tensor category

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## Idea
 {#Idea}

A _tensor category_ is usually understood to be a [[monoidal category]] equipped with further "[[linear algebra|linear algebraic]]" [[properties]] and [[structure]], hence with [[monoidal category|monoidal]]-[[structure]] given by a kind of [[tensor product]] in the original sense (i.e. actually being a universal [[bilinear map]] of sorts) whence the name.

Conventions differ, but at the very least one means

* a [[monoidal category]],

which is typically required to be

* [[symmetric monoidal category|symmetric monoidal]]

   (and if it is only [[braided monoidal category|braided monoidal]] one speaks of a _quasitensor category_),

* ([[Ab]], [[tensor product of abelian groups|$\otimes$]])-[[enriched category|enriched]] or ([[Vect]],[[tensor product of vector spaces|$\otimes$]])-[[enriched category|enriched]],

  to make an [[enriched monoidal category]]

and, in addition, often

* [[additive category|additive]] (symmetric) monoidal, so that the [[tensor product]] [[preserved limit|preserves]] [[finite product|finite]] [[direct sums]],

* [[abelian category|abelian]] (symmetric) monoidal, in which the [[tensor product]] [[preserved colimit|preserves]] [[finite colimits]] in separate arguments,

* with [[dual objects]], making a [[rigid monoidal category]].

## Properties

### Tannaka theory, Deligne's theorem, super-representation theory

[[Deligne's theorem on tensor categories]] ([Deligne 02](#Deligne02)) establishes [[Tannaka duality]] between sufficiently well-behaved linear tensor categories in [[characteristic zero]] and [[supergroups]], realizing these tensor categories as [[categories of representations]] of these supergroups.

## Related concepts

* [[fusion category]]

* [[tensor triangulated category]]

* [[tensor (∞,1)-category]]

* [[2-algebraic geometry]]

* [[tensor network]]

## References

* {#Deligne90} [[Pierre Deligne]], section 2 of _[[Catégories Tannakiennes]]_, Grothendieck Festschrift, vol. II, Birkhäuser Progress in Math. 87 (1990) pp. 111-195 ([pdf](https://publications.ias.edu/sites/default/files/60_categoriestanna.pdf))

* [[Bojko Bakalov]], [[Alexander Kirillov]], *Lectures on tensor categories and modular functors*, University Lecture Series **21**, Amer. Math. Soc. (2001) [[web](http://www.math.stonybrook.edu/~kirillov/tensor/tensor.html), [ams:ulect/21](https://bookstore.ams.org/view?ProductCode=ULECT/21)]

  > (focus on [[Reshetikhin-Turaev construction]] of [[modular functors]] from [[modular tensor categories]])

* [[Masaki Kashiwara]], [[Pierre Schapira]], Section 4 of: *[[Categories and Sheaves]]*, Grundlehren der Mathematischen Wissenschaften **332**, Springer (2006) [[doi:10.1007/3-540-27950-4](https://link.springer.com/book/10.1007/3-540-27950-4), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf)]

* [[Damien Calaque]], [[Pavel Etingof]], _Lectures on tensor categories_, IRMA Lectures in Mathematics and Theoretical Physics 12, 1-38 (2008) ([arXiv:math/0401246](https://arxiv.org/abs/math/0401246))

* [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], _Topics in Lie theory and Tensor categories -- 9 Tensor categories_, Lecture notes (spring 2009) ([pdf](http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/MIT18_769S09_lec09.pdf) [web](http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/))

* {#EGNO15} [[Pavel Etingof]], [[Shlomo Gelaki]], [[Dmitri Nikshych]], [[Victor Ostrik]], *Tensor Categories*, AMS Mathematical Surveys and Monographs **205** (2015) [[ISBN:978-1-4704-3441-0](https://bookstore.ams.org/surv-205), [pdf](http://www-math.mit.edu/~etingof/egnobookfinal.pdf)]

[[Deligne's theorem on tensor categories]] is due to

* {#Deligne02} [[Pierre Deligne]], _Catégorie Tensorielle_, Moscow Math. Journal 2 (2002) no. 2, 227-248. ([pdf](https://www.math.ias.edu/files/deligne/Tensorielles.pdf))

Review in:

* {#Ostrik04} [[Victor Ostrik]], _Tensor categories (after P. Deligne)_ ([arXiv:math/0401347](http://arxiv.org/abs/math/0401347))

[[!redirects quasitensor category]]

[[!redirects tensor categories]]


tensor contraction

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# Contraction
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## Contraction of tensors

By [[tensor]] we mean a vector in some [[tensor power]] $V^{\otimes n}$ of a vector $k$-space $V$ (or a projective $k$-module if $k$ is only a commutative ring). Let $V^* = Hom_k(V,k)$ be the dual vector space and $(V^*)^{\otimes m}$ be some tensor of $V^*$. Then one may define $(l,s)$-contraction
$$(V^*)^{\otimes m}\otimes V^{\otimes n}\to (V^*)^{\otimes (m-1)}\otimes V^{\otimes (n-1)}$$
by pairing by the evaluation map the $l$-th tensor factor of $(V^*)^{\otimes r}$ and $s$-th tensor factor of $V^{\otimes n}$. In fact as a map written, one can contract also elements of $(V^*)^{\otimes m}\otimes V^{\otimes n}$ which did not come from a product of a pair of element (i.e. which are not [[decomposable tensor]]s).

Let the rank $r$ of $V$ be finite.
If $S\in V^{\otimes n}$ is given in some basis by components $S^{i_1,\ldots, i_n}$ and $T\in (V^*)^{\otimes r}$ is given in the dual basis by components $T_{j_1,\ldots,j_r}$, then the components of the contraction will be
$$contr_{l,s}(T,S)^{i_1,\ldots, i_{s-1},i_{s+1},\ldots,i_n}_{j_1,\ldots, j_{l-1},j_{l+1},\ldots,j_m} = \sum_{u = 1}^r T_{j_1,\ldots, j_{l-1},u,j_{l+1},\ldots,j_m} S^{i_1,\ldots, i_{s-1},u,i_{s+1},\ldots,i_n}$$

More generally one can contract mixed tensors and do several contractions simultaneously. In fact it is better to think of a contraction as a tensor multiplication of two tensors and then doing the genuine contraction of one upper and one lower index of the same tensor:
$$
contr_{l,s}(A)^{i_1,\ldots, i_{s-1},i_{s+1},\ldots,i_n}_{j_1,\ldots,j_{l-1},j_{l+1},\ldots, j_m}) := \sum_{u = 1}^r A^{i_1,\ldots, i_{s-1},u,i_{s+1},\ldots,i_n}_{j_1,\ldots,j_{l-1},u,j_{l+1},\ldots, j_m}
$$

The simplest case is the [[trace]] of a $(1,1)$-tensor: $tr A = \sum_{i=1}^r A^i_i$.

These operations can be symmetrized or antisymmetrised appropriately to make sense on symmetric or antisymmetric powers.

For example, there is a contraction of a vector $X\in V$ and a $n$-form $\omega\in \Lambda V^*$:
$$(X,\omega)\mapsto \iota_X(\omega)$$
and $\iota_X: \omega\mapsto \iota_X(\omega)$ is a graded derivation of the exterior algebra of degree $-1$. This is also done for the tangent bundle which is a $C^\infty(M)$-module $V = T M$, then one gets the contraction of [[vector fields]] and [[differential forms]]. It can also be done in vector spaces, fibrewise.

## Related concepts

* [[tensor algebra]]

* [[symmetric algebra]]

* [[exterior algebra]]

* [[divided power algebra]]

* [[multilinear algebra]]

## References

* wikipedia: [tensor contraction](http://en.wikipedia.org/wiki/Tensor_contraction)

* Shlomo Sternberg, _Introduction to differential geometry_

[[!redirects tensor contraction]]
[[!redirects tensor contractions]]


tensor hierarchy

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## Idea

In [[gauged supergravity]] the [[gauge symmetry|gauging]] of a [[subgroup]] of a [[global symmetry|global]] [[U-duality]] [[symmetry group]] (via a choice of [[embedding tensor]]) in general requires the existence of a hierarchy of [[differential p-form]]-[[field (physics)|fields]] to ensure the gauge covariance of the resulting [[field strengths]]. This is known as the _[[tensor hierarchy]]_ ([de Wit-Samtleben 05](#deWitSamtleben05), [deWit-Samtleben 08](#deWitSamtleben08))

## Definition

[[!include embedding tensors and tensor hierarchy in super Lie theory -- section]]

## Related concepts

* [[gauged supergravity]]

* [[embedding tensor]]

* [[EL-∞ algebra|$E L_\infty$-algebra]]

* [[adjusted Weil algebra]]

## References

The concept originates with

* {#deWitSamtleben05} [[Bernard de Wit]], [[Henning Samtleben]], _Gauged maximal supergravities and hierarchies of nonabelian vector-tensor systems_, Fortsch. Phys. 53 (2005) 442-449 ([arXiv:hep-th/0501243](https://arxiv.org/abs/hep-th/0501243))

* {#deWitSamtleben08} [[Bernard de Wit]], [[Henning Samtleben]], _The end of the $p$-form hierarchy_, JHEP 0808:015, 2008 ([arXiv:0805.4767](https://arxiv.org/abs/0805.4767))

See also

* Jesper Greitz, [[Paul Howe]], [[Jakob Palmkvist]], _The tensor hierarchy simplified_, Class. Quantum Grav. 31 (2014) 087001 ([arXiv:1308.4972](https://arxiv.org/abs/1308.4972))

Possible relation to [[U-duality]] and [[M-theory]]:

* [[Bernard de Wit]], [[Hermann Nicolai]], [[Henning Samtleben]], _Gauged Supergravities, Tensor Hierarchies, and M-Theory_, JHEP 0802:044, 2008 ([arXiv:0801.1294](https://arxiv.org/abs/0801.1294))

Relation to [[Borcherds algebras]]:

* {#Palmkvist13} [[Jakob Palmkvist]], _The tensor hierarchy algebra_, J. Math. Phys. 55, 011701 (2014) ([arXiv:1305.0018](https://arxiv.org/abs/1305.0018))

Relation to [[Leibniz algebras]]:

* [[Sylvain Lavau]], _Tensor hierarchies and Leibniz algebras_, J. Geom. Phys. 144:147-189 (2019) ([arXiv:1708.07068](https://arxiv.org/abs/1708.07068))

A proposal for understanding tensor hierarchies as [[dg-Lie algebra]]/[[L-infinity algebra]]-refinenents of [[Leibniz algebras]]:

* {#LavauPalmkvist19} [[Sylvain Lavau]], [[Jakob Palmkvist]], _Infinity-enhancing of Leibniz algebras_ ([arXiv:1907.05752](https://arxiv.org/abs/1907.05752))

* [[Sylvain Lavau]], [[Jim Stasheff]], _$L_\infty$-algebra extensions of Leibniz algebras_ ([arXiv:2003.07838](https://arxiv.org/abs/2003.07838))

A proposal for understanding tensor hierarchies via [[higher gauge theory]] with [[adjusted Weil algebras]]:

* [[Leron Borsten]], [[Hyungrok Kim]], [[Christian Saemann]].  *$E L_\infty$-algebras, Generalized Geometry, and Tensor Hierarchies* [[arXiv:2106.00108](https://arxiv.org/abs/2106.00108)]

survey and review:

* [[Christian Saemann]], *$E L_\infty$-algebras, Generalized Geometry, and Tensor Hierarchies*, talk at [SFT@Cloud 2021](https://indico.cern.ch/event/1042834/) (2021) [[pdf](https://indico.cern.ch/event/1042834/contributions/4487418/attachments/2313514/3937748/saemann_generalized_geometry.pdf), [[Saemann-ELInfinityAndTensorHierarchies.pdf:file]]]

[[!redirects tensor hierarchies]]


tensor linear category

A [[monoidal category|monoidal]] [[linear category]], see at *[[tensor category]]*.

[[!redirects tensor linear categories]]


tensor meson

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## Idea

By a _tensor meson_ the literature typically means a type of [[meson]] whose [[Wigner classification]] is a [[Lorentz group|Lorentz]]-[[spin group]] spin-2 [[linear representation|representation]], hence given by a symmetric rank-2 [[tensor]].

Indeed, in the [[AdS/QCD correspondence]] these "tensor mesons" are [[AdS/CFT duality|dual]] to [[graviton]]-modes in the $\sim$ [[bulk spacetime]].

Beware that there are also [[mesons]] given by _anti_-symmetric rank-2 tensors (the [[h1-meson]] and [[b1-meson]]), but since these are "spin-1", they are counted as [[vector mesons]].

More generally, there are [[higher spin mesons]], which hence could also be called "tensor mesons".

