index

> This page is about the notion of index in [[analysis]]/[[operator algebra]]. For other notions see elsewhere.

***

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#### Functional analysis
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#Contents#
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## Idea
 {#Idea}

The notion of _index_ was originally defined

* _[For elliptic differential operators](#IdeaForDifferentialOperators)_

as an invariant correction of the [[kernel]] of such an operator (namely corrected by the [[cokernel]]). The definition has particularly nice properties in the special case

* _[For Dirac operators](#ForDiracOperators)_

where it coincides with the [[partition function]] of [[supersymmetric quantum mechanics]].

More generally the resulting notion is abstractly characterized as being the pairing operation ([[composition]])

* _[in KK-theory](#GeneralIdeaInKKTheory)_.

Even more generally, in [[generalized cohomology theory]] indices are given by [[genera]] and universal [[orientation in generalized cohomology]], such as for instance the [[elliptic genus]] for [[elliptic cohomology]] and the [[Witten genus]] for [[tmf]]. See at _[[genus]]_ for more on this generalized notion of indices.

There is also a quite general [[microlocal formulation of index theory]] due to Kashiwara and Schapira.

### For elliptic differential and Fredholm operators
 {#IdeaForDifferentialOperators}

The _[[analytical index]]_ of an [[elliptic operator|elliptic]] [[differential operator]] $D \colon \Gamma(E_1) \to \Gamma(E_2)$ is defined to be the the difference between the [[dimension]] of its [[kernel]] and that of its [[cokernel]].

One reason why this is an interesting invariant of an elliptic differential operator is that when deforming the operator by a [[compact operator]] then the dimension of both the kernel and the cokernel may change, but their difference remains the same. Hence one may think of the analytic index as a "corrected" version of its [[kernel]], such as to make it be more invariant.

On the other hand, the _[[topological index]]_ of an elliptic differential operator $D$, is defined to be the pairing of the [[cup product]] of its [[Chern character]] and the [[Todd class]] of the base manifold with its [[fundamental class]].

More generally such analytic and topological indices are defined for [[Fredholm operators]].

The _[[Atiyah-Singer index theorem]]_ assert that these two notios of index are in fact equal.

### For Dirac operators
 {#ForDiracOperators}

If the [[Fredholm operator]] in question happens to be
a _[[Dirac operator]]_ $D$ (such as that encoding the
[[dynamics]] of a [[spinning particle]] or more generally the [[supercharge]] of a system in [[supersymmetric quantum mechanics]]) then the index of $D$ coincides with the [[partition function]] of this quantum mechanical system, namely the [[super-trace]] of the [[heat kernel]] $\exp(-t D^2)$ of the corresponding [[Hamiltonian]] [[Laplace operator]] $D^2$ ([Berline-Getzler-Vergne 04](#BerlineGetzlerVergne04)).

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

Let $(X,g)$ be a [[compact topological space|compact]] [[Riemannian manifold]] and $\mathcal{E}$ a smooth [[super vector bundle]] and indeed a [[Clifford module bundle]] over $X$. Consider a [[Dirac operator]]

$$
  D \colon \Gamma(X,\mathcal{E}) \to \Gamma(X, \mathcal{E})
$$

with components (with respect to the $\mathbb{Z}_2$-[[graded vector space|grading]]) to be denoted

$$
  D =
  \left[
    \array{
       0 & D^-
       \\
       D^+ & 0
    }
  \right]
  \,,
$$

where $D^- = (D^+)^\ast$. Then $D^+$ is a [[Fredholm operator]] and its index is the [[supertrace]] of the [[kernel]] of $D$, as well as of the [[heat kernel]] of $D^2$:

$$
  \begin{aligned}
    ind(D^+) & \coloneqq dim(ker(D^+)) - dim(coker(D^+))
   \\
   & = dim(ker(D^+)) - dim(ker(D^-))
   \\
   & = sTr(ker(D))
   \\
   & = sTr( \exp(-t \, D^2) ) \;\;\; \forall t \gt 0
  \end{aligned}
  \,.
$$

=--

This appears as
([Berline-Getzler-Vergne 04, prop. 3.48, prop. 3.50](BerlineGetzlerVergne04)), based on ([MacKean-Singer 67](#MacKeanSinger67)).

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

The last step here follows from an argument which is as simple as it is paramount whenever anything involves [[supersymmetry]]:

the point is that if a (hermitean) operator $H$ has a [[supercharge]] $D$, in that $H = D^2$, then all its non-vanishing [[eigenstates]] appear in "[[supermultiplet]]" pairs of the same eigenvalue: if $|\psi\rangle$ has [[eigenvalue]] $E \gt 0$ under $H$, then

1. $D |\psi\rangle \neq 0$ (since $D D |\psi\rangle = H |\psi\rangle = E |\psi\rangle \neq 0$);

1. also $D |\psi\rangle$ has eignevalue $E$ (since $[H,D] = 0$).

Therefore all eigenstates for non-vanishing eigenvalues appear in pairs whose members have opposite sign under the supertrace.
So only states with $H |\psi\rangle = 0$ contribute to the supertrace. But if $H$ and $D$ are hermitean operators for a non-degenerate inner product, then it follows that $(D^2 |\psi\rangle = 0) \Leftrightarrow (D|\psi\rangle = 0)$ and hence these are precisely the states which are also annihilated by the supercharge (are in the kernel of $D$), hence are precisely only the _supersymmetric states_.

On these now the weight $\exp(- t D^2) = 1$ and hence the supertrace over this "Euclidean propagator" simply counts the number of supersymmetric states, signed by their fermion number.

=--

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

If one thinks of $D^2$ as the time-evolution [[Hamiltonian]] of a system of [[supersymmetric quantum mechanics]] with $D$ the [[supercharge]] on the [[worldline]], then $ker(D)$ is the space of supersymmetric [[quantum states]], $\exp(-t \, D^2)$ is the Euclidean time evolution operator and its [[supertrace]] is the [[partition function]] of the system. Hence we have the translation

*  index = partition function .

This kind of argument appears throughout supersymmetric quantum field theory. In dimension 2 it controls the nature of the [[Witten genus]].

=--

### General abstract definition in KK-theory
 {#GeneralIdeaInKKTheory}

The abstract [[universal property|universal]] characterization of indices is: the index is the _pairing_ in [[KK-theory]]/[[E-theory]].

More in detail, by the discussion there [[KK-theory]] ([[E-theory]]) is the [[category]] $KK$ which is the additive and split exact [[localization]] of the category [[C*Alg]] of [[C*-algebras]] at the [[compact operators]]. For $\mathbb{C}$ the base [[C*-algebra]] of [[complex numbers]] the [[morphisms]] in this category have the following equivalent meaning:

* [[morphisms]] $\mathbb{C} \to A$ are [[operator K-theory|operator K-cohomology]] classes which are represented by "[[vector bundles]] over the space represented by $A$", namely by [[Hilbert modules]] $E$ over $A$;

* [[morphisms]] $A \to \mathcal{C}$ are [[K-homology]] classes which are represented by [[Fredholm operators]] $D$;

* the [[composition]]

  $$
    ind(D_E)
      \;\colon\;
    \mathbb{C} \stackrel{E}{\to} A \stackrel{D}{\to} \mathbb{C}
    \;\;\;\;
    \in KK(\mathbb{C}, \mathbb{C}) \simeq \mathbb{Z}
  $$

  in the category $KK$ (hence the Kasparov product) is the _index_ of the Fredholm operator $D$ twisted by $E$.

More generally, if $B$ is some other chosen base [[C*-algebra]] then $KK(A,B)$ is the group of [[Fredholm operators]] $D$ on [[Hilbert module]] bundles over the [[C*-algebra]] $B$, and one takes the pairing

$$
  ind \coloneqq \circ_{\mathbb{C}, A, B} \;\colon\;
  KK(\mathbb{C},A) \times KK(A,B) \to KK(\mathbb{C}, B)
$$

to be the index map relative $B$. (See e.g. [Schick 05, section 6](#Schick05).) This is the case that the [[Mishchenko-Fomenko index theorem]] applies to.

And hence even more generally one may regard _any_ composition in $KK$ as as a generalized index map. Via the universal characterizatin of $KK$ itself, this then gives a fundamental and general abstract characterization of the notion of index:

_The index pairing is the [[composition]] operation in the [[KK-theory|KK-]][[localization of a category|localization]] of [[C*Alg]], hence in [[noncommutative stable homotopy theory]].

## Related pages

* [[topological index]]

* [[analytical index]]

* [[fiber integration in K-theory]]

* [[Fredholm index]], [[index of a Dirac operator]]

* [[index theorem]]

  * [[Atiyah-Singer index theorem]]

  * [[Mishchenko-Fomenko index theorem]]

  * [[Baum-Connes conjecture]]

* [[zeta function of an elliptic differential operator]], [[eta function]]

* [[Poincaré–Hopf index theorem]]

* [[Gauss-Bonnet theorem]]

[[!include genera and partition functions - table]]

## References

Original articles include

* [[Michael Atiyah]], [[Isadore Singer]], _Index theory for skew-adjoint Fredholm operators_  ([pdf](http://www.maths.ed.ac.uk/~aar/papers/askew.pdf))

With an eye towards application in [[mathematical physics]]:

* [[Mikio Nakahara]], Chapter 12 of: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

Lecture notes include

* [[Jean-Michel Bismut]], _Introduction to index theory_, lecture notes ([pdf](http://www.math.leidenuniv.nl/~ajavanp/Notes_Index_TheoryBotao.pdf))

A general introduction with an emphasis of indices as [[Gysin maps]]/[[fiber integration]]/[[Umkehr maps]] is in

* [[Chris Kottke]], _Talbot Workshop 2010 Talk 2: K-Theory and Index Theory_ ([arXiv:1010.5002](http://arxiv.org/abs/1010.5002))
 {#Kottke}

Textbook accounts include chapter III of

* [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], _[[Spin geometry]]_, Princeton University Press (1989)

A standard textbook account of the description of indices of [[Dirac operators]] as [[partition functions]] in [[supersymmetric quantum mechanics]] is

* [[Nicole Berline]], [[Ezra Getzler]], [[Michèle Vergne]], _Heat Kernels and Dirac Operators_, Springer Verlag Berlin (2004)
  {#BerlineGetzlerVergne04}

based on original articles including

* {#MacKeanSinger67} H. MacKean, [[Isadore Singer]], _Curvature and eigenvalues of the Laplacian_, J. Diff. Geom. 1 (1967)

* {#AtiyahBott73} [[Michael Atiyah]], [[Raoul Bott]], [[Vijay Patodi]], _On the heat equation and the index theorem_, Invent. Math. 19 (1973), 279–330

* {#AlvarezGaume83} [[Luis Alvarez-Gaumé]], _Supersymmetry and the Atiyah-Singer index theorem_, Comm. Math. Phys. Volume 90, Number 2 (1983), 161-173. ([Euclid](http://projecteuclid.org/euclid.cmp/1103940278))

