitely cocomplete category|all finite colimits]] is idempotent complete.

In an [[(∞,1)-category]] the idea is the same, except that the notion of *idempotent* is more complicated.  Instead of just requiring that $e\circ e = e$, we need an [[equivalence]] $e\circ e \simeq e$, together with higher coherence data saying that, for instance, the two derived equivalences $e\circ e\circ e \simeq e$ are equivalent, and so on up.  In particular, *being idempotent* is no longer a [[stuff, structure, property|property]] of a morphism, but *structure* on it.

It is still true that a splitting of an idempotent in an $(\infty,1)$-category is a limit or colimit of that idempotent (now regarded as a diagram with all its higher coherence data), but this limit is no longer a finite limit; thus an $(\infty,1)$-category can have all finite limits without being idempotent-complete.

## Definition

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

Let $Idem^+$ be the [[nerve]] of the [[walking structure|free 1-category containing a]] [[retraction]], with $e:X\to X$ the idempotent, $r:X\to Y$ the retraction, and $i:Y\to X$ the section (and $e = i r$ and $r i = 1_Y$).  Let $Idem$ be the similar nerve of the free 1-category containing an idempotent, which is the full sub-simplicial set of $Idem^+$ spanned by the object $X$.  Let $Ret$ be the image in $Idem^+$ of the 2-simplex $\Delta^2 \to Idem^+$ exhibiting the composite $r i = 1_Y$; thus $Ret$ is also the quotient of $\Delta^2$ that collapses the 1-face to a point.

=--

([Lurie, 4.4.5.2 p.304](#Lurie))

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

Let C be an $\infty$-category, incarnated as a [[quasi-category]].

1.  An **idempotent morphism** in C is a map of simplicial sets $Idem \to C$. We will refer to $Fun(Idem, C)$ as the
**$(\infty,1)$-category of idempotents** in $C$.

1. A **weak retraction diagram** in $C$ is a [[homomorphism]] of [[simplicial sets]] $Ret \to C$. We refer to $Fun(Ret, C)$ as
the **$(\infty,1)$-category of weak retraction diagrams** in $C$.

1. A **strong retraction diagram** in $C$ is a map of simplicial sets $Idem^+ \to C$. We will refer to $Fun(Idem+, C)$
as the **$(\infty,1)$-category of strong retraction diagrams** in $C$.

=--

([Lurie, 4.4.5.4 p.304](#Lurie))

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

An idempotent $F \colon Idem \to C$ is **effective**  if it extends to a map $Idem^+ \to C$.

=--

([Lurie, above corollary 4.4.5.14](#Lurie))

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

An idempotent diagram $F \colon Idem \to C$ is effective precisely if it admits an [[(∞,1)-limit]], equivalently if it admits an [[(∞,1)-colimit]].

=--

By ([Lurie, lemma 4.3.2.13](#Lurie)).

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

$C$ is called an **idempotent complete $(\infty,1)$** if every idempotent is effective.

=--

([Lurie, above corollary 4.4.5.14](#Lurie))

## Properties

The following properties generalize those of idempotent-complete 1-categories.

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

A [[small (∞,1)-category]] is idempotent-complete if and only if it is [[accessible (∞,1)-category|accessible]].

=--

This is [[Higher Topos Theory|HTT, 5.4.3.6]].

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

For $C$ a [[small (∞,1)-category]] and $\kappa$ a [[regular cardinal]], the [[(∞,1)-Yoneda embedding]] $C \to C' \hookrightarrow Ind_\kappa(C)$ with $C'$ the full subcategory on $\kappa$-[[compact object]]s exhibits $C'$ as the idempotent completion of $C$.

=--

This is [[Higher Topos Theory|HTT, lemma 5.4.2.4]].

## Coherent vs incoherent idempotents

We may also ask how idempotent-completeness of $C$ is related to that of its [[homotopy category]] $h C$.  An idempotent in $h C$ is an "incoherent idempotent" in $C$, i.e. a map $e:X\to X$ such that $e \sim e^2$, but without any higher coherence conditions.  In this case we have:

+-- {: .num_theorem}
###### Theorem
**([[Higher Algebra|HA]] Lemma 1.2.4.6)**
If $C$ is [[stable (infinity,1)-category|stable]], then $C$ is idempotent-complete (i.e. every coherent idempotent is effective) if and only if $h C$ is (as a 1-category).
=--

However, if $C$ is not stable, this is false.  The following counterexample in [[∞Gpd]] is constructed in Warning 1.2.4.8 of [[Higher Algebra|HA]].  Let $\lambda : G \to G$ be an injective but non-bijective group homomorphism such that $\lambda$ and $\lambda^2$ are conjugate.  (One such is obtained by letting $G$ be the group of endpoint-fixing homeomorphisms of $[0,1]$, with $\lambda(g)$ acting as a scaled version of $g$ on $[0,\frac 1 2]$ and the identity on $[\frac 1 2,1]$.  Then $\lambda(g) \circ h = h \circ \lambda^2(g)$ for any $h$ such that $h(t) = 2t$ for $t \in [0,\frac 1 4]$.)

Then $B\lambda : B G \to B G$ is homotopic to $B\lambda^2$, hence idempotent in the homotopy category.  If it could be lifted to a coherent idempotent, then the colimit of the diagram
$$B G \xrightarrow{B \lambda} B G \xrightarrow{B \lambda} BG \to \cdots$$
would be its splitting, and hence the map $B G \to \colim (B G\xrightarrow{B \lambda} B G \to\cdots)$ would have a [[section]].  Passing to fundamental groups, $G \to \colim (G\xrightarrow{\lambda} G \to\cdots)$ would also have a section; but this is impossible as $\lambda$ is injective but not surjective.

However, we do have the following:

+-- {: .num_theorem}
###### Theorem
**([[Higher Algebra|HA]] Lemma 7.3.5.14 in older versions, [[Higher Topos Theory|HTT]] Proposition 4.4.5.20 in newer versions)**
A morphism $e$ in an $(\infty,1)$-category $C$ is idempotent (i.e. $e:\Delta^1 \to C$ extends to $Idem$) if and only if there is a homotopy $h : e \sim e^2$ such that $h\circ 1 \sim 1\circ h$.
=--

In other words, an incoherent idempotent can be fully coherentified as soon as it admits one additional coherence datum.

However, there may be multiple inequivialent ways of doing so.

## Related concepts

* [[Karoubi envelope]]

* [[Cauchy completion]]

* [noncommutative motives -- as universal additive invariants](noncommutative+motive#AsUniversalAditiveInvariant)

## References

Section 4.4.5 of

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

Formalization in [[homotopy type theory]]:

* {#Shulman14} [[Mike Shulman]], _Idempotents in intensional type theory_, [arXiv:1507.03634](https://arxiv.org/abs/1507.03634), [blog post](http://homotopytypetheory.org/2014/12/08/splitting-idempotents/)

[[!redirects idempotent complete (∞,1)-category]]
[[!redirects idempotent-complete (infinity,1)-category]]
[[!redirects idempotent-complete (∞,1)-category]]

[[!redirects idempotent-complete quasi-category]]
[[!redirects idempotent-complete quasi-categories]]

[[!redirects idempotent-completion of an (∞,1)-category]]
[[!redirects idempotent-completion of an (infinity,1)-category]]

[[!redirects idempotent complete (infinity,1)-categories]]