## Related concepts

* [[scalar meson]], [[vector meson]]

## References

### General

* F. Giacosa, Th. Gutsche, V. E. Lyubovitskij, Amand Faessler, _Decays of tensor mesons and the tensor glueball in an effective field approach_, Phys. Rev. D72 (2005) 114021 ([arXiv:hep-ph/0511171](https://arxiv.org/abs/hep-ph/0511171))

* J.Y. Süngü, A. Türkan, E. Sertbakan, E. Veli Veliev, _Axial-tensor Meson Family at $T \neq 0$_ ([arXiv:2005.11526](https://arxiv.org/abs/2005.11526))

### In holographic QCD

Discussion of tensor mesons in [[holographic QCD]]:

* [[Emanuel Katz]], [[Adam Lewandowski]], [[Matthew Schwartz]], _Tensors Mesons in AdS/QCD_, Phys. Rev. D74:086004, 2006 ([arXiv:hep-ph/0510388](https://arxiv.org/abs/hep-ph/0510388))

[[!redirects tensor mesons]]


tensor network

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## Idea
 {#Idea}

The term _tensor network_ has become popular in [[quantum physics]] for essentially what in [[mathematical physics]] is known as _[[Penrose notation]]_ and in [[monoidal category|monoidal]] [[category theory]] is referred to as _[[string diagrams]]_ (see [BCJ 10](#BCJ10) for a bilingual account).

The term rose to prominence in [[quantum physics]] partly with discussion of [[finite quantum mechanics in terms of dagger-compact categories]] but mainly via its use for representing highly [[entangled states]] in the  discussion of [[renormalization]] of [[non-perturbative effects]] in [[solid state physics]] ([Swingle 09](#Swingle09), [Swingle 13](#Swingle13)) and the resulting discovery of the relation to [[holographic entanglement entropy]] and thus to the [[AdS/CFT correspondence]].

In this context, a _tensor network_ is a _[[string diagram]] [[concept with an attitude|with an attitude]], in that it is (just) a [[string diagram]] (typically regarded in the [[monoidal category]] of [[finite-dimensional vector spaces]], but possibly also in [[super vector spaces]] etc.), but with:

1. the [[tensor product]] of all its external [[objects]] regarded as a [[space of states]] of a [[quantum system]];

1. the [[element]] in that [[tensor product]] defined by the string diagram regarded as a [[state]] ([[wave function]]) of that quantum system.

For instance, if $\mathfrak{g}$ is a [[metric Lie algebra]] (with [[string diagram]]-notation as shown [there](metric+Lie+algebra#Definition)), and with each [[tensor product]]-power of its [[dual vector space]] regarded as a [[Hilbert space]], hence as a [[space of quantum states]], via the given [[inner product]] on $\mathfrak{g}$, then an example of a _[[tree tensor network state]]_ is the following:

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

The [[quantum states]] arising this way are generically highly [[entangled state|entangled]]: roughly they are the more entangled the more [[vertices]] there are in the corresponding tensor network.

The following is an example of what is called a [[matrix product state]]:

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

The [[tree tensor network states]] in the form of [[Bruhat-Tits trees]] play a special role in the [[AdS/CFT correspondence]], either as

1. a kind of [[lattice QFT]]-approximation to an actual [[boundary field theory|boundary]] [[CFT]] [[quantum state]],

1. as the [[p-adic geometry|p-adic geometric]] incarnation of [[anti de Sitter spacetime]] in the [[p-adic AdS/CFT correspondence]],

1. as a reflection of actual [[crystal]]-site quantum states in [[AdS/CFT in solid state physics]]:

> graphics from [[schreiber:Differential Cohomotopy implies intersecting brane observables|Sati-Schreiber 19c]]

(As in [HMSS 16](p-adic+AdS/CFT+correspondence#HMSS16), [HLM 19](p-adic+AdS/CFT+correspondence#HLM19). But  maybe one wants the Poincaré-dual networks, instead, as in [HMPS 18](p-adic+AdS/CFT+correspondence#HMPS18)?)

For [[tensor network states]] dual to tesselations of [[hyperbolic space]] and made from [[perfect tensors]] one finds that the [[entanglement entropy]] of the tensor subspace associated with an [[interval]] on the [[boundary]] becomes proportional, for large number of [[vertices]], to the [[hyperbolic space|hyperbolic]] bulk boundary [[length]] of the segment of the tree network that ends on this interval, according to the [[Ryu-Takayanagi formula]] ([PYHP 15, Theorem 2](#PYHP15)). Hence one speaks of _[[holographic entanglement entropy]]_. For more on this see [below](#ForHolographicEntanglementEntropy).

General tensor network states are mixtures of the above examples. For instance in ... is the suggestion that [[BTZ black holes]] are encoded by networks that look like [[Bruhat-Tits trees]] towards the boundaries and like [[matrix product states]] towards the interior:

(As in [ESZ 19](p-adic+AdS/CFT+correspondence#ESZ19).)

\linebreak

## For holographic entanglement entropy
 {#ForHolographicEntanglementEntropy}

>  under construction

Appearance in [[renormalization]]:

> from [Swingle 09](#Swingle09)

Application to [[holographic entanglement entropy]] (...)

> graphics grabbed from [Harlow 18](#Harlow18), see at *[[HaPPY code]]*

> graphics grabbed from [Harlow 18](#Harlow18), see at *[[HaPPY code]]*

In this context the [[Ryu-Takayanagi formula]] for [[holographic entanglement entropy]] has an exact proof [PYHP 15, Theorem 2](#PYHP15).

## Examples

* [[matrix product state]]

* [[tree tensor network state]]

* [[MERA state]]

## Related concepts

* [[string diagram]]/[[Penrose notation]]

* [[finite quantum mechanics in terms of dagger-compact categories]]

* [[quantum many-body physics]], [[condensed matter physics]]

* [[quantum computing]], [[quantum information theory]]

  [[quantum error correction]]

* [[tensor]], [[tensor product]], [[tensor category]]

* [[neural network]]

[[!include states and observables -- content]]

## References

### General

Review and exposition:

* {#BCJ10} [[Jacob Biamonte]], Stephen R. Clark, Dieter Jaksch, _Categorical Tensor Network States_, AIP Advances 1(4), 042172 (2011) ([arXiv:1012.0531](https://arxiv.org/abs/1012.0531))

* [[Roman Orus]], _A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States_, 	Annals of Physics 349 (2014) 117-158  ([arXiv:1306.2164](https://arxiv.org/abs/1306.2164))

* [[Jens Eisert]], _Entanglement and tensor network states_, Modeling and Simulation 3, 520 (2013) ([arXiv:1308.3318](https://arxiv.org/abs/1308.3318))

* [[Jacob Biamonte]], Ville Bergholm, _Tensor Networks in a Nutshell_, Contemporary Physics ([arxiv:1708.00006](https://arxiv.org/abs/1708.00006))

* [[Bei Zeng]], [[Xie Chen]], [[Duan-Lu Zhou]], [[Xiao-Gang Wen]]:

  Part IV of: *[[Quantum Information Meets Quantum Matter]] -- From Quantum Entanglement to Topological Phases of Many-Body Systems*, Quantum Science and Technology (QST), Springer (2019) $[$[arXiv:1508.02595](https://arxiv.org/abs/1508.02595), [doi:10.1007/978-1-4939-9084-9](https://doi.org/10.1007/978-1-4939-9084-9)$]$

* [[Roman Orus]], _Tensor networks for complex quantum systems_, Nature Reviews Physics 1, 538-550 (2019) ([arXiv:1812.04011](https://arxiv.org/abs/1812.04011), [doi:10.1038/s42254-019-0086-7](https://doi.org/10.1038/s42254-019-0086-7))

* {#Biamonte19} [[Jacob Biamonte]], _Lectures on Quantum Tensor Networks_ ([arXiv:1912.10049](https://arxiv.org/abs/1912.10049))

* Shi-Ju Ran, Emanuele Tirrito, Cheng Peng, Xi Chen, [[Luca Tagliacozzo]], Gang Su, Maciej Lewenstein, _Tensor Network Contractions -- Methods and Applications to Quantum Many-Body Systems_, Lecture Notes in Physics, Springer (2020) ([arXiv:1708.09213](https://arxiv.org/abs/1708.09213), [doi:10.1007/978-3-030-34489-4](https://link.springer.com/book/10.1007/978-3-030-34489-4))

With [[global symmetry]]:

* Sukhwinder Singh, Robert N. C. Pfeifer, [[Guifre Vidal]], _Tensor network decompositions in the presence of a global symmetry_, Phys. Rev. A82:050301, 2010 ([arXiv:0907.2994](https://arxiv.org/abs/0907.2994))

* Sukhwinder Singh, Robert N. C. Pfeifer, [[Guifre Vidal]], _Tensor network states and algorithms in the presence of a global $U(1)$ symmetry_, Phys. Rev. B 83, 115125 (2011) ([arXix:1008.4774](https://arxiv.org/abs/1008.4774))

* {#SinghVidal12} Sukhwinder Singh, [[Guifre Vidal]], _Tensor network states and algorithms in the presence of a global $SU(2)$ symmetry_, Phys. Rev. B 86, 195114 (2012) ([arXiv:1208.3919](https://arxiv.org/abs/1208.3919))

Further resources:

* [Tensor Network Initiative](https://www.perimeterinstitute.ca/research/research-initiatives/tensor-networks-initiative)

* [tensornetwork.org](https://tensornetwork.org/)

### Application in solid state physics

Discussion of [[solid state physics]] via [[tensor network states]]:

General:

* [[Jens Eisert]], M. Cramer, M.B. Plenio, _Area laws for the entanglement entropy - a review_, Rev. Mod. Phys. 82, 277 (2010) ([arXiv:0808.3773](https://arxiv.org/abs/0808.3773))

* [[Roman Orus]], _A Practical Introduction to Tensor Networks: Matrix Product States and Projected Entangled Pair States_, 	Annals of Physics 349 (2014) 117-158  ([arXiv:1306.2164](https://arxiv.org/abs/1306.2164))

* [[Jens Eisert]], _Entanglement and tensor network states_, Modeling and Simulation 3, 520 (2013) ([arXiv:1308.3318](https://arxiv.org/abs/1308.3318))

* [[Thorsten B. Wahl]], _Tensor network states for the description of quantum many-body systems_ ([arXiv:1509.05984](https://arxiv.org/abs/1509.05984))

* Kouichi Okunishi, Tomotoshi Nishino, Hiroshi Ueda, *Developments in the Tensor Network -- from Statistical Mechanics to Quantum Entanglement* ([arXiv:2111.12223](https://arxiv.org/abs/2111.12223))

Concrete materials:

* A. Kshetrimayum, C. Balz, B. Lake, [[Jens Eisert]], _Tensor network investigation of the double layer Kagome compound $Ca_{10} Cr_{7} O_{28}$_ ([arXiv:1904.00028](https://arxiv.org/abs/1904.00028))

Classification of ([[topological phases of matter|topological]]) [[phases of matter]] via [[tensor network states]]:

* C. Wille, O. Buerschaper, [[Jens Eisert]], _Fermionic topological quantum states as tensor networks_, Phys. Rev. B 95, 245127 (2017) ([arXiv:1609.02574](https://arxiv.org/abs/1609.02574))

* Andreas Bauer, [[Jens Eisert]], Carolin Wille, _Towards a mathematical formalism for classifying phases of matter_ ([arXiv:1903.05413](https://arxiv.org/abs/1903.05413))

Tensor networks for [[string-net models]] for [[topological order]]:

On [[string-net models]] via [[tensor networks]]:

* O. Buerschaper, M. Aguado, G. Vidal, *Explicit tensor network representation for the ground states of string-net models*, Phys. Rev. B **79** (2009) 085119 $[$[arXiv:0809.2393](https://arxiv.org/abs/0809.2393), [doi:10.1103/PhysRevB.95.195110](https://doi.org/10.1103/PhysRevB.95.195110)$]$

* [[Zheng-Cheng Gu]], [[Michael Levin]], [[Brian Swingle]], [[Xiao-Gang Wen]], *Tensor-product representations for string-net condensed states*, Phys. Rev. B **79** (2009) 085118 $[$[doi:10.1103/PhysRevB.79.085118](https://doi.org/10.1103/PhysRevB.79.085118)$]$

### In holographic entanglement entropy

The use of [[tensor networks]] as a tool in [[holographic entanglement entropy]] goes back to

* {#Swingle09} [[Brian Swingle]], _Entanglement Renormalization and Holography_, Phys. Rev. D 86, 065007 (2012) ([arXiv:0905.1317](https://arxiv.org/abs/0905.1317))

* {#Swingle12} [[Brian Swingle]], _Constructing holographic spacetimes using entanglement renormalization_ ([arXiv:1209.3304](https://arxiv.org/abs/1209.3304), [spire:1185813](http://inspirehep.net/record/1185813))

Review in

* [[Frank Verstraete]], J. I. Cirac, V. Murg, _Matrix Product States, Projected Entangled Pair States, and variational renormalization group methods for quantum spin systems_, Adv. Phys. 57,143 (2008) ([arXiv:0907.2796](https://arxiv.org/abs/0907.2796))

### In the $p$-adic AdS/CFT correspondence

Tensor networks associated with [[Bruhat-Tits trees]] in [[p-adic AdS/CFT correspondence]]:

* [[Matthew Heydeman]], [[Matilde Marcolli]], [[Ingmar Saberi]], [[Bogdan Stoica]], _Tensor networks, $p$-adic fields, and algebraic curves: arithmetic and the $AdS_3/CFT_2$ correspondence_ ([arXiv:1605.07639](https://arxiv.org/abs/1605.07639))