* {#Getzler83} [[Ezra Getzler]], _Pseudodifferential operators on supermanifolds and the Atiyah-Singer index theorem_, Comm. Math. Phys. 92 (1983), 163-178. ([pdf](http://math.northwestern.edu/~getzler/Papers/1103940796.pdf))

* [[D. Quillen]], _Superconnections and the Chern character_ Topology 24 (1985), no. 1, 89–95;

* [[Varghese Mathai]], [[Daniel Quillen]], _Superconnections, Thom classes, and equivariant differential forms_. Topology 25 (1986),
no. 1, 85–110;

*  [[Ezra Getzler]], _A short proof of the Atiyah-Singer index theorem_, Topology 25 (1986), 111-117 ([pdf](http://math.northwestern.edu/~getzler/Papers/local.pdf))
 {#Getzler86}

* [[D. Quillen]], _Superconnection character forms and the Cayley transform_. Topology 27
(1988), no. 2, 211–238

For the more general discussion of indices of [[elliptic complexes]] see

* [[Peter Gilkey]], _Invariance theory, the heat equation, and the Atiyah-Singer index theorem_ ([pdf](http://pages.uoregon.edu/gilkey/dirPDF/InvarianceTheory1Ed.pdf))

An explicit formula in [[Chern-Weil theory]] for indices of differential operators on [[Hilbert modules]]-bundles is discussed in detail in

* [[Thomas Schick]], _$L^2$-index, KK-theory, and connections_, New York J. Math. 11 (2005) ([arXiv:math/0306171](http://arxiv.org/abs/math/0306171))
 {#Schick05}

A standard textbook account in the context of [[KK-theory]] is in section 24.1 of

* [[Bruce Blackadar]], _[[K-Theory for Operator Algebras]]_

[[!redirects indices]]

[[!redirects index theory]]
[[!redirects index theories]]

[[!redirects Index theory]]
[[!redirects Index theories]]


index bundle

[[!redirects index bundles]]

Where a single [[linear operator]] ([[Fredholm operator]]) has an [[index]], a [[bundle]] of such operators, of sorts, has an _index bundle_.


index of a Dirac operator

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#Contents#
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## Idea

The [[index]] of a [[Dirac operator]]. [[fiber integration in K-theory]]

### As a supertrace

(...)

### As the partition function of a supersymmetric QFT

(...)

## Properties

### Index theorems

* [[Atiyah-Singer index theorem]]

## Related concepts

* [[Fredholm index]]

## References

Chapter III of

* [[H. Blaine Lawson]], [[Marie-Louise Michelsohn]], _[[Spin geometry]]_, Princeton University Press (1989)

Introductions include

* Troels Roussau Johansen, _Introduction to the index theory of Dirac operators_ ([pdf](http://johansen.math.uni-kiel.de/lecture-on-dirac-operators))

[[!redirects index of Dirac operators]]
[[!redirects indices of Dirac operators]]


index of a subgroup

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# Contents
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## Idea

For $H  \hookrightarrow G$ a [[subgroup]] inclusion, its _index_ is the number  ${\vert G: H\vert}$ of $H$-[[cosets]] in $G$, hence roughly is the number of copies of $H$ that appear in $G$.

## Definition

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

For $H  \hookrightarrow G$ a [[subgroup]], its **index** is the [[cardinality]]

$$
  {\vert G : H\vert} \coloneqq {\vert G/H\vert}
$$

of the set $G/H$ of [[cosets]].

=--

## Properties

### Multiplicativity

If $H$ is a subgroup of $G$, the coset projection $-H: G\to G/H$ sends an element $g$ of $G$ to its orbit $gH$.

If $s : G/H\to G$ is a section of the coset projection $-H: G\to G/H$, then $G/H \times H \to G$ given by $( g H , h )\mapsto s( g H ) h^{-1}$ is a bijection. Its inverse is given by the set map $G\to G/H \times H$ given by $g\mapsto ( g H , g^{-1} s( g H ) )$. Note that the induced product projections $G\to G/H$ conincides with the coset projection.

This argument can be internalized to a group object $G$ and a subgroup object $G$ in a category $C$. In this case, the coset projection $-H: G\to G/H$ is the coequalizer of the action on $G$ by multiplication of $H$. The coset projection need not have a section. However, in case such sections exist, each section $s$ of the coset projection, the above argument internalized yields an isomorphism
$$
G/H \times H \overset{\simeq}\rightarrow G
\, .
$$

Even more generally, if $H \hookrightarrow K \hookrightarrow G$ is a sequence of [[subgroup]] objects, then each section of the projection $G/H \to G/K$ yields an isomorphism
$$
  G/K \,\times \, K/H \stackrel{\simeq}{\to} G/H
\, .
$$

Returning to the case of ordinary groups, i.e. group objects internal to $Set$, where the external axiom of choice is assumed to hold, the coset projection, being a coequalizer and hence an epimorphism, has a section. This gives the multiplicative property of the indices of a sequence $H \hookrightarrow K \hookrightarrow G$ of [[subgroups]]

$$
  {\vert G : K\vert} \dot {\vert K : H\vert}
  =
  {\vert G : H\vert}
  \,.
$$

### Finite groups

The concept of index is meaningful especially for finite groups, i.e. groups internal to [[FinSet]]. See, for example, its role in the [[classification of finite simple groups]].

Multiplicativity of the index has the following corollary, which is known as **[[Lagrange's theorem]]**: If $G$ is a [[finite group]], then the index of any subgroup is the quotient

$$
  {\vert G : H\vert} = \frac{{\vert G\vert}}{\vert H\vert}
$$

of the [[order of a group|order]] ([[cardinality]] = number of elements) of $G$ by that of $H$.

## Examples

* For $n \in \mathbb{N}$ with $n \geq 1$ and $\mathbb{Z} \stackrel{\cdot n}{\hookrightarrow} \mathbb{Z}$ the subgroup of the [[integers]] given by those that are multiples of $n$, the index is $n$.

[[!redirects finite-index subgroup]]
[[!redirects finite-index subgroups]]
[[!redirects finite index subgroup]]
[[!redirects finite index subgroups]]

[[!redirects index of a subgroup]]
[[!redirects indexes of a subgroup]]
[[!redirects indices of a subgroup]]
[[!redirects indexes of subgroups]]
[[!redirects indices of subgroups]]


index theorem

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## Entries

* [[index]]

  * [[analytical index]]

  * [[topological index]]

* index theorem

  * [[Atiyah-Singer index theorem]]

  * [[Mishchenko-Fomenko index theorem]]

  * [[microlocal formulation of index theory]]

Index theory naturally takes place in [[KK-theory]].

## References

* David Edwards, _Lectures on index theory I_, [pdf](http://www.math.uga.edu/~davide/Lectures_on_Index_Theory__Lecture_I_Operator_Theory_and_Topology.pdf),
_Lectures on index theory II_, [pdf](http://www.math.uga.edu/~davide/Lectures_on_Index_Theory__Lecture_II_Index_Theory_for_Toeplitz_Operators.pdf), _The Index Program for K-Theory with Local Coefficients_, [pdf](http://www.math.uga.edu/~davide/The_Index_Program_for_K-Theory_with_Local_Coefficients.pdf)
* [[Masaki Kashiwara]], Pierre Schapira, _Hochschild homology and microlocal Euler classes_, [arxiv/1203.4869](http://arxiv.org/abs/1203.4869)

[[!redirects index theorem]]
[[!redirects index theorems]]


index theory - contents

**[[index theory]], [[KK-theory]]**

[[noncommutative topology]], [[noncommutative geometry]]

[[noncommutative stable homotopy theory]]

**[[partition function]]**

**[[genus]], [[orientation in generalized cohomology]]**

## Definitions

**[[operator K-theory]]**

* [[C*-algebra]]

* [[Hilbert module]]

**[[K-homology]]**

* [[Fredholm operator]]

* [[differential operator]], [[pseudodifferential operator]]

  * [[symbol of a differential operator]]

  * [[elliptic operator]], [[elliptic complex]]

* [[Dirac operator]]

  * [[Spin^c Dirac operator]]

## Index theorems

* [[topological index]], [[analytical index]]

* [[Atiyah-Singer index theorem]]

  * [[Gauss-Bonnet theorem]]

  * [[Hirzebruch-Riemann-Roch theorem]]

  * [[global analytic index theory]]

  * [[Hirzebruch signature theorem]]

* [[Mishchenko-Fomenko index theorem]]

* [[Baum-Connes conjecture]]

## Higher genera

* [[elliptic genus]]

* [[Witten genus]]


indexed (infinity, 1)-category

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##Idea

An **indexed (∞,1)-category** is the [[(∞,1)-category theory|(∞,1)-category theoretic]] analogue of an [[indexed category]]. If $S$ is an (∞,1)-category, then an indexed (∞,1)-category is a functor from $S^{op}$ to [[(∞,1)Cat]].

The [[(∞,1)-Grothendieck construction]] establishes an equivalence between (∞,1)-categories indexed by $S$ and [[Cartesian fibrations]] over $S$.

##Variants

Varieties of indexed (∞,1)-category can be formed by requiring that the target be monoidal, symmetric, closed, and so on.

Not all kinds of fibration of $(\infty, 1)$-category can be formed in this way. In particular, [exponentiable ∞-functors](Conduché+functor#highercategorical_versions) are treated in ([AyalaFrancis](#AyalaFrancis}) by maps to $(\infty,1)Prof$, but with extra 'flagged' structure. An $(\infty, 1)$-version of [[proarrow equipments]] should work here.

In addition, for some purposes conditions may be placed on the indexing category, $S$, such as the atomic orbital $\infty$-categories of [[Parametrized Higher Category Theory and Higher Algebra]].

##Related pages

* [[fibrations of quasi-categories]]
* [Conduché functor](Conduché+functor#highercategorical_versions)
* [[Parametrized Higher Category Theory and Higher Algebra]]

##References

* {#AyalaFrancis} [[David Ayala]] and [[John Francis]], _Fibrations of ∞-categories_, [arxiv](https://arxiv.org/abs/1702.02681)

[[!redirects indexed (∞,1)-category theory]]
[[!redirects indexed (∞,1)-category]]
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indexed adjoint functor theorem

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## Idea

The analog of the [[adjoint functor theorem]] for [[indexed categories]].

## Statement

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

Let $\mathcal{S}$ be a [[cartesian category]], let $\mathbb{C}$ and $\mathbb{D}$ be $\mathcal{S}$-[[indexed categories]] which are [[locally small category|locally small]] and have all [[colimits]], and suppose further that $\mathbb{C}$ is [[well-copowered category|well-copowered]] and has a [[separating family]]. Then an [[indexed functor]] $F: \mathbb{C} \to \mathbb{D}$ has an indexed [[right adjoint]] precisely iff it is cocontinuous.

=--

This is ([Johnstone, theorem B2.4.6](#Johnstone)).

