tensor product of distributions

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### Context
#### Functional analysis
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[[!include functional analysis - contents]]
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#Contents#
* table of contents
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## Idea

The _tensor product of distributions_ is the generalization to [[distributions]] of the [[tensor product]] of [[smooth functions]], hence it defines for two distributions $u \in \mathcal{D}'(X)$ and $v \in \mathcal{D}'(Y)$ a new distribution $u \otimes v \in \mathcal{D}'(X \times Y)$ on the [[Cartesian product]] space which, as a [[generalized function]] behaves like $(u \otimes v)(x,y) = u(x) \cdot v(y)$.

## Definition

+-- {: .num_defn #TensorProductOfSmoothFunctions}
###### Definition
**(tensor product of smooth functions)**

For $X_1, X_2$ two [[open subsets]] of some [[Cartesian space]], there is an [[injective function|injection]] from the [[tensor product of modules|tensor product]] of the [[real vector spaces]] of [[smooth functions]] on the separate spaces to that on the [[Cartesian product]] space:

$$
  \array{
    C^\infty(X_1) \otimes_{\mathbb{R}} C^\infty(X_2)
      &\overset{}{\hookrightarrow}&
    C^\infty(X_1 \times X_2)
    \\
    (f_1, f_2)
    &\mapsto&
    f_1 \otimes f_2
  }
$$

with

$$
  (f_1 \otimes f_2)(x_1, x_2) \coloneqq f_1(x_1) \cdot f_2(x_2)
  \,.
$$

=--

+-- {: .num_prop #TensorProductOfDistributions}
###### Proposition
**(tensor product of distributions)**

Let $u_1 \in \mathcal{D}'(X_1)$ and $u_2 \in \mathcal{D}'(X_2)$ be [[distributions]]. Then there is a unique [[distribution of two variables]] $u_1 \otimes u_2 \in \mathcal{D}'(X_1 \times X_2)$ such that for all pairs of [[bump functions]] $b_1 \in C_c^\infty(x)$ and $b_2 \in C^\infty(X_2)$ its value on their tensor product according to def. \ref{TensorProductOfSmoothFunctions} is

$$
  u(b_1 \otimes b_2)
  =
  u_1(b_1) \cdot u_2(b_2)
  \,.
$$

This $u_1 \otimes u_2$ is called the _tensor product_ of $u_1$ with $u_2$

=--

([Hörmander 90, theorem 5.1.1](#Hoermander90))

## Properties

+-- {: .num_example #WaveFrontOfTensorProductDistribution}
###### Example
**([[wave front set]] of tensor product distribution)**

Let $u \in \mathcal{D}'(X)$ and $v \in \mathcal{D}'(Y)$ be two distributions. then the [[wave front set]] of their tensor product distribution $u \otimes v \in \mathcal{D}'(X \times Y)$ (def. \ref{TensorProductOfDistributions}) satisfies

$$
  WF(u \otimes v)
  \;\subset\;
  \left(
     WF(u) \times WF(v)
  \right)
    \cup
  \left(
    \left( supp(u) \times \{0\} \right)
    \times WF(v)
  \right)
    \cup
   \left(
     WF(u)
     \times
     \left(
       supp(v) \times \{0\}
     \right)
   \right)
  \,,
$$

where $supp(-)$ denotes the [[support of a distribution]].

=--

([Hörmander 90, theorem 8.2.9](#Hoermander90))

## Related concepts

* [[Schwartz kernel theorem]]

## References

* {#Hoermander90} [[Lars Hörmander]], section 5.1 of _The analysis of linear partial differential operators_, vol. I, Springer 1983, 1990 ([pdf](http://www.ctr.maths.lu.se/media/MATP11/2014vt2014/distho.pdf))

[[!redirects tensor products of distributions]]

[[!redirects tensor product distribution]]
[[!redirects tensor product distributions]]


tensor product of enriched categories

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#Contents#
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## Idea

For $(\mathcal{V}, \otimes)$ a [[symmetric monoidal category]] regarded as a [[cosmos for enrichment]], there is a [[tensor product]] of [[enriched category|$\mathcal{V}$-enriched categories]] which on [[classes]] of [[objects]] is the [[Cartesian product]] and on [[hom-objects]] is given by $\otimes$, with [[composition]] defined factor-wise after using the [[braiding]] in $\mathcal{V}$ to align composable tensor factors.

This operation makes [[VCat|$\mathcal{V}Cat$]] itself into a ([[very large category|very large]]) [[symmetric monoidal category]] [[Kelly (1982), §1.4](#Kelly82)].