* Arpan Bhattacharyya, Ling-Yan Hung, Yang Lei, Wei Li, _Tensor network and ($p$-adic) AdS/CFT_, JHEP 1801 (2018) 139 ([arXiv:1703.05445](https://arxiv.org/abs/1703.05445))

* Ling-Yan Hung, Wei Li, Charles M. Melby-Thompson, _$p$-adic CFT is a holographic tensor network_ ([arXiv:1902.01411](https://arxiv.org/abs/1902.01411))

* Lin Chen, Xirong Liu, Ling-Yan Hung, _Emergent Einstein Equation in p-adic CFT Tensor Networks_ ([arXiv:2102.12022](https://arxiv.org/abs/2102.12022))

* Lin Chen, Xirong Liu, Ling-Yan Hung, _Bending the Bruhat-Tits Tree I:Tensor Network and Emergent Einstein Equations_ ([arXiv:2102.12023](https://arxiv.org/abs/2102.12023))

* Lin Chen, Xirong Liu, Ling-Yan Hung, _Bending the Bruhat-Tits Tree II: the p-adic BTZ Black hole and Local Diffeomorephism on the Bruhat-Tits Tree_ ([arXiv:2102.12024](https://arxiv.org/abs/2102.12024))

Discussion of [[BTZ black holes]] via [[tensor networks]] in the [[p-adic AdS/CFT correspondence]]:

* Matthew Heydeman, [[Matilde Marcolli]], Sarthak Parikh, [[Ingmar Saberi]], _Nonarchimedean Holographic Entropy from Networks of Perfect Tensors_ ([arXiv:1812.04057](https://arxiv.org/abs/1812.04057))

See also

* [[Alexander Jahn]], [[Zoltán Zimborás]], [[Jens Eisert]], _Central charges of aperiodic holographic tensor network models_ ([arXiv:1911.03485](https://arxiv.org/abs/1911.03485))

  ([[quasicrystal]] boundary states)

* [[Alexander Jahn]], [[Zoltán Zimborás]], [[Jens Eisert]], _Tensor network models of AdS/qCFT_ ([arXiv:2004.04173](https://arxiv.org/abs/2004.04173))

### In quantum computation

In [[quantum computation]]:

* {#KimSwingle17} [[Isaac H. Kim]], [[Brian Swingle]], *Robust entanglement renormalization on a noisy quantum computer* ([arXiv:1711.07500](https://arxiv.org/abs/1711.07500))

  > (specifically for [classically controlled quantum computation](quantum+computation#ClassicalControlQuantumData) in terms of [[holographic entanglement entropy|holographic]] [[tensor network]] states)

### As quantum error correcting codes

Further interpretation of [[tensor networks]] in terms of [[quantum error correcting codes]]:

* Andrew J. Ferris, David Poulin, *Tensor Networks and Quantum Error Correction*, Phys. Rev. Lett. 113, 030501 (2014) ([arXiv:1312.4578](https://arxiv.org/abs/1312.4578))

and specifically in the [[HaPPY code]] modelling [[holographic entanglement entropy]]:

* {#ADH14} [[Ahmed Almheiri]], [[Xi Dong]], [[Daniel Harlow]], _Bulk Locality and Quantum Error Correction in AdS/CFT_, JHEP 1504:163,2015 ([arXiv:1411.7041](https://arxiv.org/abs/1411.7041))

* {#PYHP15} Fernando Pastawski, Beni Yoshida, [[Daniel Harlow]], [[John Preskill]], _Holographic quantum error-correcting codes: Toy models for the bulk/boundary correspondence_, JHEP 06 (2015) 149 ([arXiv:1503.06237](https://arxiv.org/abs/1503.06237))

reviewed in

* {#Harlow18} [[Daniel Harlow]], _TASI Lectures on the Emergence of Bulk Physics in AdS/CFT_ ([arXiv:1802.01040](https://arxiv.org/abs/1802.01040))

Generalization to the [[Majorana dimer code]]:

* [[Alexander Jahn]], [[Marek Gluza]], [[Fernando Pastawski]], [[Jens Eisert]], _Majorana dimers and holographic quantum error-correcting codes_, Phys. Rev. Research 1, 033079 (2019) ([arXiv:1905.03268](https://arxiv.org/abs/1905.03268))

reviewed in:

* [[Alexander Jahn]], [[Jens Eisert]], _Holographic tensor network models and quantum error correction: A topical review_ ([arXiv:2102.02619](https://arxiv.org/abs/2102.02619))

Discussion of [[asymptotic boundaries]] of hyperbolic tensor networks as [[quasicrystals]]:

* Latham Boyle, Madeline Dickens, Felix Flicker, _Conformal Quasicrystals and Holography_, Phys. Rev. X 10, 011009 (2020) ([arXiv:1805.02665](https://arxiv.org/abs/1805.02665))

See also

* Ning Bao, [[Geoffrey Penington]], Jonathan Sorce, Aron C. Wall, _Beyond Toy Models: Distilling Tensor Networks in Full AdS/CFT_ ([arXiv:1812.01171](https://arxiv.org/abs/1812.01171))

* Ning Bao, [[Geoffrey Penington]], Jonathan Sorce, Aron C. Wall, _Holographic Tensor Networks in Full AdS/CFT_ ([arXiv:1902.10157](https://arxiv.org/abs/1902.10157))

* {#JGPE19} [[Alexander Jahn]], [[Marek Gluza]], [[Fernando Pastawski, [[Jens Eisert]], _Majorana dimers and holographic quantum error-correcting codes_ ([arXiv:1905.03268](https://arxiv.org/abs/1905.03268))

### Relation to spin chains

Relation to [[spin chains]]:

* Mari Carmen Banuls, Michal P. Heller, Karl Jansen, Johannes Knaute, Viktor Svensson, _From spin chains to real-time thermal field theory using tensor networks_ ([arXiv:1912.08836](https://arxiv.org/abs/1912.08836))

### Application to lattice gauge theory

Application to [[lattice gauge theory]]:

* [[Luca Tagliacozzo]], Alessio Celi, Maciej Lewenstein, _Tensor Networks for Lattice Gauge Theories with continuous groups_, Phys. Rev. X 4, 041024 (2014) ([arXiv:1405.4811](https://arxiv.org/abs/1405.4811))

* {#BBCCCDFJLMMRRTVV19} M.C. Bañuls, R. Blatt, J. Catani, A. Celi, J.I. Cirac, M. Dalmonte, L. Fallani, K. Jansen, M. Lewenstein, S. Montangero, C.A. Muschik, B. Reznik, E. Rico, [[Luca Tagliacozzo]], K. Van Acoleyen, [[Frank Verstraete]], U.-J. Wiese, M. Wingate, J. Zakrzewski, P. Zoller:

  _Simulating Lattice Gauge Theories within Quantum Technologies_

  ([arXiv:1911.00003](https://arxiv.org/abs/1911.00003))

### In higher parallel transport

Discussion in relation to [[higher parallel transport]]:

* {#Parzygnat18} [[Arthur Parzygnat]], _Two-dimensional algebra in lattice gauge theory_, Journal of Mathematical Physics 60, 043506 (2019) ([arXiv:1802.01139](https://arxiv.org/abs/1802.01139), [doi:10.1063/1.5078532](https://doi.org/10.1063/1.5078532))

### Related to language models

* [[Vasily Pestun]], [[Yiannis Vlassopoulos]], _Tensor network language model_, [arXiv:1710.10248](https://arxiv.org/abs/1710.10248)

[[!redirects tensor networks]]

[[!redirects tensor network state]]
[[!redirects tensor network states]]


tensor power

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## Definition

In a [[monoidal category]] $(C,\otimes)$ with [[tensor product]] $\otimes$ we say that for $n \in \mathbb{N}$ a [[natural number]] and $V \in C$ any [[object]], that

$$
  V^{\otimes n} := V \otimes V \otimes \cdots \otimes V \;\; (n \;factors)
$$

is the $n$th **tensor power** of $V$.

There is accordingly also the $n$th tensor power of any [[morphism]] $f : V \to W$, being a morphism $f^{\otimes n} : V^{\otimes n} \to W^{\otimes n}$.

This process defines a functor
$$
  (-)^{\otimes n} : C \to C
$$

which could be called the $n$th **tensor power functor**.

## Properties

### Schur functors

If $C$ is a suitable [[linear category]], the $n$th tensor power functor is a simple example of a [[Schur functor]].

### Tensor algebra

The [[coproduct]] of all of the tensor powers of $V$ naturally inherits the structure of a [[monoid]] in $C$. This is called the __tensor algebra__ of $V$. This is the [[free monoid]] object on $V$. For more on this see [[category of monoids]].

## Examples

Often in the literature this is considered for the case $C = $ [[Vect]] of [[vector space]]s.
Given a [[vector space]] $V$, the $n$-fold [[tensor product]] of this space with itself, $V \otimes \cdots \otimes V$, is usually denoted $V^{\otimes n}$ and called the $n$th **tensor power** of $V$.

## Related concepts

* [[free monoid]], [[list monad]]

* [[cofree coalgebra]]

* [[Fock space]], [[exponential modality]]

* [[extended power]]

[[!redirects tensor power]]
[[!redirects tensor powers]]

[[!redirects tensor algebra]]
[[!redirects tensor algebras]]


tensor product

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## Idea

The term **tensor product** has many different but closely related meanings.

* In its original sense a _tensor product_  is a representing object for a suitable sort of _[[bilinear map]]_ and _[[multilinear map]]_.  The most classical versions are for [[vector spaces]] ([[modules]] over a [[field]]), more generally [[modules]] over a [[ring]], and even more generally [[algebra over a monad|algebras]] over a [[commutative monad]].  In modern language this takes place in a [[multicategory]].

* Consequently, the [[functor]] $\otimes : C \times C \to C$ which is part of the data of any [[monoidal category]] $C$ is also often called a _tensor product_, since in many examples of monoidal categories it is induced from a tensor product in the above sense (and in fact, _any_ monoidal category underlies a multicategory in a canonical way).  In parts of the literature (certain) [[abelian category|abelian]] monoidal categories are even addressed as _tensor categories_.

* Given two objects in a [[monoidal category]] $(C,\otimes)$ with a right and left [[action]], respectively, of some [[monoid]] $A$, their _tensor product over $A$_ is the quotient of their tensor product in $C$ by this action.  If $A$ is commutative, then this is a special case of the tensor product in a multicategory.

* This generalizes to modules over [[monad|monads]] in a bicategory, which includes the notion of _[[tensor product of functors]]_.

* Finally, tensor products in a multicategory and tensor products over monads in a bicategory are both special cases of tensor products in a [[virtual double category]].

## Definitions

### In a multicategory

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

For $M$ a [[multicategory]] and $A$ and $B$ [[objects]] in $M$, the **tensor product** $A \otimes B$ is defined to be an object equipped with a [[universal construction|universal]] [[multimorphism]]  $A,B\to A \otimes B$ in that any multimorphism  $A,B\to C$ factors uniquely through $A,B\to A \otimes B$ via a (1-ary) morphism $A \otimes B\to C$.

=--

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

$M$ is the category [[Ab]] of abelian groups, made into a [[multicategory]] using [[multilinear maps]] as the multimorphisms, then we get the usual [[tensor product of abelian groups]]. That is, $A \otimes B$ is equipped with a universal map from $A \times B$ (as a set) to $C$ such that this map is linear (a group homomorphism) in each argument separately.  This tensor product can also be constructed explicitly by

1. starting with the cartesian product $A\times B$ in sets,
1. generating a _free_ abelian group from it, and then
1. quotienting by relations $(a_1,b)+(a_2,b)\sim (a_1+a_2,b)$ and $(a,b_1)+(a,b_2)\sim (a,b_1+b_2)$.  (The 0-ary relations $(0,b)\sim 0$ and $(a,0)\sim 0$ follow automatically; you need them explicitly if you generalise to [[abelian monoid]]s.)

=--

Note that in this case, $A\otimes B$ is not a [[subobject]] _or_ a [[quotient]] of the [[cartesian product]] $A\times B$.  However, in many other cases the tensor product in a multicategory _can_ be obtained as a quotient of some other pre-existing product; see _[[tensor product of modules]]_ below.

Other examples of tensor products in multicategories:

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

The [[Gray tensor product]] of [[strict 2-category|strict 2-categories]] is a tensor product in the multicategory of 2-categories and [[cubical functor]]s.  Likewise for Sjoerd Crans' tensor product of Gray-categories.

=--

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

In particular, any [[closed category]] (even if not monoidal) has an underlying multicategory.  Tensor products in this multicategory are characterized by the adjointness relation
$$ \hom(A\otimes B, C) \cong \hom(A, \hom(B,C)). $$
This may be the oldest notion of tensor product, since the definition of the internal-hom of abelian groups and vector spaces, unlike that of their tensor product, is intuitively obvious.

=--

While the universal property referred to above (every bilinear map $A,B\to C$ factors uniquely through $A,B\to A\otimes B$ via a map $A\otimes B \to C$) suffices to define the tensor product, it does not suffice to prove that it is associative and unital.  For this we need the stronger property that any multilinear map $D_1,\dots,D_m,A,B,E_1,\dots, E_n \to C$ factors uniquely through $A,B\to A\otimes B$ via a multilinear map $D_1,\dots,D_m, A\otimes B ,E_1,\dots, E_n \to C$.