## References

* [[Robert Paré]], Dietmar Schumacher, *Abstract families and the adjoint functor theorems*, in _Indexed categories and their applications_, Lecture Notes in Math. **661** Springer (1978) [[doi:10.1007/BFb0061361](https://doi.org/10.1007/BFb0061361)]

* {#Johnstone} [[Peter Johnstone]], Section B2.4 in: _[[Sketches of an Elephant]]_


indexed category

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## Idea

An _indexed category_ is a [[2-presheaf]].

When doing [[category theory]] relative to a  [[base topos]] $\mathcal{S}$ (or other more general sort of [[category]]), the objects of $\mathcal{S}$ are thought of as replacements for [[sets]].  Since often in category theory we need to speak of "a set-indexed family of objects" of some category, we need a corresponding notion in "category theory over $\mathcal{S}$."  An **$\mathcal{S}$-indexed category** is a [[category]] $\mathbb{C}$ together with, for every [[object]] $X\in \mathcal{S}$, a notion of "$X$-indexed family of objects of $\mathbb{C}$."

## Definition

Let $\mathcal{S}$ be a [[category]].

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

An **$\mathcal{S}$-indexed category** $\tilde \mathbb{C}$ is a category defined by a [[pseudofunctor]]

$$
  \mathbb{C} : \mathcal{S}^{op}\to Cat
$$

from the [[opposite category]] of $\mathcal{S}$ to the [[2-category]] [[Cat]] of categories.

Under the [[Grothendieck construction]] [[equivalence of categories|equivalence]] this is equivalently a [[fibered category]]

$$
  \array{
    \tilde \mathbb{C} \\ \downarrow \\ \mathcal{S}
  }
$$

over $\mathcal{S}$.

Similarly, an $\mathcal{S}$-[[indexed functor]] $\mathbb{C} \to \mathbb{D}$ is a [[pseudonatural transformation]] of pseudofunctors, and an indexed natural transformation is a [[modification]].

This defines the [[2-category]] $\mathcal{S} IndCat \coloneqq [\mathcal{S}^{op}, Cat]$ of $\mathcal{S}$-indexed categories.

=--

This appears for instance as ([Johnstone, def. B1.2.1](#Johnstone)).

One may also call $\mathbb{C}$ a [[prestack]] in categories over $\mathcal{S}$.

Traditionally one writes the image of an object $X \in \mathcal{S}$ under $\mathbb{C}$ as $\mathbb{C}^X$ and calls it _the category of $X$-indexed families of objects of $\mathbb{C}$_.  Similarly, one writes the image of a [[morphism]] $u\colon X\to Y$ as $u^*\colon \mathbb{C}^Y\to \mathbb{C}^X$.

If $\mathcal{S}$ has a [[terminal object]] $*$ we think of $\mathbb{C}^*$ as the **underlying ordinary category** of the $\mathcal{S}$-indexed category $\mathbb{C}$. Part of the theory of indexed categories is about when and how to extend structures on $\mathbb{C}^*$ to all of $\mathbb{C}$.

A morphism of $S$-indexed categories is an [[indexed functor]].

## Examples

### Self indexing

+-- {: .num_example #CanonicalSelfIndexing}
###### Example
**(canonical self-indexing)**

If $\mathcal{S}$ has [[pullbacks]], then its [[codomain fibration]] is an $\mathcal{S}$-indexed category denoted $\mathbb{S}$.

This assigns to an object $I$ the corresponding [[over-category]]

$$
  \mathbb{S}^I \coloneqq \mathcal{S}/I
$$

and to a morphism $f : I \to J$ the functor $f^*$ that sends every $s \to I$ to its [[pullback]] $f^*$ along $f$.

=--

This indexed category represents $\mathcal{S}$ itself (or rather its [[codomain fibration]]) in the world of $\mathcal{S}$-indexed categories.

### Base change

+-- {: .num_example #BaseChange}
###### Example
**(change of base)**

If $F\colon \mathcal{S}\to \mathcal{T}$ is a [[functor]] and $\mathbb{C}$ is a $\mathcal{T}$-indexed category, then we have an $\mathcal{S}$-indexed category $F^*\mathbb{C}$ defined by

* $(F^*\mathbb{C})^I = \mathbb{C}^{F(I)}$ for every object $I \in \mathcal{S}$;

* and $x^* = F(x)^*$ for every morphism $x : I \to J$ in $\mathcal{S}$.

=--

### Indexed category of a functor

Combining these previous examples we get

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

For $F : \mathcal{S} \to \mathcal{C}$ a functor and $\mathcal{C}$ a [[finitely complete category]], there is the $\mathcal{S}$-indexed category $F^* \mathbb{C}$ given by

* $(F^* \mathbb{C})^I = \mathcal{C}/F(I)$.

If the functor $F$ preserves [[pullback]]s then this induces a morphism $\mathbb{S} \to F^* \mathbb{C}$ of $\mathcal{S}$-indexed categories.

=--

### Indexed category of a topos over a base topos

This situation frequently arises when $\mathcal{S}$ and $\mathcal{C}$ are [[toposes]] and $F \coloneqq f^*$ is the [[inverse image]] part of a [[geometric morphism]].

$$
  f
  :
  \mathcal{C}
   \stackrel{\overset{f^*}{\leftarrow}}{\underset{f_*}{\to}}
  \mathcal{S}
 \,.
$$

In this way, if $\mathcal{S}$ is a topos, then to be thought of as a _[[base topos]]_, then any topos *over* $\mathcal{S}$ (i.e. an object of the [[slice 2-category]] [[Topos]]$/S$) gives rise to a topos *relative to* $\mathcal{S}$, i.e. a "topos object" in the 2-category of $\mathcal{S}$-indexed categories, and this operation can be shown to be fully faithful.

See [[base topos]] for more on this.

Also, via this indexed category, $f$ exhibits $\mathcal{C}$ as a [[2-sheaf]] (see there) over $\mathcal{C}$, with respect to the [[canonical topology]].

### Hyperdoctrine

* [[hyperdoctrine]]

### Indexed monoidal category

See also _[[indexed monoidal category]]_, _[[indexed closed monoidal category]]_ and  _[[dependent linear type theory]]_.

### Indexed $(\infty, 1)$-category

See _[[indexed (infinity, 1)-category]]_.

## Properties

### Extensions of adjunctions to indexed categories

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

Let

$$
  (L \dashv R) :
   \mathcal{C} \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} \mathcal{S}
$$

be a pair of [[adjoint functor]]s between
[[finitely complete categories]]. Then $R$ extends to an $\mathcal{S}$-indexed functor

$$
  \mathbb{R} : \mathbb{C} \to \mathbb{S}
$$

where $\mathbb{S}$ is the self-indexing of $\mathcal{S}$ from example \ref{CanonicalSelfIndexing} and $\mathbb{C}$ is the base change indexing of $\mathcal{C}$ from example \ref{CartesianFunctorIndexing}.

By the general properties of adjunctions on [[overcategories]] (see there) we get for each $I \in \mathcal{S}$ an adjunction

$$
  (L/I \dashv R/I) : \mathbb{C}^I = \mathcal{C}/R(I) \to \mathcal{S}/I = \mathbb{S}^I
  \,.
$$

Here $\mathbb{R} : I \mapsto R/I$ is always a $\mathcal{S}$-indexed functor $\mathbb{C} \to \mathbb{S}$, and $\mathbb{L} : I \mapsto L/I$ is if $L$ preserves [[pullback]]s (by example \ref{CartesianFunctorIndexing}). If so, we have an $\mathcal{S}$-indexed adjunction

$$
  (\mathbb{L} \dashv \mathbb{R}) :
  \mathbb{C} \to \mathbb{S}
$$

=--

This appears as ([Johnstone, lemma B1.2.3](#Johnstone)).

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

(...)

=--

### Well-powered indexed categories
 {#WellPoweredness}

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

An $\mathcal{S}$-indexed category $\mathbb{C}$ is called **well-powered** if the [[fibered category]] $\tilde \mathbb{C} \to \mathcal{S}$ corresponding to it under the [[Grothendieck construction]] has the property that the [[forgetful functor]]

$$
  U : Q(2, \tilde \mathbb{C})
   \to
  Rect(*,\tilde \mathbb{C})
$$

has a [[right adjoint]], where $Q(2,\tilde \mathbb{C})$ is the [[full subcategory]] of $Rect(2, \tilde \mathbb{C})$ on vertical [[monomorphism]]s.

=--

This appears as ([Johnstone, example. B1.3.14](#Johnstone)).

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

Let $(L \dashv R) : \mathcal{C} \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} \mathcal{S}$  be a pair of [[adjoint functor]]s such that $L$ preserves pullbacks. Then the $\mathcal{S}$-indexed category $\mathbb{C}$ is well powered if $\mathbb{S}$ is.

=--

This is ([Johnstone, prop. B1.3.17](#Johnstone)).

## Related concepts

* [[indexed functor]]

* [[indexed topos]]

* [[indexed Grothendieck construction]]

* [[indexed monoidal category]]

* [[hyperdoctrine]]

## References

The notion of *indexed categories* was introduced in

* {#Grothendieck60} [[Alexander Grothendieck]], §A.1 ([p. 3](http://www.numdam.org/item/SB_1958-1960__5__299_0.pdf#page=3)) of *Technique de descente et théorèmes d’existence en géométrie algébrique. I. Généralités. Descente par morphismes fidèlement plats*, Séminaire [[N. Bourbaki]] exp. no190 (1960) 299-327 [[numdam:SB_1958-1960__5__299_0](http://www.numdam.org/item/?id=SB_1958-1960__5__299_0)]

but under the name "[[fibered category]]" (*catégorie fibrée*) which later became the standard term, instead, for the ([equivalent](Grothendieck+construction#EquivalenceBetweenFiberedCategoriesAndPsuedofunctors)) [[Grothendieck construction]] on an indexed category, cf.:

* {#Bénabou1975} [[Jean Bénabou]], p. 898 (2 of 41) of: _Fibrations petites et localement petites_, C. R. Acad. Sci. Paris **281** Série A (1975) 897-900 [[gallica](http://gallica.bnf.fr/ark:/12148/bpt6k6228235m/f171.image#)]

An early monograph:

* [[P. T. Johnstone]], [[R. Paré]], [[R. D. Rosebrugh]], [[D. Schumacher]], [[R. J. Wood]], [[G. C. Wraith]], _Indexed Categories and Their Applications_, Lecture Notes in Mathematics **661** (1978) [[doi:10.1007/bfb0061362](https://doi.org/10.1007/bfb0061362)]

Early discussion in the context of [[categorical semantics]] in [[computer science]]:

* {#TBG91} [[Andrzej Tarlecki]], [[Rod M. Burstall]] [[Joseph A. Goguen]], *Some fundamental algebraic tools for the semantics of computation: Part 3. indexed categories*, Theoretical Computer Science **91** 2 (1991) 239-264 [doi:10.1016/0304-3975(91)90085-G]

Discussion in a context of [[topos theory]]:

* {#Johnstone} [[Peter Johnstone]], Section B1.2  in: _[[Sketches of an Elephant]]_ (2002)

[[!redirects indexed categories]]


indexed functor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Category theory
+--{: .hide}
[[!include category theory - contents]]
=--
#### $(\infty,2)$-Topos theory
+--{: .hide}
[[!include (infinity,2)-topos theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

An _[[indexed category]]_ is a [[2-presheaf]]. An _indexed functor_ is a [[morphism]] of 2-presheaves, the "indexed"-terminology here is traditional in 1-[[topos theory]] and hence indexed functors are usually considered only between [[pseudofunctors]] (as opposed to more general [[2-functors]]).