If $\mathcal{V}$ is furthermore [[closed monoidal category|closed]] and [[complete category|complete]] so that [[enriched functor category|$\mathcal{V}$-enriched functor categories]] exist, then these constitute the [[internal hom]] [[right adjoint]] to the operation of forming the enriched product category with a given enriched category [[Kelly (1982), §2.3](#Kelly82)].

## Examples

\begin{example}
  For $\mathcal{V} = $ [[Set]] equipped with its [[cartesian product]] $\otimes \,\coloneqq\, \times$, the construction of enriched product categories reduces to the ordinary notion of [[product categories]].
\end{example}

## Related concepts

* [[bifunctor]]

## References

Textbook accounts:

* {#EilenbergKelly65} [[Samuel Eilenberg]], [[G. Max Kelly]],  §III.3 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)]

* {#Kelly82} [[Max Kelly]], §2.3 of: _Basic concepts of enriched category theory_, London Math. Soc. Lec. Note Series __64__, Cambridge Univ. Press (1982), Reprints in Theory and Applications of Categories **10** (2005) 1-136  [[ISBN:9780521287029](https://www.cambridge.org/de/academic/subjects/mathematics/logic-categories-and-sets/basic-concepts-enriched-category-theory?format=PB&isbn=9780521287029), [tac:tr10](http://www.tac.mta.ca/tac/reprints/articles/10/tr10abs.html), [pdf](http://www.tac.mta.ca/tac/reprints/articles/10/tr10.pdf)]

See also

* [[Enrico Ghiorzi]], §3.6 *Internal enriched categories*, Applied Categorical Structures **30** (2022) 947–968  [[arXiv:2006.07997](https://arxiv.org/abs/2006.07997), [doi:10.1007/s10485-022-09678-w](https://doi.org/10.1007/s10485-022-09678-w)]

* [[Haskell]]: [Data-Category-Enriched.html](https://hackage.haskell.org/package/data-category-0.10/docs/Data-Category-Enriched.html)

The tensor product of enriched categories is called the *commuting tensor product* of enriched categories in:

* [[Richard Garner]], [[Ignacio López Franco]], _Commutativity_ [[arXiv:1507.08710](http://arxiv.org/abs/1507.08710)]

where the authors show how to construct the tensor product even when $V$ is not [[braided monoidal category|braided monoidal]], but merely [[duoidal category|duoidal]].

[[!redirects enriched product category]]
[[!redirects enriched product categories]]

[[!redirects tensor product of enriched categories]]
[[!redirects tensor products of enriched categories]]


tensor product of functors

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## Idea

Just like a right $A$-module can be [[tensor product of modules|tensored]] (over $A$) with a left $A$-module to obtain an abelian group, a [[functor]] $T : \mathcal{C}^\op \to \mathcal{V}$ can be tensored (over $\mathcal{C}$) with a functor $S : \mathcal{C} \to \mathcal{V}$ to obtain an object of $\mathcal{V}$. Here, $\mathcal{V}$ can be an arbitrary [[monoidal category]]. A simple case is $\mathcal{V} = Set$ with the [[cartesian product]] as monoidal structure.

## Definition

Let $\mathcal{C}$ be a category. Let $(\mathcal{V},{\otimes})$ be a monoidal category. Let $T : \mathcal{C}^\op \to \mathcal{V}$ and $S : \mathcal{C} \to \mathcal{V}$ be functors. Then their tensor product $T \otimes_{\mathcal{C}} S$ is defined (if it exists) as the [[coend]]
$$ T \otimes_{\mathcal{C}} S \coloneqq \int^c T(c) \otimes S(c). $$

The following slight variation is also important. Let $\mathcal{C}$ be a category. Let $\mathcal{D}$ be a category with all [[coproducts]], so that the [[copower]] $X \cdot d = \coprod_{x \in X} d \in \mathcal{D}$ exists for any set $X$ and any object $d \in \mathcal{D}$. Let $T : \mathcal{C}^\op \to Set$ and $S : \mathcal{C} \to \mathcal{D}$ be functors. Then their tensor product is (if it exists) the coend
$$ T \otimes_{\mathcal{C}} S \coloneqq \int^c T(c) \cdot S(c) \in \mathcal{D}. $$

Intuition for the tensor product of functors can be gained by relating it to the tensor product of modules (see the first example below) and by a picture involving gluing specifications (see below).