### In terms of heteromorphisms

An alternative approach is to define the tensor product via an inter-categorical universal property involving [[heteromorphism|heteromorphisms]]. Tensor products do not always arise via an adjunction, but we can observe that $ hom (a \otimes b, c) \simeq het (\langle a, b \rangle, c) $ in general. That is to say, any morphism from $a \otimes b$ to $c$ in some category $C$ corresponds to a heteromorphism from $\langle a, b \rangle$ in $C \times C$ to $c$ in $C$. In other words, the tensor product is a left representation of $het (\langle a, b \rangle, c)$.

When tensor products exist, we have a canonical het $\eta_{\langle a, b \rangle} \colon \langle a, b \rangle \to a \otimes b$ from $id_{a \otimes b} \in hom (a \otimes b, a \otimes b)$. Given another het $\phi \colon \langle a, b \rangle \to c$, we get the following commutative diagram.

$$
\begin{array}{cccC}
                                                 & {\langle a, b \rangle} &  &   & \\
          \eta_{\langle a, b \rangle} & \downarrow           & \overset{\phi}\searrow       &     & \\
                                                 & a \otimes b            & \underset{f}\dashrightarrow & c &  \\ \end{array}
$$

This represents an example of a more general method for translating universal properties in multicategories into ones involving heteromorphisms.

### Of modules in a monoidal category

Let $R$ be a [[commutative ring]] and consider the multicategory $R$[[Mod]] of $R$-[[modules]] and $R$-[[multilinear maps]].  In this case the
[[tensor product of modules]] $A\otimes_R B$ of $R$-modules $A$ and $B$ can be constructed as the [[quotient]] of the [[tensor product of abelian groups]] $A\otimes B$ underlying them by the [[action]] of $R$; that is,
$$ A\otimes_R B = A\otimes B / (a,r\cdot b) \sim (a\cdot r,b). $$

More category-theoretically, this can be constructed as the [[coequalizer]] of the two maps
$$ A\otimes R \otimes B \;\rightrightarrows\; A\otimes B $$
given by the action of $R$ on $A$ and on $B$.

If $R$ is a [[field]], then $R$-modules are vector spaces; this gives probably the most familiar case of a tensor product spaces, which is also probably the situation where the concept was first conceived.

This tensor product can be generalized to the case when $R$ is not commutative, as long as $A$ is a right $R$-module and $B$ is a left $R$-module.  More generally yet, if $R$ is a [[monoid]] in any [[monoidal category]] (a ring being a monoid in [[Ab]] with its tensor product), we can define the tensor product of a left and a right $R$-module in an analogous way.  If $R$ is a commutative monoid in a [[symmetric monoidal category]], so that left and right $R$-modules coincide, then $A\otimes_R B$ is again an $R$-module, while if $R$ is not commutative then $A\otimes_R B$ will no longer be an $R$-module of any sort.

* Not all tensor products in multicategories are instances of this construction.  In particular, the tensor product in [[Ab]] is not the tensor product of modules over any monoid in the cartesian monoidal category [[Set]].  Abelian groups _can_ be considered as "sets with an action by something," but that something is more complicated than a monoid: it is a special sort of [[monad]] called a [[commutative theory]].

* Conversely, if $R$ is a _commutative_ monoid in a symmetric monoidal category, there is a multicategory of $R$-modules whose tensor product agrees with the coequalizer defined above, but if $R$ is not commutative this is impossible.  However, see the section on tensor products in virtual double categories, below.

### Of modules in a bicategory

The tensor product of left and right modules over a noncommutative monoid in a monoidal category is a special case of the tensor product of modules for a [[monad]] in a [[bicategory]].  If $R: x\to x$ is a monad in a bicategory $B$, a right $R$-module is a 1-cell $A: y\to x$ with an action by $R$, a left $R$-module is a 1-cell $B: x\to z$ with an action by $R$, and their tensor product, if it exists, is a 1-cell $y\to z$ given by a similar coequalizer.  Regarding a monoidal category as a 1-object bicategory, this recovers the above definition.

For example, consider the bicategory $V-Mat$ of $V$-valued [[matrix|matrices]] for some monoidal category $V$.  A monad in $V-Mat$ is a $V$-[[enriched category]] $A$, an $(A,I)$-bimodule is a functor $A\to V$, an $(I,A)$-bimodule is a functor $A^{op}\to V$, and their tensor product in $V-Mat$ is a classical construction called the **[[tensor product of functors]]**.  It can also be defined as a [[coend]].

### In a virtual double category

A [[virtual double category]] is a common generalization of a multicategory and a bicategory (and actually of a [[double category]]).  Among other things, it has objects, 1-cells, and "multi-2-cells."  We leave it to the reader to define a notion of tensor product of 1-cells in such a context, analogous to the tensor product of objects in a multicategory.  A multicategory can be regarded as a 1-object virtual double category, so this generalizes the notion of tensor product in a multicategory.

On the other hand, in any bicategory (in fact, any double category) there is a virtual double category whose objects are monads and whose 1-cells are bimodules, and the tensor product in this virtual double category is the tensor product of modules in a bicategory defined above.  Thus, tensor products in a virtual double category include all notions of tensor product discussed above.

## Examples

* [[tensor product of abelian groups]]

* [[tensor product of groups]]

* [[tensor product of modules]]

* [[tensor product of vector spaces]]

  * [[inductive tensor product]]

* [[smash product]] of [[pointed sets]]

* [[tensor product of representations]]

* [[tensor product of vector bundles]]

* [[tensor product of algebras]]

* [[tensor product of algebras over a commutative monad]]

* [[tensor product of Lie algebras]]

* [[tensor product of chain complexes]]

* [[tensor product of Banach spaces]]

* [[tensor product of functors]]

* [[enriched product category|tensor product of enriched categories]]

* [[Deligne tensor product of abelian categories]]

* [[tensor product of presentable (infinity,1)-categories]]

* in terms of [[linear type theory]] the tensor product is the [[categorical semantics]] of the [[multiplicative conjunction]].

## Related concepts

* [[tensor calculus]]

* [[cartesian product]]

* [[external tensor product]]

* [[internal hom]]

* [[cotensor product]]

* [[pushout product]]

* [[multiplicative conjunction]], [[quantum entanglement]]

* [[composite system]]

[[!include homotopy-homology-cohomology]]

## References

Tensor products were introduced by [[Hassler Whitney]] in

* [[Hassler Whitney]], _Tensor products of Abelian groups_, Duke Mathematical Journal 4:3 (1938), 495-528.  [doi](https://doi.org/10.1215/s0012-7094-38-00442-9).

Lecture notes:

* [[Keith Conrad]], *Tensor products* [part I:[pdf](http://www.math.uconn.edu/~kconrad/blurbs/linmultialg/tensorprod.pdf), [[Conrad-TensorProductsI:file]]; part II: [pdf](https://kconrad.math.uconn.edu/blurbs/linmultialg/tensorprod2.pdf), [[Conrad-TensorProductsII:file]]]

For more on the

* [[tensor product of modules]]

* [[tensor product of vector spaces]]

see there.

For the [[category theory|category theoretic]] formalization of tensor products see the references at

* *[[monoidal category]]* and *[[tensor category]]*

Formalization of tensor products in [[dependent linear type theory]]:

* {#Riley22Thesis} [[Mitchell Riley]], *A Bunched Homotopy Type Theory for Synthetic Stable Homotopy Theory*, PhD Thesis (2022) [[doi:10.14418/wes01.3.139](https://doi.org/10.14418/wes01.3.139), [ir:3269](https://digitalcollections.wesleyan.edu/object/ir%3A3269), [pdf](https://mvr.hosting.nyu.edu/pubs/thesis.pdf)]

[[!redirects tensor products]]


tensor product of abelian groups

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## Idea

For $A$ and $B$ two [[abelian groups]],
their _tensor product_ $A \otimes B$ is a new abelian group
such that a [[group homomorphism]] $A \otimes B \to C$
is equivalently a [[bilinear map]] out of $A$ and $B$.

Tensor products of [[abelian groups]] were defined by [[Hassler Whitney]] in 1938.

## Definition
 {#Definition}

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

Let [[Ab]] be the collection of [[abelian groups]], regarded as a [[multicategory]] whose [[multimorphisms]] are the [[multilinear maps]] $A_1, \cdots, A_n \to B$.

The [[tensor product]] $A, B \mapsto A \otimes B$ in this multicategory is the **tensor product of abelian groups**.

=--

Equivalently this means explicitly:

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

For $A, B$ two [[abelian groups]], their **tensor product of abelian groups** is the abelian group $A \otimes B$ which is the [[quotient]] of the [[free abelian group]] on the [[product]] of their underlying sets $A \times B$
by the [[generators and relations|relations]]

* $(a_1,b)+(a_2,b)\sim (a_1+a_2,b)$

* $(a,b_1)+(a,b_2)\sim (a,b_1+b_2)$

for all $a, a_1, a_2 \in A$ and $b, b_1, b_2 \in B$.

=--

In words: it is the group whose elements are presented by pairs of elements in $A$ and $B$ and such that the group operation for one argument fixed is that of the other group in the other argument.

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

If one generalises to abelian [[semigroups]], this definition of the tensor product also defines the __tensor product of abelian semigroups__.

=--

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

The 0-ary relations $(0,b)\sim 0$ and $(a,0)\sim 0$ follow automatically; one needs them explicitly only if one generalises to [[abelian monoids]].

=--

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

By definition of the [[free construction]] and the [[quotient]] there is a canonical [[function]] of the underlying sets

$$
  p_{A,B}
  \;\colon\;
  U(A) \times U(B)
  \overset{}{\longrightarrow}
  U(A \otimes B)
$$

(where $U \colon Ab \to Set$ is the [[forgetful functor]]).

On elements this sends $(a,b)$ to the equivalence class that it represents under the above equivalence relations.

=--

The following relates the tensor product to [[bilinear functions]]. It is a definition or a proposition dependening on whether one takes the notion of bilinear function to be defined before or after that of tensor product of abelian groups.

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

A [[function]] of underlying sets $f : A \times B \to C$
is a [[bilinear function]] precisely if it factors by the morphism of \ref{TheCanonicalMap} through a [[group homomorphism]] $\phi : A \otimes B \to C$ out of the tensor product:

$$
  f : A \times B \stackrel{\otimes}{\to}
      A \otimes B
      \stackrel{\phi}{\to}
      C
  \,.
$$

=--

### As a quotient inductive type

In [[dependent type theory]], the tensor product of two abelian groups $A$ and $B$ can be expressed as a [[quotient inductive type]] $A \otimes B$, generated by a function $f:A \times B \to A \otimes B$, terms, functions, and dependent [[identifications]] guaranteeing that $A \otimes B$ is an [[abelian group]]

* $0:A \otimes B$
* $+:(A \otimes B) \times (A \otimes B) \to (A \otimes B)$
* $-:(A \otimes B) \to (A \otimes B)$
* $\mathrm{assoc}(a, b, c):a + (b + c) =_{A \otimes B} (a + b) + c$ for all $a, b, c:A \otimes B$
* $\mathrm{lunit}(a):0 + a =_{A \otimes B} a$ for all $a:A \otimes B$
* $\mathrm{runit}(a):a + 0 =_{A \otimes B} a$ for all $a:A \otimes B$
* $\mathrm{linv}(a):-a + a =_{A \otimes B} 0$ for all $a:A \otimes B$
* $\mathrm{rinv}(a):a + (-a) =_{A \otimes B} 0$ for all $a:A \otimes B$
* $\mathrm{settrunc}(a, b):\mathrm{isProp}(a =_{A \otimes B} b)$ for all $a, b:A \otimes B$

and dependent identifications stating that $f$ is a [[bilinear map]]

* $\mathrm{ldist}(a_1, a_2, b):f(a_1,b)+f(a_2,b) =_{A \otimes B} f(a_1+a_2,b)$

* $\mathrm{rdist}(a, b_1, b_2):f(a, b_1)+f(a, b_2) =_{A \otimes B} f(a,b_1+b_2)$

for all $a, a_1, a_2:A$ and $b, b_1, b_2:B$.

## Properties

### Symmetric monoidal category structure

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

Equipped with the tensor product $\otimes$ of def. \ref{ExplicitTensorProduct} and the exchange map $\sigma_{A, B}: A\otimes B \to B \otimes A$ generated by $\sigma_{A, B}(a, b) = (b, a)$, [[Ab]] becomes a [[symmetric monoidal category]].

The [[unit object]] in $(Ab, \otimes)$ is the additive group of [[integers]] $\mathbb{Z}$.

=--

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

To see that $\mathbb{Z}$ is the unit object, consider for any abelian group $A$ the map

$$
  A \otimes \mathbb{Z} \to A
$$

which sends for $n \in \mathbb{N} \subset \mathbb{Z}$

$$
  (a, n) \mapsto n \cdot a \coloneqq \underbrace{a + a + \cdots + a}_{n\;summands}
  \,.
$$

Due to the quotient relation defining the tensor product, the element on the left is also equal to

$$
  (a, n) = (a, \underbrace{1 + 1 \cdots + 1}_{n\; summands})
  =
  \underbrace{ (a,1) + (a,1) + \cdots + (a,1) }_{n\; summands}
  \,.
$$

This shows that $A \otimes \mathbb{Z} \to A$ is in fact an [[isomorphism]].