## Definition

\begin{definition}
Let $\mathcal{S}$ be a category. Let  $\mathbb{C}$ and $\mathbb{D}$ be $\mathcal{S}$-[[indexed categories]], that is, [[pseudofunctors]] $\mathcal{S}^\mathrm{op} \to \mathbf{Cat}$, then an **$\mathcal{S}$-indexed functor** $F:{\mathbb{C}}\to{\mathbb{D}}$ is a [[pseudonatural transformation]] $F \colon \mathbb{C} \Rightarrow \mathbb{D}$: it assigns to each [[object]] $A$ of $\mathcal{S}$ a [[functor]] $F_A:{\mathbb{C}}(A) \to {\mathbb{D}}(A)$ and to each [[morphism]] $f:A\to B$ of $\mathcal{S}$ a [[natural isomorphism]] $\mathbb{D}(f) \circ F_B \cong F_A \circ \mathbb{C}(f)$ that is coherent with respect to the structural isomorphisms of $\mathbb{C}$ and $\mathbb{D}$ (see [[pseudonatural transformation]] for details).
\end{definition}

One can also consider indexed functors between categories indexed by different bases, i.e. morphisms of indexed categories.

\begin{definition}
Let $\mathbb{C} : \mathcal{S}^{op} \to \mathbf{Cat}$ and $\mathbb{D} : \mathcal{T}^{op} \to \mathbf{Cat}$ be [[indexed categories]]. A morphism between them is a pair of a functor $\Phi : \mathcal{S} \to \mathcal{T}$ and a pseudonatural transformation $\Phi^\flat : \Phi^*\mathbb{C} \Rightarrow \mathbb{D}$, where $\Phi^* \mathbb{C} = \mathbb{C} \circ \Phi^{op}$ is reindexing of $\mathbb{C}$ along $\Phi$.
\end{definition}

In other words, it's a lax commutative triangle over $\mathbf{Cat}$.

**Remark**. These morphisms correspond to morphisms of [[fibrations]] through the [[Grothendieck construction]].

## Related concepts

* [[indexed adjoint functor theorem]]

* [[indexed monoidal category]]

## References

Section B1 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_

[[!redirects indexed functors]]


indexed Grothendieck construction

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

#Contents#
* table of contents
{:toc}

## Idea

The generalization of the [[Grothendieck construction]] from [[categories]] to [[indexed categories]]:

## References

* [[Elena Caviglia]], [[Luca Mesiti]], *Indexed Grothendieck construction* [[arXiv:2307.16076](https://arxiv.org/abs/2307.16076)]

[[!redirects indexed Grothendieck constructions]]


indexed monoidal (infinity,1)-category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
#### Linear algebra
+-- {: .hide}
[[!include homotopy - contents]]
=--
=--
=--

# Indexed monoidal $(\infty,1)$-categories

* table of contents
{:toc}

## Idea

An _indexed monoidal $(\infty,1)$-category_ is the [[(∞,1)-category|(∞,1)-categorical]] version of an [[indexed monoidal category]].  That is, it consists of a "base" $(\infty,1)$-category $\mathcal{C}$ together with, for each $X\in \mathcal{C}$, a [[monoidal (∞,1)-category]] $Mod(X)$ varying functorially with $X$.  In one of the fundamental examples, $\mathcal{C}$ is the $(\infty,1)$-category of [[∞-groupoids]] ("spaces"), while $Mod(X)$ is that of [[parametrized spectra]].

Indexed monoidal $(\infty,1)$-categories are conjectured to be the [[categorical semantics]] of [[linear homotopy type theory]].

## Definition

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

An __indexed monoidal $(\infty,1)$-category__ is

1. an [[(∞,1)-category]] $\mathcal{C}$ with [[finite (∞,1)-limits]];

1. an [[(∞,1)-functor]] $Mod \colon \mathcal{C}^{op} \to SymMonCat_\infty$ to [[symmetric monoidal (∞,1)-categories]];

such that

1. each $Mod(X)$ is [[closed monoidal (infinity,1)-category|closed]] (with [[internal hom]] to be denoted $[-,-]$);

1. for each $f \colon \Gamma_1 \to \Gamma_2$ in $Mor(\mathcal{C})$ the assigned [[(∞,1)-functor]] $f^\ast \colon Mod(\Gamma_2) \to Mod(\Gamma_1)$ has a [[left adjoint]] $f_!$ and a [[right adjoint]] $f_\ast$;

1. The adjunction $(f_! \dashv f^\ast)$ satisfies [[Frobenius reciprocity]] and the [[Beck-Chevalley condition]].

=--

In the conjectural syntax of dependent linear type theory, the objects of $\mathcal{C}$ correspond to [[contexts]], which is why we sometimes write them as $\Gamma$.  Similarly, we may sometimes write $f_!$ and $f_\ast$ as $\sum_f$ and $\prod_f$ respectively.

The statement of [[Frobenius reciprocity]] then is that

$$
  \underset{f}{\sum}
  \left(
    X \otimes f^\ast Y
  \right)
  \simeq
  \left(
    \underset{f}{\sum} X
  \right)
  \otimes
  Y
  \,.
$$

## Examples

For each example we also spell out some of the abstract constructions (discussed in _[Structures](#Structures)_ below) realized in that model.  We also include some examples of [[indexed monoidal category|indexed monoidal 1-categories]] (which are a special case).

+--{: .query}
Although probably those examples should be moved to [[indexed monoidal category]].
=--

### Slices of a topos

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

For $\mathbf{H}$ a [[topos]], then its system $\mathbf{H}_{/(-)} \colon \mathbf{H}^{op} \to CartMonCat \to MonCat$ of [[slice toposes]] is an indexed monoidal category, hence an indexed monodial $(\infty,1)$-category.

=--

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

This example for dependent linear type theory is extremely "non-linear".  For instance, it has almost no [[dualizable objects]]; the only one is the terminal object in each slice.  Given that the formulas below for secondary integral transforms (def.\ref{SIT}) assume dualizable objects, this means that in this non-linear context there will be no nontrivial secondary integral transforms.

=--

We now pass gradually to more and more linear examples.x

### Parameterized pointed objects
 {#PointedObjects}

A first step away from the Cartesian example above towards more genuinely linear types is the following.

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

Let $\mathbf{H}$ be a [[topos]]. For $X \in \mathbf{H}$ any [[object]], write

$$
  \mathcal{C}_X \coloneqq \mathbf{H}_{/X}^{X/}
$$

for the [[category of pointed objects]] in the [[slice topos]] $\mathbf{H}_{/X}$. Equipped with the [[smash product]] $\wedge_X$ this is a [[closed monoidal category|closed]] [[symmetric monoidal category]] $(\mathcal{C}_X, \wedge_X, X \coprod X)$.

=--

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

For $f \colon X \longrightarrow Y$ any [[morphism]] in $\mathbf{H}$, the [[base change]] [[inverse image]] $f^\ast$ restricts to a functor $f^\ast \colon \mathcal{C}_Y \longrightarrow \mathcal{C}_X$ and this makes

$$
  \mathbf{H}_{/(-)}^{(-)/}
  \;\colon\;
  \mathbf{H}^{op}
  \longrightarrow
  MonCat
$$

an indexed monoidal category.

=--

This appears as ([Shulman 08, examples 12.13 and 13.7](#Shulman08)) and ([Shulman 12, example 2.33](#Shulman12)).

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

For $f \colon X \longrightarrow Y$ any [[morphism]] in $\mathbf{H}$ then the [[base change]] [[inverse image]] $f^\ast \colon \mathbf{H}_{/Y} \longrightarrow \mathbf{H}_{/X}$ preserves pointedness, and the [[pushout]] functor $f_! \colon \mathbf{H}^{X/} \longrightarrow \mathbf{H}^{Y/}$ preserves co-pointedness. These two functors hence form an [[adjoint pair]]
$(f_! \dashv f^\ast) \colon \mathcal{C}_X \longrightarrow \mathcal{C}_Y$.
Moreover, since [[colimits]] in the under-over category $\mathbf{H}_{/X}^{X/}$ are computed as colimits in $\mathbf{H}$ of [[diagrams]] with an [[initial object]] adjoined, and since by the [[Giraud axioms]] in the [[topos]] $\mathbf{H}$ [[pullback]] preserves these colimits, it follows that $f^\ast \colon \mathcal{C}_Y \to \mathcal{C}_X$ preserves colimits.
Finally by the discussion at _[[category of pointed objects]]_ we have that $\mathcal{C}_X$ and $\mathcal{C}_Y$ are [[locally presentable categories]], so that by the [[adjoint functor theorem]] it follows that $f^\ast$ has also a [[right adjoint]] $f_\ast \colon \mathcal{C}_X \to \mathcal{C}_Y$.

To see that $f^\ast$ is a [[strong monoidal functor]] observe that the [[smash product]] is, by the discussion there, given by a [[pushout]] over [[coproducts]] and [[products]] in the [[slice topos]]. As above these are all preserved by [[pullback]]. Finally to see that $f^\ast$ is also a [[strong closed functor]] observe that the [[internal hom]] on [[pointed objects]] is, by the discussion there, a [[fiber product]] of cartesian internal homs. These are preserved by the above case, and the fiber product is preserved since $f^\ast$ preserves all limits. Hence $f^\ast$ preserves also the internal homs of pointed objects.

=--

### Parameterized modules
 {#ParameterizedModules}

The examples of genuinely linear objects in the sense of [[linear algebra]] are the following.

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

Let $E$ be a [[commutative ring]]. Write $E Mod$ for its [[category of modules]]. For $X\in $ [[Set]] write

$$
  E Mod(X) \coloneqq Func(X, E Mod) \simeq (E Mod)^{\times_{\vert X \vert}}
$$

for the [[functor category]] from $X$ (regarded as a [[category]]) to $E Mod$, hence for the [[bundles]] of $E$-[[modules]] over the [[discrete space]] $X$. Equipped with the [[tensor product of modules]] this is a [[symmetric monoidal category]] $E Mod^{\otimes}$. This construction yields a [[functor]]

$$
  E Mod \;\colon\; Set^{op} \longrightarrow SymMonCat
  \,.
$$

which is an indexed monoidal category.