Showing that $\sigma_{A, B}$ is natural in $A, B$ is trivial, so $\sigma$ is a braiding. $\sigma^2$ is identity, so it gives Ab a symmetric monoidal structure.

=--

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

The tensor product of abelian groups [[distributivity|distributes]] over the [[direct sum]] of abelian groups

$$
  A \otimes \oplus_{s \in S} B_s
  \simeq
  \oplus_{s \in S} ( A \otimes B_s )
  \,.
$$

=--

### Monoids

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

A [[monoid]] in $(Ab, \otimes)$ is equivalently a [[ring]].

=--

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

Let $(A, \cdot)$ be a [[monoid]] in $(Ab, \otimes)$. The fact that the multiplication

$$
  \cdot : A \otimes A \to A
$$

is bilinear means by the above that for all $a_1, a_2, b \in A$ we have

$$
  (a_1 +  a_2) \cdot b = a_1 \cdot b  + a_2 \cdot b
$$

and

$$
  b \cdot (a_1 +  a_2) = b \cdot a_1 + b \cdot a_2
  \,.
$$

This is precisely the [[distributivity law]] of the ring.

=--

## Examples

For $n \in \mathbb{N}$ positive we write $\mathbb{Z}_n$ for the [[cyclic group]] of [[order]] $n$, as usual.

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

For $a,b \in \mathbb{N}$ and positive, we have

$$
  \mathbb{Z}_a \otimes \mathbb{Z}_b \simeq \mathbb{Z}_{(a,b)}
  \,,
$$

where $(-,-)$ denotes the [[greatest common divisor]].

=--

A proof is spelled out for instance as ([Conrad, theorem 4.1](#Conrad)).

## Related concepts

* [[tensor product of commutative monoids]]

* [[tensor product of chain complexes]]

* [[tensor product of modules]]

* [[tensor product of algebras over a commutative monad]]

## References

The original definition is due to [[Hassler Whitney]]:

* [[Hassler Whitney]], *Tensor products of Abelian groups*, Duke Mathematical Journal **4** 3 (1938) 495-528
[[doi:10.1215/S0012-7094-38-00442-9](https://doi.org/10.1215/S0012-7094-38-00442-9)]

An exposition (in the case of vector spaces) is in

* Tim Gowers, _How to lose your fear of tensor products_, ([web](https://www.dpmms.cam.ac.uk/~wtg10/tensors3.html))

and, in the further generality of the [[tensor product of modules]], in

* Keith Conrad, _Tensor products_ ([pdf](https://kconrad.math.uconn.edu/blurbs/linmultialg/tensorprod.pdf))
 {#Conrad}

[[!redirects tensor product of abelian groups]]

[[!redirects tensor products of abelian groups]]


tensor product of algebras

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

# Contents
* table of contents
{: toc}

## Idea

Let $R$ be a [[commutative ring]]. The [[category]] of [[associative algebras]] over $R$ is the category

$$
  \mathsf{Alg}_R
  =
  {R \downarrow \mathsf{Ring}}
$$

of [[rings]] [[undercategory|under]] $R$. If $R$ is a commutative [[rig]], we can do the same with

$$ \mathsf{Alg}_R = {R \downarrow \mathsf{Rig}} .
$$

The _tensor product of $R$-algebras_ has as [[underlying]] $R$-[[module]] just the [[tensor product of modules]] of the underlying modules, $A \otimes_R B$. On homogeneous elements $(a,b) \in A \times B \stackrel{\otimes}{\to} A \otimes_R B$ the algebra structure is given by

$$
  (a_1, b_1) \cdot (a_2, b_2) = (a_1 \cdot a_2, b_1 \cdot b_2)
  \,.
$$

We write also $A \otimes_R B$ for the tensor product of algebras.

For [[commutative algebra|commutative]] $R$-algebras, the tensor product is the [[coproduct]] in $Comm Alg_R$:

$$
  A \otimes_R B
  \simeq
  A \coprod B
  \in
  Comm Alg_R
  =
  Comm R \downarrow \mathsf{Rig}
  ;
$$

hence the [[pushout]] in [[CRing]]

$$
  \array{
     && R
     \\
     & \swarrow && \searrow
     \\
     A &&&& B
     \\
     & \searrow && \swarrow
     \\
     && A \otimes_R B
  }
  \,.
$$

See at _[[category of monoids#PushoutOfCommutativeMonoids|pushouts of commutative monoids]]_.

## Properties

### Relation to tensor product of categories of modules

For $A$ an associative algebra over a [[field]] $k$, write $A$[[Mod]] for its [[category of modules]] of finite [[dimension]]. Then the tensor product of algebras corresponds to the [[Deligne tensor product of abelian categories]] $\boxtimes \colon Ab \times Ab \to Ab$:

$$
  (A \otimes_k B) Mod \simeq  (A Mod) \otimes (B Mod)
  \,.
$$

See at _[[tensor product of abelian categories]]_ for more.

## Related concepts

* [[tensor product]]

* [[tensor product of abelian groups]]

* [[tensor product of modules]]

* [[tensor product of algebras over a commutative monad]]

* [[Hochschild cohomology]]

[[!redirects tensor product of algebras]]
[[!redirects tensor products of algebras]]

[[!redirects tensor product of commutative algebras]]
[[!redirects tensor products of commutative algebras]]


tensor product of algebras over a commutative monad

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

#Contents#
* table of contents
{:toc}

## Idea

[[algebra over a monad|Algebras]] over a [[commutative monad]], in many cases, admit a [[tensor product]] analogous to the traditional [[tensor product of modules]] over a [[ring]].

This tensor product satisfies a universal property analogous to the one of the tensor product of modules, namely it [[representable multicategory|represents]] [[multimorphisms]] in the same way in which the tensor product of modules represents [[bilinear maps]].

## Construction of the tensor product

Let $(T,\mu,\eta)$ be a commutative monad on a symmetric monoidal category $(C,\otimes,1)$.
Denote the [[monoidal functor|monoidal multiplication]] of $T$ by $\nabla$.

Given [[algebra over a monad|$T$-algebras]] $(A,a)$ and $(B,b)$, their **tensor product** is, if it exists, the object $A\otimes_T B$ given by the coequalizer in the [[Eilenberg-Moore category]] $C^T$

 \begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
]
  T(TA \otimes TB) \ar[bend right=10]{rr}[swap]{T(a\otimes b)} \ar{r}{T\nabla} & TT(A\otimes B) \ar{r}{\mu} & T(A\otimes B) \ar{r} & A\otimes_T B.
 \end{tikzcd}

We say that $C^T$ **has tensors** if such equalizers exist for all $(A,a)$ and $(B,b)$. In that case, $\otimes_T$ is a functor $C^T\times C^T\to C^T$.

+-- {: .num_theorem #sealthms}
###### Theorem
([Seal '12, Corollary 2.5.6 and Theorem 2.6.4](#Seal12))
Suppose that either:

* $C^T$ has tensors, and $\otimes_T$ preserves [[reflexive coequalizers]] in both variables separately ([[reflexive coequalizer#properties|hence jointly]]);

or that

* $C$ has [[reflexive coequalizers]], and that the functor $T(-\otimes -)$ preserves them in both variables separately ([[reflexive coequalizer#properties|hence jointly]]).

Then $\otimes_T$ makes the [[Eilenberg-Moore category]] $C^T$ a [[monoidal category]], with

* [[unit object]] given by $T1$, where $1$ is the unit object of $C$;
* [[unitors]] and [[associators]] induced by those of $\otimes$.

Moreover, if $\otimes$ is [[symmetric monoidal category|symmetric]], $\otimes_T$ is symmetric too, with [[braiding]] induced by the one of $\otimes$.
=--

## Bimorphisms and universal property

[[commutative monad|Commutative monads]] admit a notion of [[multimorphism]] of algebras analogous to the notion of [[bilinear]] and [[multilinear map]].

**Caveat.** Some authors call the analogue of a bilinear map a _bimorphism_. This is a distinct notion from the one given [[bimorphism|in this page]] (a morphism which is both epi and mono).
A less ambiguous term is [[binary morphism]].

Let $(A,a)$, $(B,b)$ and $(R,r)$ be $T$-algebras. A **binary morphism** of algebras, or **$T$-balanced map**, or **$T$-bilinear** (or **bimorphism**, see the caveat above) from $A$ and $B$ to $R$ is a morphism $f:A\otimes B\to R$ of $C$ such that the following diagram commutes.

\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
]
  TA \otimes TB \ar{d}{a\otimes b} \ar{r}{\nabla} & T(A\otimes B) \ar{r}{Tf} & TR \ar{d}{r} \\
  A\otimes B \ar{rr}{f} && R
\end{tikzcd}

For $n$ inputs, we can define a **multimorphism of algebras** in the same way.

We can view a binary morphism as a morphism which is a "morphism of algebras in each variable separately". This intuition can be made precise as follows.

+-- {: .num_theorem #bimorphs}
###### Proposition
([Kock '71, Theorem 1.1](#Kock71bilin), [Seal '12, Proposition 2.1.2](#Seal12))
A morphism $f:A\otimes B\to R$ of $C$ is a binary morphism of algebras if and only if both the following two diagrams commute. (The maps $s$ and $t$ denote the [[strong monad|strength and costrength]] of the monad $T$.)
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
]
A \otimes TB \ar{d}{1\otimes b} \ar{r}{s} & T(A\otimes B) \ar{r}{Tf} & TR \ar{d}{r}  && TA \otimes B \ar{d}{a\otimes 1} \ar{r}{t} & T(A\otimes B) \ar{r}{Tf} & TR \ar{d}{r}  \\
A \otimes B \ar{rr}{f} && R && A \otimes B \ar{rr}{f} && R
\end{tikzcd}
=--

We can denote by $C^T(A,B;C)$ the set of binary morphisms from $A$ and $B$ to $C$ this is a functor $(C^T)^{op}\times(C^T)^{op}\times C^T\to Set$. Analogous assignments can be given for $n$ inputs; this equips $C^T$ with the structure of a [[multicategory]] extending its usual category structure.
The most prominent example of this is [[multilinear maps]] extending [[linear maps]].

As in the case of multilinear maps, we have that the tensor product of algebras, if it exists, represent multimorphisms.

+-- {: .num_theorem #uniprop}
###### Theorem
([Seal '12, Proposition 2.3.4](#Seal12))
Suppose that $C^T$ has tensors. For each $T$-algebras $(A,a)$, $(B,b)$ and $(R,r)$ there is a bijection
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
]
C^T(A\otimes_T B, C) \ar{r}{\cong} & C^T(A,B; C) ,
\end{tikzcd}
natural in $A$, $B$ and $C$.
=--
In other words, $C^T$ with $\otimes_R$ is a [[representable multicategory]].

The proof (see the reference above) follows closely the classical case of modules over a ring.

## For monoidal closed categories

The treatment of the closed case goes back to [the work done in the 70s by Anders Kock](#Kock71closed). A treatment of that case has been recently given also in [the thesis of Martin Brandenburg](#Brandenburg2014).

If $C$ is a [[monoidal closed category]] and has [[equalizers]], and $C^T$ has tensors, the first hypothesis of [the theorem above](#sealthms) is satisfied, and so $(C^T,\otimes_T,T1)$ is a monoidal category. Moreover, $C^T$ can be equipped with an [[internal hom]] $[A,B]_T$ which makes $C^T$ closed (see [[internal hom of algebras over a commutative monad]]).
In that case the natural bijection
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
]
C(A\otimes B, C)  \ar{r}{\cong} & C(A ,{[B,C]})
\end{tikzcd}
given by the [[hom-tensor adjunction]] of $C$ induces a natural bijection
\begin{tikzcd}[%
nodes={scale=1.25}, arrows={thick},%
]
C^T(A\otimes_T B, C) \ar{r}{\cong} & C^T(A, {[B,C]}_T)
\end{tikzcd}
which makes $C^T$ itself a [[closed monoidal category]].

This generalizes the [[hom-tensor adjunction]] of [[modules]], [[abelian groups]] and [[vector spaces]].

See also [[commutative algebraic theory#ClosedMonoidalStructureOnAlgebras|closed monoidal structure on algebras over a commutative algebraic theory]].

## Examples

* [[tensor product of modules]]
* [[tensor product of abelian groups]]
* [[tensor product of algebras]]

## See also

* [[monoidal monad]], [[commutative monad]], [[strong monad]], [[commutative algebraic theory]]
* [[tensor product]], [[tensor product of modules]], [[multilinear map]], [[representable multicategory]]
* [[internal hom of algebras over a commutative monad]]
* [[monoidal category]], [[closed category]], [[closed monoidal category]]

## References

* {#Brandenburg2014} [[Martin Brandenburg]], _Tensor categorical foundations of algebraic geometry_ ([arXiv:1410.1716](https://arxiv.org/abs/1410.1716))

* {#Keigher78} [[William Keigher]], _Symmetric monoidal closed categories generated by commutative adjoint monads_, Cahiers de Topologie et Géométrie Différentielle Catégoriques, 19 no. 3 (1978), p. 269-293  ([NUMDAM](http://www.numdam.org/numdam-bin/fitem?id=CTGDC_1978__19_3_269_0), [pdf](http://archive.numdam.org/article/CTGDC_1978__19_3_269_0.pdf))

* {#Kock70monads} [[Anders Kock]], _Monads on symmetric monoidal closed categories_, Arch. Math. 21 (1970), 1--10.