=--

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

In the context of prop. \ref{IndexedMonoidalCategoryOfModuleBundles}
consider $E = k$ a [[field]]. Then $k Mod \simeq Vect_k$ is the [[category]] [[Vect]] of $k$-[[vector spaces]].

For $X \in Set$ a set, an $X$-dependent object $A \in Mod(X)\simeq Vect_k(X)$ is just a collection of ${\vert X\vert}$ vector spaces $A_x$ for $x\in X$.

For $X \in FinSet \hookrightarrow Set$ a [[finite set]], the [[dependent sum]] and [[dependent product]] operations coincide and produce the [[direct sum]] of vector spaces:

$$
  \underset{X}{\sum} A \simeq \underset{X}{\prod} A \simeq
  \underset{x\in X}{\oplus} A_x
  \in Vect_k(\ast)
  \,.
$$

Under this identification every morphism $f \in Mor(Set)$ with finite fibers carries a canonical untwisted fiberwise fundamental class, def. \ref{FiberwiseFundamentalClass}, $[f]_{canonical}$.

For

$$
  \array{
    && X_1 \times X_2
    \\
    & {}^{\mathllap{p_1}}\swarrow && \searrow^{\mathrlap{p_2}}
    \\
    X_1 && && X_2
  }
$$

a product corrrespondence of [[finite sets]], then an [[integral kernel]] $K$ on this, according to def. \ref{IntegralKernel}, with $A_i = 1_{X_i}$, is equivalently an ${\vert X_1\vert}\times {\vert X_2\vert}$-array of elements in $k$, hence a [[matrix]] $K_{\bullet,\bullet}$.

=--

### Parametrized module spectra
 {#ParameterizedModuleSpectra}

The example of parameterized modules [above](#ParameterizedModules) has an evident generalization from [[linear algebra]] to [[stable homotopy theory]] with [[abelian categories|abelian]] [[categories of modules]] refined to [[stable (∞,1)-categories|stable]] [[(∞,1)-categories of ∞-modules]]. Despite what this [[higher category theory]]-terminology might make the reader feel, this refinement flows naturally along the same lines as the 1-categorical situation. One may view the axiomatics of an indexed monoidal $(\infty,1)$-category as neatly characterizing precisely this intimate similarity.

| [[linear algebra]] | [[brave new algebra]] |
|--------------------|-----------------------|
| [[abelian group]]  | [[spectrum]] |
| [[commutative ring]] | [[E-∞ ring|E-∞]] [[ring spectrum]] |
| [[module]] | [[∞-module]] spectrum |
| [[abelian category|abelian]] [[category of modules]] | [[stable (∞,1)-category|stable]] [[(∞,1)-category of ∞-modules]] |

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

Let $E$ be an [[E-∞ ring]] spectrum and write $E Mod$ for its [[(∞,1)-category of ∞-modules]]. For $X \in $ [[∞Grpd]] write

$$
  E Mod(X) \coloneqq Func(X,E Mod)
$$

for the [[(∞,1)-category of (∞,1)-functors]] from $X$ (regarded as an [[(∞,1)-category]]) to $E Mod$, hence for the [[parameterized spectra]] over $X$ with $E$-[[∞-module]] structure. Equipped with the [[smash product of spectra]] this is a [[symmetric monoidal (∞,1)-category]] $E Mod^\otimes$ This construction yields an [[(∞,1)-functor]]

$$
  E Mod \;\colon\; \infty Grpd^{op} \longrightarrow SymMonCat_\infty
$$

This is an indexed monoidal $(\infty,1)$-category.

=--

([Hopkins-Lurie](#HopkinsLurie), [Schreiber 14](#Schreiber14))

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

In the case that $E = \mathbb{S}$ is the [[sphere spectrum]], then $\mathbb{S}Mod \simeq Spectra$ is just the plain [[(∞,1)-category of spectra]] and then the above is the theory of plain [[parameterized spectra]].

In this case the corresponding [[(∞,1)-Grothendieck construction]] is the [[tangent (∞,1)-topos]] $T(\infty Grpd)$ of [[∞Grpd]]. This happens to be itself an [[(∞,1)-topos]], and as such is conjecturally semantics for *ordinary* [[homotopy type theory]].  If ordinary homotopy type theory is enhanced with rules and axioms motivated by such models, it also becomes able to speak about stable homotopy theory and parametrized stable objects; the relation between this theory and [[dependent linear type theory]] remains to be explored.

=--

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

The $\Sigma$-functor, def. \ref{Sigma}, in the model of parameterized $E$-module spectra,
prop. \ref{ParameterizedModuleSpectraAreAnIndexedMonoidalCategory}, is the [[suspension spectrum]] construction $\Sigma^\infty$ followed by [[smash product]] of spectra with $E$:

$$
  \Sigma(X) \simeq \Sigma^\infty(X)\wedge E
  \,.
$$

This is the $E$-[[generalized homology]]-spectrum of the [[∞-groupoid]] $X$.

This construction does have a [[right adjoint]] $\Omega^\infty$, where $(\Sigma^\infty \dashv \Omega^\infty)$ is the $E$-[[stabilization]]-adjunction for [[∞Grpd]]. Hence [[parameterized spectra]] have an [[exponential modality]], def. \ref{SemanticsForExponentialModality}.

=--

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

In the class of models of prop. \ref{ParameterizedModuleSpectraAreAnIndexedMonoidalCategory}, an indexed monoidal $(\infty,1)$-category encodes the theory of [[twisted generalized cohomology]].

[[!include twisted generalized cohomology in linear homotopy type theory -- table]]

=--

### Parameterized formal moduli problems

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

For $\mathbf{H}$ a [[differential cohesive (∞,1)-topos]] with [[infinitesimal shape modality]] $\Pi_{inf}$ write for any object $X\in \mathbf{H}$

$$
  Mod(X)
  \hookrightarrow
   \mathbf{H}_{/X}^{X/}
$$

for the [[full sub-(∞,1)-category]] of that of pointed objects over $X$, def. \ref{PointedObjectsInSlice}, on those that are in the kernel of $\Pi_{inf}$.

=--

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

The construction in \ref{ParameterizedFormalModuliProblems} has the interpretation as  the category of generalized [[formal moduli problems]] parameterized over $X$. The genuine formal moduli problems satisfy one an extra exactness property of the kind discussed at _[[cohesive (∞,1)-presheaf on E-∞ rings]]_.

=--

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

Parameterized formal moduli problems as in def. \ref{ParameterizedFormalModuliProblems} conjecturally form semantics for non-unital
linear homotopy-type theory.

=--

### Quasicoherent sheaves of modules
 {#ForQuasicoherentSheaves}

Pull-push of [[quasicoherent sheaves]] is usually discussed as a [[Grothendieck context]] of [[six operations]], but under some conditions it also becomes a [[Wirthmüller context]] and hence an indexed monoidal $(\infty,1)$-category

Using results of Lurie this follows in the full generality of [[E-∞ geometry]] ([[spectral geometry]]).

Consider quasi-compact and quasi-separated [[E-∞ scheme|E-∞ algebraic spaces]] ([[spectral geometry|spectral algebraic spaces]]). (This includes precisely those [[spectral Deligne-Mumford stacks]] which have a [[scallop decomposition]], see [here](derived+Deligne-Mumford+stack#RelationToDerivedAlgebraicSpaces).)

If $f \;\colon\; X \longrightarrow Y$ is a map between these which is

1. locally almost of finite presentation;

1. strongly proper;

1. has finite [[Tor-amplitude]]

then the left adjoint to pullback of [[quasicoherent sheaves]] exists

$$
  (f_! \dashv f^\ast)
    \;\colon\;
  QCoh(X)
    \stackrel{\overset{f_!}{\longrightarrow}}{\underset{f^\ast}{\longleftarrow}}
  QCoh(Y)
  \,.
$$

([[Proper Morphisms, Completions, and the Grothendieck Existence Theorem|LurieProper, proposition 3.3.23]])

If $f$ is

* quasi-affine

then the right adjoint exists

$$
  (f^\ast \dashv f_\ast)

    \;\colon\;
  QCoh(X)
    \stackrel{\overset{f^\ast}{\longleftarrow}}{\underset{f_\ast}{\longrightarrow}}
  QCoh(Y)
  \,.
$$

([[Quasi-Coherent Sheaves and Tannaka Duality Theorems|LurieQC, prop. 2.5.12]], [[Proper Morphisms, Completions, and the Grothendieck Existence Theorem|LurieProper, proposition 2.5.12]])

The [[projection formula]] in the dual form

$$
  f_\ast A \otimes B \longrightarrow f_\ast (A\otimes f^\ast B)
$$

for $f$ quasi-compact and quasi-separated appears as ([[Proper Morphisms, Completions, and the Grothendieck Existence Theorem|LurieProper, remark 1.3.14]]).

Now if all the conditions on $f$ hold, so that $(f_! \dashv f^\ast \dashv f_\ast) \;\colon\; QCoh(X) \longrightarrow QCoh(Y)$, then passing to opposite categories $QCoh(X)^{op} \longrightarrow QCoh(Y)^{op}$ exchanges the roles of $f_!$ and $f_\ast$, makes the projection formula be as in the above discussion and hence yields a Wirthmüller context.

The existence of [[dualizing modules]] $K$

$$
  D X = [X,K]
$$

is discussed in ([[Representability theorems|Lurie, Representability theorems, section 4.2]].)

### Stable homotopy theory of $G$-equivariant spectra

For a finite group, $G$, consider the spectral [[Mackey functors]] on the category of finite $G$-sets. These are [[G-spectrum|genuine G-spectra]] (see [equivariant stable homotopy theory](equivariant+stable+homotopy+theory#in_terms_of_mackeyfunctors)) which together constitute a $G$-equivariant $\infty$-category (or $G$-$\infty$-category, see [[Parametrized Higher Category Theory and Higher Algebra]]). This sends any orbit $G/H$, for $H$ a subgroup of $G$, to the monoidal $\infty$-category of genuine $H$-spectra.

This forms an indexend monoidal $(\infty,1)$-category, as do other instances of $(\infty, 1)$-categories of $T$-spectra, for $T$ an [orbital](Parametrized+Higher+Category+Theory+and+Higher+Algebra#AtomicOrbital) $\infty$-category.

## Structures in an indexed monoidal $(\infty,1)$-category
 {#Structures}

We discuss here structures (constructions) that may be defined and studied within an indexed monoidal $(\infty,1)$-category.

### Exponential modality and Fock spaces
 {#TheCanonicalComodality}

The original axiomatics for [[linear type theory]] in ([Girard 87](#Girard87)) contain in addition to the structures corresponding to a ([[star-autonomous category|star-autonomous]]) [[symmetric monoidal category|symmetric]] [[closed monoidal category]] a certain (co-)[[modality]] traditionally denoted "$!$", the _[[exponential modality]]_.