* {#Kock70closed} [[Anders Kock]], _Strong functors and monoidal monads_, Arhus Universitet, Various Publications Series No. 11 (1970).  [PDF](http://home.imf.au.dk/kock/SFMM.pdf).

* {#Kock71closed} [[Anders Kock]], _Closed categories generated by commutative monads_, 1971 ([[KockMonoidalMonads.pdf:file]])

* {#Kock71bilin} [[Anders Kock]], _Bilinearity and cartesian closed monads_, Mathematica Scandinavica, 29, 1971.

* {#Seal12} Gavin J. Seal, _Tensors, monads and actions_ ([arXiv:1205.0101](http://arxiv.org/abs/1205.0101))

[[!redirects tensor product of algebras over commutative monads]]
[[!redirects tensor product of algebras of a commutative monad]]
[[!redirects tensor product of algebras of commutative monads]]
[[!redirects tensor product of algebras for a commutative monad]]
[[!redirects tensor product of algebras for commutative monads]]
[[!redirects tensor product for algebras over a commutative monad]]
[[!redirects tensor product for algebras over commutative monads]]
[[!redirects tensor product for algebras of a commutative monad]]
[[!redirects tensor product for algebras of commutative monads]]
[[!redirects tensor product for algebras for a commutative monad]]
[[!redirects tensor product for algebras for commutative monads]]


tensor product of Banach spaces

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

# Tensor products of Banach spaces
* table of contents
{: toc}

## Idea

There are various [[norms]] that may be placed on the [[tensor product]] of the underlying [[vector spaces]] of two [[Banach spaces]]; the result is not usually [[complete space|complete]], but of course we may take its [[completion]].  One of these, the _projective tensor product_, makes [[Ban]] (the category of Banach spaces and [[short linear maps]]) into a [[closed symmetric monoidal category]], but there are others that still put useful structures on $Ban$.  If we start with [[Hilbert spaces]], then there is a choice of norm that will make the result into a Hilbert space; then [[Hilb]] also becomes a symmetric monoidal category.

## Definitions

Let $V$ and $W$ be [[Banach spaces]], and let $V \otimes W$ be their [[tensor product]] as [[vector spaces]].  To define a tensor product of $V$ and $W$ as Banach spaces, we will place a [[norm]] on $V \otimes W$, making a [[normed vector space]]; the only difference in the following definitions is which norm to use.  Then we take the completion $V {\displaystyle\hat{\otimes}} W$, which is a Banach space.

+-- {: .num_defn}
###### Definition (projective tensor product)

Every element of $V \otimes W$ may be written (in many different ways) as a formal [[linear combination]] of formal tensor products of elements of $V$ and $W$ (suppressing the symbol $\otimes$):
$$ \sum_i \alpha_i v_i w_i .$$
Let the __projective cross norm__ ${\|x\|_\pi}$ of an element $x$ of $V \otimes W$ be
$$ {\|x\|_\pi} \coloneqq \inf \{ \sum_i {|\alpha_i|} {\|v_i\|_V} {\|w_i\|_W} \;|\; x = \sum_i \alpha_i v_i w_i \} .$$
Then the __projective tensor product__ $V {\displaystyle\hat{\otimes}_\pi} W$ of $V$ and $W$ is the completion of $V \otimes W$ under the projective cross norm.
=--

+-- {: .num_defn}
###### Definition (injective tensor product)

If $\lambda$ and $\mu$ are [[linear functionals]] on $V$ and $W$ (respectively), then $\lambda \otimes \mu$ is a linear functional on $V \otimes W$.  Let the __injective cross norm__ ${\|x\|_\epsilon}$ of an element $x$ of $V \otimes W$ be
$$ {\|x\|_\epsilon} \coloneqq \sup \{ {|(\lambda \otimes \mu)x|} \;|\; {\|\lambda\|_{V^*}}, {\|\mu\|_{W^*}} \leq 1 \} .$$
Then the __injective tensor product__ $V {\displaystyle\hat{\otimes}_\epsilon} W$ of $V$ and $W$ is the completion of $V \otimes W$ under the injective cross norm.
=--

+-- {: .num_defn}
###### Definition (tensor product of Hilbert spaces)

If $V$ and $W$ are [[Hilbert spaces]], then their norms determine and are determined by their [[inner products]], so let us discuss inner products.  The elements of $V \otimes W$ are generated by elements of the form $v w$, so set
$$ \langle{v_1 w_1, v_2 w_2}\rangle \coloneqq \langle{v_1, v_2}\rangle \langle{w_1, w_2}\rangle $$
and extend by linearity.  We write the norm of an element $x$ of the [[inner product space]] $V \otimes W$ as ${\|x\|_\sigma}$.  Then the __tensor product__ $V {\displaystyle\hat{\otimes}_\sigma} W$ of the Hilbert spaces $V$ and $W$ is the completion of $V \otimes W$ under this norm (or inner product).
=--

## Cross norms

Besides the specific norms defined above, we can define axioms of a reasonable norm on $V \otimes W$.

+-- {: .num_defn}
###### Definition (cross norm)

A __cross norm__ on $V$ and $W$ is *any* [[norm]] $\chi$ on $V \otimes W$ such that:

*  ${\|v \otimes w\|_\chi} = {\|v\|_V} {\|w\|_W}$ for any elements $v$ and $w$ of $V$ and $W$ (respectively);
*  ${\|\lambda \otimes \mu\|_{\chi^*}} = {\|\lambda\|_{V^*}} {\|\mu\|_{W^*}}$ for any [[bounded linear functionals]] $\lambda$ and $\mu$ on $V$ and $W$ (respectively).
=--

+-- {: .num_defn}
###### Definition (uniform cross norm)

A __uniform cross norm__ is an operation that takes two Banach spaces and returns a norm on their algebraic tensor product, [[natural transformation|naturally]] in the two spaces.  Equivalently, it\'s a [[functor]] $\chi\colon Ban \times Ban \to NVect$ that makes the following diagram commute (or fills it with a [[natural isomorphism]]):
$$ \array { Ban \times Ban   & \overset{\chi}\rightarrow     & NVect \\
            \downarrow       &                               & \downarrow \\
            Vect \times Vect & \underset{\otimes}\rightarrow & Vect
} $$
=--

A uniform cross norm is obviously desirable from the [[nPOV]], but does it meet the analysts\' needs for a cross norm?  Yes:

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

A uniform cross norm assigns a cross norm to any two Banach spaces.
=--

The specific cross norms from the previous section qualify as much as possible:

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

The projective and injective cross norms are uniform cross norms (and hence are in fact cross norms).  The norm on the algebraic tensor product of two Hilbert spaces is also a cross norm.
=--

As far as I can tell, the Hilbert-space cross norm $\sigma$ doesn\'t apply to arbitrary Banach spaces, so it doesn\'t define a uniform cross norm as defined above; however, it does define a functor on $Hilb \times Hilb$, so it\'s as uniform as could be expected.

Looking only at the general theory of cross norms, the projective and injective cross norms appear naturally:

+-- {: .num_prop #topbot}
###### Proposition [(Ryan 2002, Prop. 6.1)](#Ryan02)

Let $V$ and $W$ be Banach spaces and $\chi$ be any norm on $V \otimes W$. Then $\chi$ is a cross norm if and only if
$$ {\|x\|_\epsilon} \leq {\|x\|_\chi} \leq {\|x\|_\pi},$$
for every $x \in V \otimes W$.
=--

That is, we have a [[poset]] of cross norms, and the projective and injective cross norms are (respectively) the [[top]] and [[bottom]] of this poset.

We therefore obtain the following relationship between $\epsilon$, $\sigma$, and $\pi$:

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

If $V$ and $W$ are Hilbert spaces and $x$ is an element of $V \otimes W$, then
$$ {\|x\|_\epsilon} \leq {\|x\|_\sigma} \leq {\|x\|_\pi} .$$
=--

Of course, any cross norm $\chi$ on $V$ and $W$ allows us to form the Banach space $V {\displaystyle\hat{\otimes}_\chi} W$, which may reasonably be called a __tensor product__ of $V$ and $W$; that\'s why we care.

## Schmidt decomposition

The __Schmidt decomposition__ is a way of expressing a pure state in the tensor product of two Hilbert spaces in terms of states of the two components:

+-- {: .num_theorem}
###### Theorem [(Nielsen and Chuang Theorem 2.7)](#NielsenChuang)

Let $A$ and $B$ be finite-dimensional Hilbert spaces.  Let $|\psi\rangle$ be a [[pure state]] of $A \otimes B$.  Then there exist [[orthonormal family|orthonormal families]] $\{ |i_A \rangle \}_i$ in $A$ and $\{ |i_B \rangle \}_i$ in $B$, and non-negative real numbers $\lambda_i$, such that
\[ |\psi\rangle = \sum_i \lambda_i |i_A \rangle \otimes |i_B\rangle \]
and $\sum_i \lambda_i^2 = 1$.
=--

The numbers $\lambda_i$ are called the __Schmidt co-efficients__ of $|\psi\rangle$, and the families $\{ |i_A\rangle \}$ and $\{ |i_B\rangle \}$ the __Schmidt bases__ for $A$ and $B$.

+-- {: .num_defn}
###### Definition
The __Schmidt number__ of $|\psi\rangle$ is the number of non-zero Schmidt coefficients of $|\psi\rangle$.
=--

\begin{remark}
Although most often stated for finite dimensional Hilbert spaces, the Schmidt decomposition theorem stated as above works for $A$ and $B$ Hilbert spaces of arbitrary dimension (even uncountable).
A proof follows through a straightforward application of the polar decomposition and spectral theorems to the positive trace-class operator given by the partial trace of the projection operator onto $\psi$ (see [MO: Schmidt Decomposition on infinite-dimensional Hilbert spaces](#MO20960).)
\end{remark}

## Foundational issues

We need the [[Hahn–Banach theorem]] for $\epsilon$ to be a cross norm; but $\sigma$ and $\pi$ work regardless.  Possibly some of the other propositions rely on some other form of the [[axiom of choice]]; I haven\'t seen their proofs.

## Related pages

* [[spatial tensor product]]
* [[inductive tensor product]]

## References

* M. Nielsen and I. Chuang.  _Quantum Computation and Quantum Information_.  Cambridge University Press.  2000.  {#NielsenChuang}

* {#Ryan02} [[Raymond A. Ryan]],  _Introduction to Tensor Products of Banach Spaces_.  Springer Monographs in Mathematics. Springer-Verlag London. 2002.   ([doi:10.1007/978-1-4471-3903-4](https://www.springer.com/gp/book/9781852334376), [toc pdf](https://d-nb.info/962673145/04))

* J. M. Egger, *Notes on Banach spaces and Hilbert spaces* ([pdf](https://ms.mcmaster.ca/~hanj18/jme-files/jme-banachnotes.pdf))

* {#M20960} MathOverflow: [Schmidt decomposition on infinite-dimensional Hilbert spaces](https://mathoverflow.net/q/281012)

See also:

* Wikipedia, *[Topological tensor product](https://en.wikipedia.org/wiki/Topological_tensor_product)*

  (which also discusses tensor products of [[locally convex spaces]]),

* Wikipedia, *[Tensor product of Hilbert spaces](https://en.wikipedia.org/wiki/Tensor_product_of_Hilbert_spaces)*

Exposition in

* [[Qiaochu Yuan]], *Banach spaces (and Lawvere metrics, and closed categories)*

[[!redirects tensor product of a Banach space]]
[[!redirects tensor products of a Banach space]]
[[!redirects tensor product of Banach spaces]]
[[!redirects tensor products of Banach spaces]]

[[!redirects tensor product of a Hilbert space]]
[[!redirects tensor products of a Hilbert space]]
[[!redirects tensor product of Hilbert spaces]]
[[!redirects tensor products of Hilbert spaces]]

[[!redirects projective tensor product]]
[[!redirects projective tensor products]]
[[!redirects projective tensor norm]]
[[!redirects projective tensor norms]]
[[!redirects projective cross norm]]
[[!redirects projective cross norms]]

[[!redirects injective tensor product]]
[[!redirects injective tensor products]]
[[!redirects injective tensor norm]]
[[!redirects injective tensor norms]]
[[!redirects injective cross norm]]
[[!redirects injective cross norms]]

[[!redirects cross norm]]
[[!redirects cross norms]]
[[!redirects reasonable cross norm]]
[[!redirects reasonable cross norms]]
[[!redirects uniform cross norm]]
[[!redirects uniform cross norms]]

[[!redirects Schmidt decomposition]]
[[!redirects Schmidt decompositions]]
[[!redirects Schmidt coefficient]]
[[!redirects Schmidt coefficients]]
[[!redirects Schmidt co-efficient]]
[[!redirects Schmidt co-efficients]]
[[!redirects Schmidt basis]]
[[!redirects Schmidt bases]]
[[!redirects Schmidt basises]]
[[!redirects Schmidt number]]
[[!redirects Schmidt numbers]]


tensor product of chain complexes

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

#Contents#
* table of contents
{:toc}

## Idea

A natural [[tensor product]] of [[chain complexes]] that makes the [[category of chain complexes]] into a [[closed monoidal category]].