In ([Benton 95, p.9,15](#Benton95), [Bierman 95](#Bierman95))
it is noticed (reviewed in ([Barber 97, p. 21 (26)](#Barber97))) that a natural [[categorical semantics]] for this modality identifies it with the [[comonad]] that is induced from a [[strong monoidal adjunction]]

$$
  (\Sigma \dashv \Omega)
  \;\colon\;
  Mod(\ast)
  \stackrel{\overset{\Sigma}{\leftarrow}}{\underset{\Omega}{\longrightarrow}}
  \mathcal{C}
$$

between the [[closed monoidal category|closed]] [[symmetric monoidal category]] $Mod(\ast)$ which interprets the given [[linear type theory]] and a [[cartesian monoidal category]] $\mathcal{C}$.

If there is only the [[strong monoidal functor]] $\Sigma \;\colon\; \mathcal{C} \longrightarrow Mod(\ast)$ without possibly a [[right adjoint]] $\Omega$, then ([Barber 97, p. 21 (27)](#Barber97)) speaks of the _structural [[fragment]]_ of [[linear type theory]].

In ([Ponto-Shulman 12](#PontoShulman12)) it is observed that this in turn is canonically induced if $Mod(\ast)$ is the [[linear type theory]] over the trivial context $\ast$ of a dependent linear type theory ([[indexed closed monoidal category]]) with category of contexts being $\mathcal{C}$:

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

Let $Mod \colon \mathcal{C}^{op} \to MonCat$ be an indexed monoidal $(\infty,1)$-category. Then for $X \in \mathcal{C}$ an object, set

$$
  \Sigma(X) \coloneqq \underset{X}{\sum} 1_X
  \in Mod(\ast)
$$

and for $f \;\colon\; Y \longrightarrow X$ a [[morphism]] in $\mathcal{C}$ set

$$
  \Sigma(f)
    \coloneqq
   \Sigma(Y)
    =
  \underset{Y}{\sum} 1_Y
  \stackrel{\simeq}{\to}
  \underset{X}{\sum} \underset{f}{\sum} f^\ast 1_X
  \stackrel{\underset{X}{\sum}(\epsilon_f)}{\longrightarrow}
  \underset{X}{\sum} 1_X
  =
  \Sigma(X)
  \,.
$$

=--

(In the typical kind of model this means to assign to a [[space]] $X$ the linear space of [[sections]] of the trivial [[line bundle]] over it.)

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

The construction in def. \ref{Sigma} gives a [[strong monoidal functor]]

$$
  \Sigma
  \;\colon\;
  \mathcal{C}
  \longrightarrow
  Mod(\ast)
$$

=--

This is ([Ponto-Shulman 12, (4.3)](#PontoShulman12)).

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

Below in example \ref{SigmaFunctorAsSecondaryTransform} we see that the functor $\Sigma$ in prop. \ref{SigmaFunctor} is a special case of a general construction of secondary integral transforms in an indexed monoidal $(\infty,1)$-category.

=--

This suggests the following definition.

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

Given an indexed monoidal $(\infty,1)$-category such that the functor $\Sigma$ from prop. \ref{SigmaFunctor} has a strong monoidal [[right adjoint]]

$$
  \Omega \colon Mod(\ast) \longrightarrow \mathcal{C}
$$

we refer to the  [[comonad]] of the $(\Sigma \dashv \Omega)$-[[adjunction]]

$$
  ! \coloneqq \Sigma \circ \Omega \colon Mod(\ast) \to Mod(\ast)
  \,.
$$

as the _[[exponential modality]]_ in the conjectural [[linear type theory]] over the point whose semantics is $Mod(\ast)$.

=--

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

The condition in def. \ref{SemanticsForExponentialModality} that $\Sigma$ (and its relative/dependent versions) has a [[right adjoint]] $\Omega$ may also be understood as saying that the dependent linear type theory satisfies the _[[axiom of comprehension]]_. See at _[comprehension -- Examples -- In dependent linear type theory](http://ncatlab.org/nlab/show/axiom+of+separation#ExamplesDependentLinearTypeTheory)_ for more.

=--

### Dependent linear deMorgan duality

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

For $\mathcal{C}^\otimes$ a [[closed monoidal category]] with [[unit object]] $1$ and [[internal hom]] $[-,-]$ write

$$
  \mathbb{D} \coloneqq [-,1]
$$

for the [[functor]] given by internal hom into the unit object.  Syntactically this corresponds to the _linear [[negation]]_ operation.

=--

+-- {: .num_prop #DependentLinearDeMorganDuality}
###### Proposition
**(dependent linear de Morgan duality)**

In an indexed monoidal $(\infty,1)$-category, linear negation, def. \ref{LinearNegation}, intertwines dependent sum and dependent product:

$$
  \underset{f}{\prod} \mathbb{D}
  \simeq
  \mathbb{D} \underset{f}{\sum}
$$

=--

For proof see [here](Wirthmüller%20context#ComparisonOfPushForwardsAndWirthmuelleriso) at _[[Wirthmüller context]]_.

### Primary integral transforms
 {#PrimaryIntegralTransform}

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

Given an indexed monoidal $(\infty,1)$-category $Mod\colon \mathcal{C}^{op}\to SymMonCat$, and given objects $X_1, X_2$ of $\mathcal{C}$ then a **linear [[polynomial functor]]**

$$
  P \colon Mod(X_1) \to Mod(X_2)
$$

is a functor of the form

$$
  P \simeq \underset{f_2}{\sum} \underset{g}{\prod} f_1^\ast
$$

for a [[diagram]] in $\mathcal{C}$ of the form

$$
  \array{
    && Y &\stackrel{g}{\longrightarrow}& Z
    \\
    & {}^{\mathllap{f_1}}\swarrow  & && & \searrow^{\mathrlap{f_2}}
    \\
    X_1
    &&
    &&
    &&
    X_2
  }
  \,.
$$

If here $g = id$ then this diagram is a [[correspondence]]

$$
  \array{
    &&  Z
    \\
    & {}^{\mathllap{f_1}}\swarrow  & & \searrow^{\mathrlap{f_2}}
    \\
    X_1
    &&
    &&
    X_2
  }
$$

and the resulting $P \simeq \underset{f_2}{\sum}f_1 ^\ast$ is called a **linear polynomial functor** or **primary [[integral transform]]**.

=--

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

Given a [[correspondence]] as above, then the primary integral transform through it is equivalent to the pull-tensor-push operation through the [[product]]

$$
  \underset{f_2}{\sum} \circ f_1^\ast
  \simeq
  \underset{p_2}{\sum}
  \circ
  \left( \left(f_1,f_2\right)_!\left(1_Z\right) \otimes \left(-\right)\right)
  \circ
  p_1^\ast
  \,.
$$

=--

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

By [[Frobenius reciprocity]].

=--

### Fundamental classes
 {#FundamentalClasses}

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

A _fiberwise twisted fundamental class_ $[f]$ on a morphism $f \colon X\to Y$ in $\mathcal{C}$ is

1. a choice of dualizable object $\tau \in Mod(Y)$ (the _twist_);

1. a choice of equivalence

   $$
     \underset{f}{\sum} f^\ast 1_X
     \stackrel{\simeq}{\longrightarrow}
     \underset{f}{\prod} f^\ast \tau
     \,.
   $$

=--

In this form this is stated in ([Schreiber 14](#Schreiber14)).

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

A special case of def. \ref{FiberwiseFundamentalClass} is obtained when the twist vanishes, $\tau = 1$, and when dependent sum and dependent product are naturally equivalent for all objects, $\sum_f \simeq \prod_f$. In this case the two adjoints to $f^\ast$ coincide to form an [[ambidextrous adjunction]]. This case is considered in ([Hopkins-Lurie](#HopkinsLurie)).

=--

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

In view of dependent linear de Morgan duality, prop. \ref{DependentLinearDeMorganDuality}, a fiberwise twisted fundamental class in def. \ref{FiberwiseFundamentalClass} is equivalently a choice of equivalence

$$
  \underset{f}{\sum} f^\ast 1_X
  \stackrel{\simeq}{\longrightarrow}
  \mathbb{D}\underset{f}{\sum} f^\ast \mathbb{D}\tau
  \,,
$$

hence an identification of the dependent sum of the unit with the dual of the dependent sum of the twist.

=--

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

A twisted fiberwise fundamental class $[f]$, def. \ref{FiberwiseFundamentalClass}, induces for all dualizable $A\in Mod(X)$ a [[natural equivalence]]

$$
  \mathbb{D} \underset{f}{\sum} f^\ast \mathbb{D}(A\otimes \tau)
  \simeq
  \underset{f}{\sum}f^\ast A
  \,.
$$

=--

This is the "[Wirthmüller isomorphism](Wirthmüller%20context#ComparisonOfPushForwardsAndWirthmuelleriso)".

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

For $[f]$ a twisted fiberwise fundamental class and for $A$ dualizable, write

$$
  [f]_A
  \;\colon\;
  A \otimes \tau
  \stackrel{\mathbb{D}_{\epsilon_{\mathbb{D}(A \otimes \tau)}}}{\longrightarrow}
  \mathbb{D} \underset{f}{\sum} f^\ast (\mathbb{D}(A\otimes \tau))
  \stackrel{\simeq}{\longrightarrow}
  \underset{f}{\sum}f^\ast A
$$

for the composite of the linear dual of the $(\sum_f \dashv f^\ast)$-[[counit of an adjunction|adjunction counit]] and the Wirthmüller isomorphism, prop. \ref{WirthmullerIsomorphism}.

=--

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

The key point is that the morphism in def. \ref{ComponentOfFiberwiseClass} goes in the reverse direction of the [[counit of an adjunction|adjunction counit]]. In this way it plays a role in the construction of secondary integral transforms [below](#SecondaryIntegralTransforms).