## Definition

Let $R$ be a [[commutative ring]] and $\mathcal{A} = R$[[Mod]] the [[category]] of [[modules]] over $R$. Write $Ch_\bullet(\mathcal{A})$ for the [[category of chain complexes]] of $R$-modules.

\begin{definition}
\label{TheTensorProdComplex}
For $X, Y \in Ch_\bullet(\mathcal{A})$ write  $(X \otimes Y)_\bullet \in Ch_\bullet(\mathcal{A})$ for the [[chain complex]] whose component in degree $n$ is given by the [[direct sum]] (the [[coproduct]]) of [[tensor products of modules]] whose degrees adds to $n$:

$$
  (X \otimes Y)_n
    \;\coloneqq\;
  \bigoplus_{i + j = n}
    X_i \otimes_R Y_j
$$

and whose [[differential]] is given on elements $(x,y)$ of homogeneous degree by the following formula:

$$
  \partial^{X \otimes Y} (x, y)
  \;\coloneqq\;
  \big(\partial^X x, y\big)
    \,+\,
  (-1)^{deg(x)} \big(x, \partial^Y y\big)
  \,.
$$

Its [[tensor unit]] $1_{Ch}$ is $R$ considered as a chain complex concentrated in degree 0.
\end{definition}

## Properties

### As a total complex of a double complex

The tensor product of chain complexes is equivalently the [[total complex]] of the [[double complex]] which is the objectwise tensor product:

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

For $X, Y \in Ch_\bullet(\mathcal{A})$ write  $X_\bullet \otimes Y_\bullet \in Ch_\bullet(Ch_\bullet(\mathcal{A}))$ for the [[double complex]] whose component in degree $(n_1, n_2)$ is given by the [[tensor product]]

$$
  X_{n_1} \otimes Y_{n_2} \in \mathcal{A}
$$

whose horizontal differential is $\partial^{hor} \coloneqq \partial^X \otimes id_Y$ and whose vertical differential is $\partial^{vert} \coloneqq id_{X} \otimes \partial^Y$.

=--

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

The tensor product of chain complexes, def. \ref{TheTensorProdComplex} is isomorphic to the [[total complex]] of the [[double complex]] of def. \ref{DoubleComplexTensor}:

$$
  Tot_\bullet (X_\bullet \otimes Y_\bullet)
  \simeq
  (X \otimes Y)_\bullet
  \,.
$$

=--

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

By direct unwinding of the definitions.

=--

### As Day convolution

The following section is copied form an answer by [[Alexander Campbell]]
on [MathOverflow](https://mathoverflow.net/revisions/359316/6).

The tensor product of chain complexes is a [[Day convolution]] product. The important thing to note is that, to define a [[Day convolution]] [[monoidal structure]] on the $\mathcal{V}$-[[enriched functor]] category $[\mathcal{C},\mathcal{V}]$ (where $\mathcal{V}$ is a complete and cocomplete [[symmetric monoidal closed category]], e.g. $\mathbf{Ab}$), we needn't demand $\mathcal{C}$ to be a monoidal $\mathcal{V}$-category: it suffices for $\mathcal{C}$ to be a **[[promonoidal]]** $\mathcal{V}$-category. This is the generality at which [[Day convolution]] was originally defined in [Day's thesis](http://web.science.mq.edu.au/~street/DayPhD.pdf) (see also [his earlier paper in the Reports of the Midwest Category Seminar IV](https://web.math.rochester.edu/people/faculty/doug/otherpapers/DayReport.pdf), where the word "premonoidal" was used).

A [[promonoidal structure]] on a small $\mathcal{V}$-category $\mathcal{C}$ consists of tensor product and unit "[[profunctors]]", i.e. $\mathcal{V}$-functors $P \colon \mathcal{C}^\mathrm{op}\times\mathcal{C}^\mathrm{op} \times \mathcal{C} \to \mathcal{V}$ and $J \colon \mathcal{C} \to \mathcal{V}$, together with associativity and unit constraints subject to the usual two
"[[pentagon]]" and "triangle" axioms. Given a [[promonoidal structure]] on $\mathcal{C}$, we may construct the **[[Day convolution]]** [[monoidal structure]] on $[\mathcal{C},\mathcal{V}]$, whose tensor product is given at a pair of $\mathcal{V}$-functors $F,G \in [\mathcal{C},\mathcal{V}]$ by the coend
$$F\ast G = \int^{A,B \in \mathcal{C}} P(A,B;-) \otimes FA \otimes GB$$
in $\mathcal{V}$,
and whose unit object is the $\mathcal{V}$-functor $J \in [\mathcal{C},\mathcal{V}]$, and so on. This [[monoidal structure]] on $[\mathcal{C},\mathcal{V}]$ is biclosed (i.e., the tensor product $\mathcal{V}$-functor has a right $\mathcal{V}$-adjoint -- equivalently, preserves (weighted) [[colimits]] -- in each variable). In fact, every [[biclosed monoidal structure]] on $[\mathcal{C},\mathcal{V}]$ arises in this way from some [[promonoidal structure]] on $\mathcal{C}$. (For instance, one recovers the $\mathcal{V}$-functor $P$ from the tensor product $\ast$ by $P(A,B;C) = (\mathcal{C}(A,-) \ast \mathcal{C}(B,-))C$.)

So, since the $\mathbf{Ab}$-category $\mathbf{Ch}$ of chain complexes is (equivalent to) an $\mathbf{Ab}$-[[enriched functor]] category $[\mathcal{C},\mathbf{Ab}]$ (for the $\mathbf{Ab}$-category $\mathcal{C}$ described in the question to which you linked), and since the standard [[monoidal structure]] on $\mathbf{Ch}$ is $\mathbf{Ab}$-enriched and biclosed, this [[monoidal structure]] must be the [[Day convolution]] [[monoidal structure]] for some [[promonoidal]] structure on $\mathcal{C}$. And it isn't too hard to describe that [[promonoidal]] structure. For instance, (presuming I haven't bungled the calculation) the functor $P$ is defined on objects by

$$
P(i,j;k) = \begin{cases}
\mathbb{Z}  & \mathrm{if } \, i+j=k, \\
\mathbb{Z} \oplus \mathbb{Z} &  \mathrm{if } \, i+j=k+1, \\
\mathbb{Z} &  \mathrm{if } \, i+j=k+2, \\
0 & \mathrm{else}.
\end{cases}
$$

### Canonical Forgetful Functor
 {#CanonicalForgetfulFunctor}

Every $\mathcal{V}$-[[enriched monoidal category]] $(\mathcal{C}, \otimes_{\mathcal{C}}, 1_{\mathcal{C}})$ has a canonical [[forgetful functor]]

$$
  U_{\mathcal{C}}
  \;\coloneqq\;
  \mathcal{C}(1_{\mathcal{C}},-)
  \;\colon\;
  \mathcal{C} \longrightarrow \mathcal{V}
$$

given by forming [[hom-objects]] out of the [[tensor unit]].

This functor is automatically [[lax monoidal functor|lax monoidal]] and if $\mathcal{C}$ is [[closed monoidal category|closed monoidal]] with [[internal hom]] $[-,-]_{\mathcal{C}}$ then we further have $U_{\mathcal{C}}[-,-]_{\mathcal{C}} \cong \mathcal{C}(-,-)$. This functor is particularly important to the theory of enriched categories.

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

$U_{Ch} \colon Ch_\bullet(R) \longrightarrow R\text{Mod}$ takes a chain complex to its [[cycles]]/closed elements of degree 0.

=--

\begin{proof}
  Since the [[tensor unit]] $I \,\in\, Ch_\bullet(R)$ is the chain complex concentrated on $R$ in degree 0, hence with all [[differentials]] being 0, a [[chain map]] out of $I$

1. is fixed by its value on $1 \in R$ (by $R$-[[linear map|linearity]])

1. must take values in elements of degree 0 whose differential also vanishes (by the chain map property).

\end{proof}

\begin{remark}
In particular $U_{Ch}$ does not simply take a chain complex to its term at degree 0. That would of course give a different forgetful functor.
\end{remark}

## Examples

### Square as tensor product of interval with itself
 {#SquareAsIntervalSquared}

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

For $R$ some [[ring]], let $I_\bullet \in Ch_\bullet(R Mod)$ be the chain complex given by

$$
  I_\bullet =
  \left[
     \cdots \to 0 \to 0 \to R \stackrel{\partial^{I}_0}{\to} R \oplus R
  \right]
  \,,
$$

where $\partial^I_0 = (-id, id)$.

This is the [[normalized chain complex]] of the [[chain on a simplicial set|simplicial chain complex]] of the standard simplicial interval, the 1-[[simplex]] $\Delta_1$, as follows: we may think of

$$
 I_0 = R \oplus R \simeq R[ \{(0), (1)\} ]
$$

as the $R$-[[linear span]] of two [[basis]] elements labelled "$(0)$" and "$(1)$", to be thought of as the two 0-[[chain on a simplicial set|chains]] on the endpoints of the interval. Similarly we may think of

$$
  I_1 = R \simeq R[\{(0 \to 1)\}]
$$

as the free $R$-module on the single basis element which is the unique non-degenerate 1-[[simplex]] $(0 \to 1)$ in $\Delta^1$.

Accordingly, the [[differential]] $\partial^I_0$ is the oriented [[boundary]] map of the interval, taking this basis element to

$$
  \partial^I_0 : (0 \to 1) \mapsto (1) - (0)
$$

and hence a general element $r\cdot(0 \to 1)$ for some $r \in R$ to

$$
  \partial^I_0 : r\cdot(0 \to 1) \mapsto r\cdot (1) - r\cdot(0)
  \,.
$$

We now write out in full details the tensor product of chain complexes of $I_\bullet$ with itself, according to def. \ref{TheTensorProdComplex}:

$$
  S_\bullet \coloneqq I_\bullet \otimes I_\bullet
  \,.
$$

By definition and using the above choice of [[basis]] element, this is in low degree given as follows:

$$
  \begin{aligned}
    S_0
    &=
    I_0 \otimes I_0
    \\
    & =
    (R \oplus R) \otimes (R \oplus R)
    \\
    & \simeq R \oplus R \oplus R \oplus R
    \\
    & =
    \left\{
       r_{00} \cdot ((0),(0)')
       +
       r_{01} \cdot ((0),(1)')
       +
       r_{10} \cdot ((1),(0)')
       +
       r_{11} \cdot ((1),(1)')
       |
       r_{\cdot, \cdot} \in R
    \right\}
  \end{aligned}
  \,,
$$

where in the last line we express a general element as a linear combination of the canonical basis elements which are obtained as tensor products $(a,b) \in R\otimes R$ of the previous basis elements. Notice that by the definition of [[tensor product of modules]] we have relations like

$$
  r ( (0), (1)') = (r(0), (1)') = ((0), r(1)')
$$

etc.

Similarly then, in degree-1 the tensor product chain complex is

$$
  \begin{aligned}
     (I \otimes I)_1
     & =
     (I_0 \otimes I_1) \oplus (I_1 \otimes I_0)
     \\
     & \simeq
     R \otimes (R \oplus R) \oplus (R \oplus R) \otimes R
     \\
     & \simeq
     R \oplus R \oplus R \oplus R
     \\
     & \simeq
     \left\{
       r_{0} \cdot ((0),(0\to 1)')
       +
       r_{1} \cdot ((1), (0 \to 1)')
       +
       \bar r_0 \cdot ((0\to 1), (0)')
       +
       \bar r_1 \cdot ((0 \to 1), (1)')
       |
       r_{\cdot}, \bar r_{\cdot} \in R
     \right\}
  \end{aligned}
  \,.
$$

And finally in degree 2 it is

$$
  \begin{aligned}
     (I \otimes I)_2
     & \simeq
     I_1 \otimes I_1
     \\
     & \simeq
     R \otimes R
     \\
     & \simeq R
     \\
     & \simeq
     \left\{
       r\cdot ((0 \to 1), (0 \to 1)')
       |
       r \in R
     \right\}
  \end{aligned}
  \,.
$$

All other contributions that are potentially present in $(I \otimes I)_\bullet$ vanish (are the [[trivial group|0-module]]) because all higher terms in $I_\bullet$ are.

The tensor product basis elements appearing in the above expressions have a clear [[geometry|geometric]] interpretation: we can label a square with them as follows

$$
  \array{
     ((0),(1)')
     &&\underset{((0\to 1),(1)')}{\to}&&
     ((1),(1)')
     \\
     \\
     {}^{\mathllap{((0),(0\to 1)')}}\uparrow
     &&\righttoleftarrow^{((0 \to 1), (0\to 1)')}&&
     \uparrow^{\mathrlap{((1),(0 \to 1)')}}
     \\
     \\
     ((0),(0)')
     &&\underset{((0\to 1),(0)')}{\to}&&
     ((1),(0)')
  }
  \,.
$$

This diagram indicates a [[cellular complex|cellular]] square and identifies its canonical [[singular chains]] with the elements of $(I \otimes I)_\bullet$. The arrows indicate the orientation. For instance the fact that

$$
  \begin{aligned}
    \partial^{I \otimes I} ((0 \to 1), (0)')
    & =
    (\partial^I (0 \to 1), (0)')
    + (-1)^1
    ((0\to 1), \partial^I (0))
    \\
    & =
    ( (1) - (0), \;(0)' )
    - 0
    \\
    & =
    ((1), (0)') - ((0), (0)')
  \end{aligned}
$$

says that the oriented [[boundary]] of the bottom morphism is the bottom right element (its target) minus the bottom left element (its source), as indicated. Here we used that the differential of a degree-0 element in $I_\bullet$ is 0, and hence so is any tensor product with it.