=--

### Secondary integral transforms
 {#SecondaryIntegralTransforms}

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

For

$$
  \array{
    && Z
    \\
    & {}^{\mathllap{f_1}}\swarrow && \searrow^{\mathrlap{f_2}}
    \\
    X_1
    && &&
    X_2
  }
$$

a [[correspondence]] as in def. \ref{PolynomialFunctorsAndCorrespondences}, then an _[[integral kernel]]_ for it is the choice of

1. two dualizable objects $A_i \in Mod(X_i)$;

1. a morphism between their pullbacks to the correspondence space:

   $$
     f_1^\ast A_1 \stackrel{K}{\longleftarrow} f_2^\ast A_2
     \,.
   $$

=--

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

Given

1. a [[correspondence]] $X_1 \stackrel{f_1}{\longleftarrow} Z \stackrel{f_2}{\longrightarrow} X_2$ as in def. \ref{PolynomialFunctorsAndCorrespondences};

1. an [[integral kernel]] $K$ as in def. \ref{IntegralKernel};

1. a $\tau$-twisted fundamental class $[f_2]$ as in def. \ref{FiberwiseFundamentalClass}

we say that the induced _secondary integral transform_ is the morphism

$$
  \int_Z \Xi d \mu_f
  \;\colon\;
  \mathbb{D} \underset{X_1}{\sum} A_1
  \longrightarrow
  \mathbb{D}\underset{X_2}{\sum} (A_2\otimes \tau)
$$

which is the dual (the image under $\mathbb{D}(-)$) of the following composite:

$$
  \underset{X_1}{\sum} A_1
  \stackrel{\underset{X_1}{\sum}\epsilon_{A_1}}{\longleftarrow}
  \underset{X_1}{\sum}
  \underset{f_1}{\sum}
  f_1^\ast A_1
  \stackrel{\simeq}{\leftarrow}
  \underset{Z}{\sum}f_1^\ast A_1
  \stackrel{\underset{Z}{\sum} K}{\longleftarrow}
  \underset{Z}{\sum}f_2^\ast A_2
  \stackrel{\simeq}{\leftarrow}
  \underset{X_2}{\sum}\underset{f_2}{\sum} f_2^\ast A_2
  \stackrel{\underset{X_2}{\sum}[f_2]_{A_2}}{\longleftarrow}
  \underset{X_2}{\sum} (A_2\otimes \tau)
  \,,
$$

where the morphism on the left is the [[counit of an adjunction|adjunction counit]], the morphism in the middle is the given [[integral kernel]], and the morphism on the right is the given fundamental class.

=--

In this form this appears in ([Schreiber 14](#Schreiber14)). Specialized to the ambidextrous case of remark \ref{FundamentalClassFromAmbidexterity}, this is equivalent to the construction in ([Hopkins-Lurie](#HopkinsLurie)).

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

A [[correspondence]] $X_1 \stackrel{f_1}{\longleftarrow} Z \stackrel{f_2}{\longrightarrow} X_2$ may be thought of as a space $Z$ of "paths" or "trajectories" that connect points in $X_2$ to points in $X_1$. From this point of view definition \ref{SIT} is a linear map that takes functions on $X_2$ to functions on $X_1$ by pointwise forming a sum over paths connecting these points and adding all the contributions of the given [[integral kernel]] along these paths.

This is conceptually just how the [[path integral]] in [[physics]] is supposed to work, only that mostly it doesn't, due to lack of a proper definition. However, at least some path integrals for [[topological field theories]] may be realized as secondary integral transforms of the above kind.

Notice that while in [[modal type theory]] the ([[comonad|co-]])[[monads]] $(f^\ast \sum_f \dashv f^\ast \prod_f)$ are pronounced as _[[possibility]]_ and _[[necessity]]_, the monad $\prod_f f^\ast$ appearing above, via def. \ref{FiberwiseFundamentalClass}, may be pronounced _randomness_, see at _[[function monad]]_ for more.

=--

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

Consider the special case def. \ref{SIT} where the right leg of the correspondence is the identity

$$
  \array{
    && Z
    \\
    & {}^{\mathllap{f}}\swarrow && \searrow^{\mathrlap{=}}
    \\
    X && && Z
  }
$$

and where the integral kernel is the identity

$$
  K = id_{1_Z} \colon id_Z^\ast 1_Z = 1_Z \to 1_Z = f^\ast 1_X
  \,.
$$

Then the associated secondary integral transform, def. \ref{SIT}, is the morphism

$$
  \underset{X}{\sum} 1_X
  \stackrel{\underset{\epsilon_f}{\sum}}{\longleftarrow}
  \underset{X}{\sum} \underset{f}{\sum} f^\ast 1_X
  \stackrel{\simeq}{\longleftarrow}
  \underset{Z}{\sum} 1_Z
  \,.
$$

This is just the $\Sigma$-functor of def. \ref{Sigma}:

$$
  \mathbb{D}\int_Z id \, d\mu_f
  =
  \Sigma(f)
  \;
  \colon
  \;
  \Sigma(Z)
  \stackrel{\Sigma(f)}{\longleftarrow}
  \Sigma(X)
  \,.
$$

=--

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

Given a correspondence with integral kernel as in example \ref{MatrixAsIntegralKernel}, then the induced secondary integral transform according to def. \ref{SIT} is the [[linear function]]

$$
  k^{\vert X_1\vert}
  \longleftarrow
  k^{\vert X_2\vert}
$$

represented by that matrix.

=--

## Related concepts

* [[type theory]], [[homotopy type theory]]

* [[modal type theory]]

* [[stable homotopy type]]

* [[geometric homotopy type theory]]

* [[cohesive homotopy type theory]]

## References
 {#References}

Plain [[linear type theory]] originates in

*  [[Jean-Yves Girard]], _Linear logic_,   Theoretical Computer Science 50:1, 1987.  ([pdf](http://iml.univ-mrs.fr/~girard/linear.pdf))
 {#Girard87}

The [[categorical semantics|categorical interpretation]] of Girard's $!$-[[modality]] as a [[comonad]] is due to

* P. N. Benton, G. M. Bierman, [[Martin Hyland]], [[Valeria de Paiva]], _Term assignment for intuitionistic linear logic_, Technical Report 262,
Computer Laboratory, University of Cambridge, August 1992.

and that this is naturally to be thought of as arising from a [[monoidal adjunction]] between the closed [[symmetric monoidal category]] and a [[cartesian closed category]] is due to

* {#Bierman95} G. Bierman, _On Intuitionistic Linear Logic_ PhD thesis, Computing
Laboratory, University of Cambridge, 1995 ([pdf](http://research.microsoft.com/~gmb/papers/thesis.pdf))

* {#Benton95} [[Nick Benton]], *A mixed linear and non-linear logic: Proofs, terms and models*, in *Computer Science Logic. CSL 1994*, Lecture Notes in Computer Science **933** [[doi:10.1007/BFb0022251](https://doi.org/10.1007/BFb0022251), [[BentonLinearLogic.pdf:file]]]

A review of all this and further discussion is in

* {#Barber97} Andrew Graham Barber, _Linear Type Theories, Semantics and Action Calculi_, 1997 ([web](http://www.lfcs.inf.ed.ac.uk/reports/97/ECS-LFCS-97-371/), [pdf](http://www.lfcs.inf.ed.ac.uk/reports/97/ECS-LFCS-97-371/ECS-LFCS-97-371.pdf))

The 1-categorical version of [[indexed closed monoidal categories]] is discussed in

* {#Shulman08} [[Mike Shulman]], _Framed bicategories and monoidal fibrations_, in  Theory and Applications of Categories,  Vol. 20, 2008, No. 18, pp 650-738.  ([arXiv:0706.1286](http://arxiv.org/abs/0706.1286), [TAC](http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html))

* {#PontoShulman12} [[Kate Ponto]], [[Mike Shulman]], _Duality and traces for indexed monoidal categories_, Theory and Applications of Categories, Vol. 26, 2012, No. 23, pp 582-659 ([arXiv:1211.1555](http://arxiv.org/abs/1211.1555))

* {#Shulman12} [[Mike Shulman]], _Enriched indexed categories_ ([arXiv:1212.3914](http://arxiv.org/abs/1212.3914))

Comments on the formalization of secondary [[integral transforms]] and [[path integral]] [[quantization]] in an indexed monoidal $(\infty,1)$-category are in

* {#Schreiber14} [[Urs Schreiber]], _[[schreiber:Quantization via Linear homotopy types]]_ ([arXiv:1402.7041](http://arxiv.org/abs/1402.7041))

* {#HopkinsLurie} [[Michael Hopkins]], [[Jacob Lurie]], _[[Ambidexterity in K(n)-Local Stable Homotopy Theory]]_

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


indexed monoidal category

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

# Indexed monoidal categories
* table of contents
{: toc}

## Definition

### Indexed monoidal category

An **indexed monoidal category** is a kind of [[indexed category]], consisting of a base [[category]] $S$ and a [[pseudofunctor]] $S^{op} \to MonCat$ to the [[2-category]] of [[monoidal categories]] and [[strong monoidal functors]] between them. We write this as $A\mapsto (C^A, \otimes_A, I_A)$.

By the usual [[Grothendieck construction]], this pseudofunctor can be regarded as a [[fibration]].  And if $S$ has [[finite products]], then the "fiberwise" monoidal structures $\otimes_A$ can also be "Grothendieckified" into an "[[external tensor product]]"
$$ \boxtimes\colon C^A \times C^B \to C^{A\times B}$$
defined by $M\boxtimes N = \pi_2^\ast M \otimes_{A\times B} \pi_1^\ast N$.  This makes the total category of the fibration a [[monoidal category]] and the fibration itself a strong [[monoidal functor]] (where $S$ is regarded as equipped with its [[cartesian monoidal category|cartesian monoidal structure]]); this is called a **[[monoidal fibration]]**.  Moreover, we can recover $\otimes_A$ from $\boxtimes$ via $M\otimes_A N = \Delta_A^\ast (M\boxtimes N)$, so the two structures have the same information. ([Shulman 08](#Shulman08)).

### Indexed closed monoidal category
 {#Closed}

If all the fibers are not just monoidal but [[closed monoidal categories]] and the [[base change]] morphisms are not just strong monoidal but also strong [[closed monoidal functors]], then the indexed monoidal category is an indexed _closed monoidal category_ ([Shulman 08, def. 13.1](#Shulman08), [Shulman 12, theorem 2.14](#Shulman12)).

If in addition all fibers are [[symmetric monoidal category|symmetric monoidal]] one might also call this a system of [[Wirthmüller contexts]] of [[six operations]]. If furthermore all fibers have all [[dual objects|duals]], then this is also what should be called [[categorical semantics]] for [[dependent linear type theory]].

## Examples

* $S=Sets$, $C^A=$ $A$-indexed families of objects of $C$, for any monoidal category $C$.
* $S=Gpd$, $C^A=$ $A$-diagrams of objects of $C$
* $S=$ any category with pullbacks, $C^A = S/A$
* $S=Top$, $C^A=$ [[parametrized spectra]] over $A$
* $S=Grp$ or $TopGrp$, $C^A=$ sets or spaces with an action by $A$
* The homotopy category of any of the above equipped with a homotopy theory

In many cases, the reindexing functors $f^\ast\colon C^B \to C^A$ induced by a morphism $f\colon A\to B$ in $S$ all have left adjoints $f_!$.  If these left adjoints satisfy the [[Beck-Chevalley condition]] for all pullback squares in $S$, then the indexed category is traditionally said to have **indexed coproducts**.  For many applications, though, we only need this condition for a few pullback squares, which coincidentally (?) happen to be those that are pullbacks in any category with finite products (whether or not it even has all pullbacks).

## Related concepts

* [[external tensor product]]

* [[dependent linear type theory]], [[Wirthmüller context]]

* [[monoidal Grothendieck construction]]

## References

The notion of indexed monoidal category has been rediscovered many times by different people in different contexts.  Some references include:

* M. F. Gouzou and R. Grunig, *Fibrations relatives*, Seminaire de Theorie des
Categories, dirige par J. Benabou, November 1976

* A. Asperti and A. Corradini. A categorical modal for logic programs: indexed monoidal categories. In:J.W. de Bakker and W.P. de Roever, Ed. REX Workshop, 110–137, Lecture Notes in ComputerScience, Vol. 666. Springer Verlag, New York-Berlin, 1993.

* G. Amato and J. Lipton, Indexed categories and bottom-up semantics of logic programs. Lecture Notesin Computer Science, Vol. 2250. Springer Verlag, New York-Berlin, 2001

* [[Pieter Hofstra]], Federico De Marchi, _Descent for Monads_, in  Theory and Applications of Categories, **16** 24 (2006) 668–699.  ([tac:16-24](http://www.tac.mta.ca/tac/volumes/16/24/16-24abs.html))

* {#Shulman08} [[Mike Shulman]], _Framed bicategories and monoidal fibrations_, in  Theory and Applications of Categories,  Vol. 20, 2008, No. 18, pp 650-738.  ([TAC](http://www.tac.mta.ca/tac/volumes/20/18/20-18abs.html))

* {#Shulman12} [[Mike Shulman]], *Enriched indexed categories*, Theory Appl. Categ. **28** (2013) 616-695 [[arXiv:1212.3914](http://arxiv.org/abs/1212.3914), [tac:28-21](http://www.tac.mta.ca/tac/volumes/28/21/28-21abs.html)]

Discussion of [[traces]] and of [[dual objects]] in indexed monoidal categories is in

* {#PontoShulman12} [[Kate Ponto]], [[Mike Shulman]], *Duality and traces in indexed monoidal categories*, Theory and Applications of Categories **26** 23 (2012)  [[arXiv:1211.1555](http://arxiv.org/abs/1211.1555), [tac:26-23](http://www.tac.mta.ca/tac/volumes/26/23/26-23abs.html),  [blog](http://golem.ph.utexas.edu/category/2011/11/traces_in_indexed_monoidal_cat.html)]

This also presents a [[string diagram]] calculus for indexed monoidal categories, extending that for monoidal [[hyperdoctrines]] in

* {#BradyTrimble98} Geraldine Brady, [[Todd Trimble]], _[[A string diagram calculus for predicate logic]]_ (1998)

which is made rigorous in:

* [[Cary Malkiewich]], [[Kate Ponto]], _Coherence for indexed symmetric monoidal categories_ $[$[arxiv:1811.12873](https://arxiv.org/abs/1811.12873)$]$

On the [[monoidal Grothendieck construction]]:

* {#Moeller_Vasilakopoulou19} [[Joe Moeller]], [[Christina Vasilakopoulou]], _Monoidal Grothendieck Construction_, Theory and Applications of Categories, **35** 31 (2020) 1159-1207 $[$[arXiv:1809.00727](http://arxiv.org/abs/1809.00727), [tac:35-31](www.tac.mta.ca/tac/volumes/35/31/35-31abs.html)$]$

[[!redirects indexed monoidal category]]
[[!redirects indexed monoidal categories]]

[[!redirects indexed closed monoidal category]]
[[!redirects indexed closed monoidal categories]]


indexed topos

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

#Contents#
* table of contents
{:toc}

## Definition

Let $\mathcal{S}$ be a [[topos]], regarded as a [[base topos]].

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

An **$\mathcal{S}$-indexed topos** $\mathbb{E}$ is an $\mathcal{S}$-[[indexed category]] such that

* for each object $I \in \mathcal{S}$ the fiber $\mathbb{E}^I$ is a [[topos]];

* for each morphism $x : I \to J$ in $\mathcal{S}$ the corresponding transition functor $x^* : \mathbb{E}^J \to \mathbb{E}^I$ is a [[logical morphism]].

An **$\mathcal{S}$-indexed geometric morphism** is an $\mathcal{S}$-indexed [[adjunction]] $(f^* \dashv f_*)$ between $\mathcal{S}$-indexed toposes, such that $f^*$ is left exact.

This yields a [[2-category]] $Topos_{\mathcal{S}}$ of $\mathcal{S}$-indexed toposes.

=--

This appears at ([Johnstone, p. 369](#Johnstone)).

## Examples

* For $p : \mathcal{E} \to \mathcal{S}$ a [[geometric morphism]], the induced morphism $\mathbb{E} \to \mathbb{S}$ (discussed at [[base topos]]) is an $\mathcal{S}$-indexed topos.

## Properties

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

Write [[Topos]]$/\mathcal{S}$ for the [[slice 2-category]] of toposes over $\mathcal{S}$. This is a [[full sub-2-category]] $\mathcal{S}$-indexed toposes

$$
  Topos/{\mathcal{S}} \hookrightarrow Topos_{\mathcal{S}}
  \,.
$$

=--

This appears as ([Johnstone, prop. 3.1.3](#Johnstone)).

## Related concepts

* [[topos]], [[base topos]]

* [[indexed category]]

* [[hyperdoctrine]]

## References

Section B3.1 of

* [[Peter Johnstone]], _[[Sketches of an Elephant]]_
 {#Johnstone}

[[!redirects indexed toposes]]
[[!redirects indexed topoi]]


indiscrete category

# Indiscrete categories
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## Definitions

An **indiscrete category** is a [[category]] $C$ in which there is a unique [[morphism]] from each [[object]] $x$ to each object $y$:
$$
  \forall x,y \in Obj(C) : C(x,y) = *
  \,,
$$
where $*$ is the [[point]].

The terms **chaotic category**, and **codiscrete category** are also used.

## Properties

This means that

* an indiscrete category is in fact a [[groupoid]], in fact a [[codiscrete groupoid]];
* any [[inhabited set|inhabited]] indiscrete category is [[equivalence of categories|equivalent]] to the [[terminal category]].

Therefore, up to [[equivalence of categories|equivalence]], an indiscrete category is simply a [[truth value]].

The functor $Ind\colon Set \to Str Cat$ sending a [[set]] $X$ to the indiscrete category with $X$ as its set of objects (viewed as a [[strict category]], that is up to [[isomorphism of categories|isomorphism]]) is [[adjoint functor|right adjoint]] to the [[forgetful functor]] $Ob\colon Str Cat \to Set$ sending a category to its set of objects.  (The [[adjoint functor|left adjoint]] $Disc$ to this forgetful functor sends a set $X$ to the [[discrete category]] on $X$.)

Of course, we can compose $Ind$ (or $Disc$) with the forgetful functor from $Str Cat$ to the [[2-category]] $Cat$, in which we consider categories up to [[equivalence of categories|equivalence]], as usual.  Then the composite
$$ Set \overset{Ind}\to Str Cat \to Cat $$
is [[natural equivalence|naturally equivalent]] to the composite
$$ Set \overset{Inh}\to TV \overset{Subs}\to Set \overset{Disc}\to Str Cat \to Cat ,$$
where $TV$ is the set (viewed a $2$-category) of [[truth values]], $Inh$ takes a set to the truth value of the statement that it is [[inhabited set|inhabited]], $Pt$ takes a truth value to a [[subsingleton]] ([[left adjoint]] to $Inh$), and $Disc$ is as above.

##Related concepts

* [[codiscrete groupoid]]

* [[category enriched in a bicategory]]

[[!redirects indiscrete category]]
[[!redirects indiscrete categories]]
[[!redirects chaotic category]]
[[!redirects chaotic categories]]


induced character

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### Context
#### Group Theory
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[[!include group theory - contents]]
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# Induced characters
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## Definition

Let $\phi\colon H\to G$ be a [[group homomorphism]], $V$ a [[representation]] of $H$, and $\chi$ the [[character]] of $V$.  The **induced character** $\phi_!(\chi)$ of $f$ is the character of the [[induced representation|induced]] $G$-representation
$$\phi_!(V) = Ind^G_H(V) = V\otimes_{k[H]} k[G].$$

## Formula

There is a formula for the induced character:

$$ \phi_!(\chi)(g) = \frac{1}{|H|} \sum_{k^{-1} g k = \phi(h)} \chi(h) $$

where the sum is over all pairs $(k\in G, h\in H)$ such that $k^{-1} g k = \phi(h)$.

This formula is usually given only in the case when $\phi$ is injective, when it can be re-expressed as a sum over [[cosets]].  The case when $\phi$ is surjective is [Exercise 7.1 of (Serre)](http://books.google.com/books?id=NCfZgr54TJ4C&pg=PA57&lpg=PA57) and the general case is easy to put together from these.  It can also be derived abstractly using [[bicategorical trace]].

## References

* [[J-P. Serre]], *Linear Representations of Finite Groups*

* [MO question](http://mathoverflow.net/questions/89928/induced-character-for-non-injective-homomorphisms) about the non-injective case.

[[!redirects induced characters]]


induced comodule

Given a commutative unital ring $k$ and a morphism $D\to C$ of $k$-coalgebras, one can consider the dualized notion of induced module, using the [[cotensor product]] instead of tensor product.

If $D$ is [[flat module|flat]] as a $k$-module (e.g. $k$ is a field),
$N$ is a left $D$- right $C$-bicomodule and $M$ is a left $C$-comodule,
then the cotensor product $N \Box_C M$ is a $D$-subcomodule of
$N \otimes_k M$. In particular, under the flatness assumption, if $\pi : D \rightarrow C$ is a surjection of
coalgebras then $D$ is a left $D$- right $C$-bicomodule
via $\Delta_D$ and $(\id \otimes \pi) \circ \Delta_D$
respectively, hence $\mathrm{Ind}^D_C := D \Box^C -$
is a functor from left $C$- to left
$D$-comodules called the induction functor for left comodules from $C$ to $D$.

One can consider this construction more generally for [[coring]]s.

* K.-H. Ulbrich, _On modules indued or coinduced from Hopf subalgebras_, Mathematica Scandinavica __67__:2 (1990) 177-182
* S. Caenepeel, S. Raianu, F. van Oystaeyen, _Induction and coinduction for Hopf algebras: applications_, J. Algebra __165__:1 (1994) 204-222, [doi](https://doi.org/10.1006/jabr.1994.1107)
* [[T. Brzeziński]], _The structure of corings. Induction functors, Maschke-type theorem, and Frobenius and Galois properties_, Algebras Represent. Theory __5__ (2002) 389–410
* Johan Kustermans, _Induced corepresentations of locally compact quantum groups_, J. Funct. Analysis __194__:2 (2002) 410-459 [doi](https://doi.org/10.1006/jfan.2002.3953)
* Stefaan Vaes, _A new approach to induction and imprimitivity results_, J. Funct. Analysis __229__:2 (2005) 317-374 [doi](https://doi.org/10.1016/j.jfa.2004.11.016)


induced metric

A [[subset]] of a [[metric space]] canonically inherits a [[metric]] by restriction. This is called the _induced metric_.

The corresponding [[metric topology]] is the [[subspace topology]].

[[!redirects induced metric]]

[[!redirects induced metric space]]
[[!redirects induced metric spaces]]