Similarly the oriented boundary of the square itself is computed to

$$
  \begin{aligned}
    \partial^{I \otimes I} ((0 \to 1), (0 \to 1)')
    &=
    (\partial^I (0 \to 1), (0 \to 1)')
    -
    ((0 \to 1), \partial^I(0 \to 1))
    \\
    & =
    ((1)- (0), (0 \to 1)')
    -
    ((0 \to 1), (1)' - (0)')
    \\
    & =
    ((1), (0 \to 1)')
    -
    ((0), (0 \to 1)')
    -
    ((0 \to 1), (1)')
    +
    ((0 \to 1), (0)')
  \end{aligned}
  \,,
$$

which can be read as saying that the boundary is the evident boundary thought of as oriented by drawing it _counterclockwise_ into the plane, so that the right arrow (which points up) contributes with a +1 prefactor, while the left arrow (which also points up) contributes with a -1 prefactor.

=--

### Singular chain complex

For $X,Y \in $ [[Top]] two [[topological spaces]], the [[Eilenberg-Zilber theorem]] asserts a [[quasi-isomorphism]]

$$
  C_\bullet(X \times Y)
  \to
  (C(X) \otimes C(Y))_\bullet
$$

between the [[singular chain complex]] of the [[product topological space]] and the tensor product of chain complexes of the separate singular chain complexes.

### Filtering and spectral sequences

As for any [[total complex]] of a double complex, the tensor product of chain complexes is naturally a [[filtered chain complex]], either by the degree of the first of by that of the second chain complex factor.

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

Let $R$ be a [[commutative ring]]. For $A, B \in R$[[Mod]], the two ways of computing the [[Tor]] [[left derived functor]] coincide

$$
  (L_n ((-)\otimes_R B))(A) \simeq (L_n (A \otimes_R (-)))(B)
$$

and hence we can consistently write $Tor_n(A,B)$ for either.

=--

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

Let $Q^A_\bullet \stackrel{\simeq_{qi}}{\to} A$ and $Q^B_\bullet \stackrel{\simeq_{qi}}{\to} B$ be [[projective resolutions]] of $A$ and $B$, respectively. The corresponding [[tensor product of chain complexes]]
$Tot (Q^A_\bullet\otimes Q^B_\bullet)$, hence by prop. \ref{AsTotalComplex} the [[total complex]] of the degreewise [[tensor product of modules]] [[double complex]] carries the [[filtered chain complex|filtration]] by horizontal degree as well as that by vertical degree.

Accordingly there are the corresponding two [[spectral sequences of a double complex]], to be denoted here $\{{}^{A}E^r_{p,q}\}_{r,p,q}$ (for the filtering by $A$-degree) and $\{{}^{B}E^r_{p,q}\}_{r,p,q}$ (for the filtering by $B$-degree). By the discussion there, both converge to the chain homology of the total complex.

We find the value of both spectral sequences on low degree pages according to the general discussion at _[spectral sequence of a double complex - low degree pages](spectral+sequence+of+a+double%20complex#LowDegreePages)_.

The 0th page for both is

$$
  {}^A E^0_{p,q} = {}^B E^0_{p,q} \coloneqq Q^A_p \otimes_R Q^B_q
  \,.
$$

For the first page we have

$$
  \begin{aligned}
    {}^A E^1_{p,q}
    & \simeq H_q(C_{p,\bullet})
    \\
    & \simeq  H_q( Q^A_p \otimes Q^B_\bullet )
  \end{aligned}
$$

and

$$
  \begin{aligned}
    {}^B E^1_{p,q}
    & \simeq H_q(C_{\bullet,p})
    \\
    & \simeq  H_q( Q^A_\bullet \otimes Q^B_p )
  \end{aligned}
  \,.
$$

Now using the [[universal coefficient theorem]] [in homology](universal%20coefficient%20theorem#InHomology) and the fact that $Q^A_\bullet$ and $Q^B_\bullet$ is a [[resolution]] by [[projective objects]], by construction, hence of tensor [[acyclic objects]] for which all [[Tor]]-modules vanish, this simplifies to

$$
  \begin{aligned}
     {}^A E^1_{p,q}
     & \simeq Q^A_p \otimes H_q(Q^B_\bullet)
     \\
     & \simeq \left\{
       \array{
           Q^A_p \otimes_R B & if\; q = 0
           \\
           0 & otherwise
       }
     \right.
  \end{aligned}
$$

and similarly

$$
  \begin{aligned}
     {}^B E^1_{p,q}
     & \simeq H_q(Q^A_\bullet) \otimes_R Q^B_p
     \\
     & \simeq \left\{
       \array{
           A \otimes_R Q^B_p & if\; q = 0
           \\
           0 & otherwise
       }
     \right.
  \end{aligned}
  \,.
$$

It follows for the second pages that

$$
  \begin{aligned}
    {}^A E^2_{p,q}
    & \simeq
    H_p(H^{vert}_q(Q^A_\bullet \otimes Q^B_\bullet))
    \\
   & \simeq
   \left\{
     \array{
     (L_p( (-)\otimes_R B ))(A) & if \; q = 0
     \\
     0 & otherwise
     }
  \right.
  \end{aligned}
$$

and

$$
  \begin{aligned}
    {}^B E^2_{p,q} & \simeq
    H_p(H^{hor}_q(Q^A_\bullet \otimes Q^B_\bullet))
    \\
   & \simeq
   \left\{
     \array{
       (L_p ( A \otimes_R (-) ))(B) & if \; q = 0
       \\
       0 \; otherwise
     }
   \right.
  \end{aligned}
  \,.
$$

Now both of these second pages are concentrated in a single row and hence have converged on that page already. Therefore, since they both converge to the same value:

$$
  L_p((-)\otimes_R B)(A)
  \simeq
  {}^A E^2_{p,0}
  \simeq
  {}^A E^\infty_{p,0}
  \simeq
  {}^B E^2_{p,0}
  \simeq
  L_p(A \otimes_R (-))(B)
  \,.
$$

=--

## Applications

* A [[monoid]] with respect to the tensor product of chain complexes is a [[differential graded algebra]].

## Related concepts

* [[internal hom of chain complexes]]

* [[tensor product]]

  * [[tensor product of abelian groups]]

  * [[tensor product of modules]]

## References

Early discussion of the tensor product of chain complexes:

* [[Henri Cartan]], [[Samuel Eilenberg]], §IV.5 in: *[[Homological Algebra]]*, Princeton University Press (1956) [[ISBN:9780691049915](https://press.princeton.edu/books/paperback/9780691049915/homological-algebra-pms-19-volume-19), [pdf](http://www.math.stonybrook.edu/~mmovshev/BOOKS/homologicalalgeb033541mbp.pdf)]

and in relation to the [[closed monoidal category|closed monoidal]]-structure that it makes with the [[internal hom of chain complexes]]:

* {#EilenbergKelly65} [[Samuel Eilenberg]], [[G. Max Kelly]],  §IV.6 of: *Closed Categories*, in: *[[Proceedings of the Conference on Categorical Algebra - La Jolla 1965]]*, Springer (1966) 421-562 [[doi:10.1007/978-3-642-99902-4](https://doi.org/10.1007/978-3-642-99902-4)]

Textbook accounts:

* [[Charles Weibel]], Section 2.7 of: *[[An Introduction to Homological Algebra]]*, Cambridge University Press (1994) [[doi:10.1017/CBO9781139644136](https://doi.org/10.1017/CBO9781139644136), [pdf](https://web.math.rochester.edu/people/faculty/doug//otherpapers/weibel-hom.pdf)]

[[!redirects tensor products of chain compexes]]
[[!redirects tensor product of cochain complexes]]
[[!redirects tensor products of cochain complexes]]


tensor product of commutative monoids

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

#Contents#
* table of contents
{:toc}

## Idea

For $A$ and $B$ two [[commutative monoids]],
their _tensor product_ $A \otimes B$ is a new commutative monoid
such that a [[monoid homomorphism]] $A \otimes B \to C$
is equivalently a [[bilinear map]] out of $A$ and $B$.

## Definition
 {#Definition}

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

Let [[CMon]] be the collection of [[commutative monoids]], regarded as a [[multicategory]] whose [[multimorphisms]] are the [[multilinear maps]] $A_1, \cdots, A_n \to B$.

The [[tensor product]] $A, B \mapsto A \otimes B$ in this multicategory is the **tensor product of commutative monoids**.

=--

Equivalently this means explicitly:

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

For $A, B$ two [[commutative monoids]], their **tensor product of commutative monoids** is the commutative monoid $A \otimes B$ which is the [[quotient]] of the [[free commutative monoid]] on the [[product]] of their underlying sets $A \times B$
by the [[generators and relations|relations]]

* $(a_1,b)+(a_2,b)\sim (a_1+a_2,b)$

* $(a,b_1)+(a,b_2)\sim (a,b_1+b_2)$

* $(0,b)\sim 0$

* $(a,0)\sim 0$

for all $a, a_1, a_2 \in A$ and $b, b_1, b_2 \in B$.

=--

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

By definition of the [[free construction]] and the [[quotient]] there is a canonical [[function]] of the underlying sets

$$
  p_{A,B}
  \;\colon\;
  U(A) \times U(B)
  \overset{}{\longrightarrow}
  U(A \otimes B)
$$

(where $U \colon CMon \to Set$ is the [[forgetful functor]]).

On elements this sends $(a,b)$ to the equivalence class that it represents under the above equivalence relations.

=--

The following relates the tensor product to [[bilinear functions]]. It is a definition or a proposition dependening on whether one takes the notion of bilinear function to be defined before or after that of tensor product of commutative monoids.

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

A [[function]] of underlying sets $f : A \times B \to C$
is a [[bilinear function]] precisely if it factors by the morphism of \ref{TheCanonicalMap} through a [[monoid homomorphism]] $\phi : A \otimes B \to C$ out of the tensor product:

$$
  f : A \times B \stackrel{\otimes}{\to}
      A \otimes B
      \stackrel{\phi}{\to}
      C
  \,.
$$

=--

## Properties

### Symmetric monoidal category structure

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

Equipped with the tensor product $\otimes$ of def. \ref{ExplicitTensorProduct} and the exchange map $\sigma_{A, B}: A\otimes B \to B \otimes A$ generated by $\sigma_{A, B}(a, b) = (b, a)$, [[CMon]] becomes a [[symmetric monoidal category]].

The [[unit object]] in $(CMon, \otimes)$ is the additive monoid of [[natural numbers]] $\mathbb{N}$.

=--

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

To see that $\mathbb{N}$ is the unit object, consider for any commutative monoid $A$ the map

$$
  A \otimes \mathbb{N} \to A
$$

which sends for $n \in \mathbb{N}$

$$
  (a, n) \mapsto n \cdot a \coloneqq \underbrace{a + a + \cdots + a}_{n\;summands}
  \,.
$$

Due to the quotient relation defining the tensor product, the element on the left is also equal to

$$
  (a, n) = (a, \underbrace{1 + 1 \cdots + 1}_{n\; summands})
  =
  \underbrace{ (a,1) + (a,1) + \cdots + (a,1) }_{n\; summands}
  \,.
$$

This shows that $A \otimes \mathbb{N} \to A$ is in fact an [[isomorphism]].

Showing that $\sigma_{A, B}$ is natural in $A, B$ is trivial, so $\sigma$ is a braiding. $\sigma^2$ is identity, so it gives CMon a symmetric monoidal structure.

=--

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

The tensor product of commutative monoids [[distributivity|distributes]] over the [[biproduct]] of commutative monoids

$$
  A \otimes \oplus_{s \in S} B_s
  \simeq
  \oplus_{s \in S} ( A \otimes B_s )
  \,.
$$

=--

### Monoids

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

A [[monoid]] in $(CMon, \otimes)$ is equivalently a [[rig]].

=--

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

Let $(A, \cdot)$ be a [[monoid]] in $(CMon, \otimes)$. The fact that the multiplication

$$
  \cdot : A \otimes A \to A
$$

is bilinear means by the above that for all $a_1, a_2, b \in A$ we have

$$
  (a_1 +  a_2) \cdot b = a_1 \cdot b  + a_2 \cdot b
$$

$$
  b \cdot (a_1 +  a_2) = b \cdot a_1 + b \cdot a_2
  \,.
$$

$$
  0 \cdot b = 0
  \,.
$$

$$
  b \cdot 0 = 0
  \,.
$$

This is precisely the [[distributivity law]] and [[absorption law]] of the rig.

=--

## Related concepts

* [[tensor product of abelian groups]]

* [[tensor product of chain complexes]]

* [[tensor product of modules]]

* [[tensor product of algebras over a commutative monad]]

[[!redirects tensor product of commutative monoids]]

[[!redirects tensor products of commutative monoids]]