## Examples

* If $T : \mathscr{C}^{op} \to Set$ and $S : \mathscr{C} \to Set$, then $T \otimes_{\mathscr{C}} S$ can be presented as set of formal symbols $t \otimes_c s$ (or $t \otimes s$ for short) where $c \in Ob(\mathscr{C})$, $s \in S(c)$, and $t \in T(c)$ modulo the equivalence relation generated by the instances $(t f) \otimes_c s \sim t \otimes_d (f s)$ when defined, where $f \in \mathscr{C}(c,d)$, and the 'action' of $\mathscr{C}$ is the shorthand $t f = T(f)(t)$ and $f s = S(f)(s)$.

* Recall that a ring $A$ can be considered as an [[Ab-enriched category]] and that a right and a left module gives rise to an additive functor $A^\op \to Ab$ respectively $A \to Ab$. Then their tensor product as functors, calculated in the Ab-enriched setting and using the ordinary [[tensor product of abelian groups]] as monoidal structure on $Ab$, coincides with their usual tensor product.

* Let $X$ be a [[simplicial set]] and $st : \Delta \to Top$ be the functor which associates to $[n]$ the topological standard $n$-simplex. Then the [[geometric realization]] of $X$ can be expressed as the tensor product $|X| = X \otimes_\Delta st$.

* Let $\mathcal{C}$ be a small category and let $F : \mathcal{C} \to \mathcal{D}$ be a functor into a [[cocomplete]] category $\mathcal{D}$. Since the category $PSh(\mathcal{C})$ of [[presheaves]] on $\mathcal{C}$ is the [[free cocompletion]] of $\mathcal{C}$, the functor $F$ induces a functor $\hat F : PSh(\mathcal{C}) \to \mathcal{D}$. This functor can be explicitly described as $\hat F(X) = X \otimes_\mathcal{C} F$.

* Let $Y : \mathcal{C} \to PSh(\mathcal{C})$ denote the [[Yoneda embedding]]. Let $F : \mathcal{C}^\op \to Set$ be a presheaf on $\mathcal{C}$. Then $F \otimes Y = F$. This fact is the [[co-Yoneda lemma]] (also referred to as the _ninja Yoneda lemma_ in some circles).

* In some sense, [[representable functors]] generalize [[free modules]]: Recall $A^n \otimes_A M \cong M^n$. Similarly,
$$ \Hom_\mathcal{C}(\cdot, c) \otimes_\mathcal{C} S = S(c). $$
This follows from some coend manipulations:
$$ \begin{array}{rcl}
  \Hom(\Hom_\mathcal{C}(\cdot, c) \otimes_\mathcal{C} S, t) &=&
  \int_{c' \in \mathcal{C}} \Hom(\Hom_\mathcal{C}(c',c) \cdot S(c'), t) \\
  &=& \int_{c' \in \mathcal{C}} \Hom(\Hom_\mathcal{C}(c',c), \Hom_{\mathcal{D}}(S(c'), t)) \\
  &=& Nat(\Hom_\mathcal{C}(\cdot, c), \Hom_\mathcal{D}(S(\cdot), t)) \\
  &=& \Hom_\mathcal{D}(S(c),t).
\end{array} $$
From this perspective, the representable functor $Hom_\mathcal{C}(\cdot, c)$ looks like a [[distribution|delta distribution]] concentrated at $c$.

## Intuition using gluing specifications

Recall that a presheaf $F : \mathcal{C}^\mathrm{op} \to \mathrm{Set}$ can be seen as "gluing specification": If $G : \mathcal{C} \to \mathcal{D}$ is some functor into a cocomplete category $\mathcal{D}$, this gluing specification can be realized as $\operatorname{colim}_{s \in F(X)} G(X)$. This colimit can also be written as the coend
$$ \int^{X \in \mathcal{C}} F(X) \cdot G(X), $$
that is as the tensor product $F \otimes_\mathcal{C} G$. The tensor product can therefore be pictured as the $G(X)$'s, glued as specified by $F$.

* Since the Yoneda embedding $Y$ includes an object into the category of formal gluing specifications, gluing the $Y(X)$'s as specified by $F$ simply yields in $F$; thus $F \otimes_\mathcal{C} Y = F$.

* Let $\mathcal{C}$ be specifically the [[simplex category]] $\Delta$. Then $F$ is just a simplicial set, so the intuition of $F$ as a gluing specification is even more vivid. Tensoring with $st : \Delta \to Top$ realizes this specification in the category of topological spaces, using standard $n$-simplices as building blocks, and therefore yields the geometric realization $|F|$.

## Related concepts

* [[tensor product of modules]]
* [[coend]]
* [[weighted colimit]]

## References

A basic reference is [[Categories Work]], Section IX.6.


tensor product of groups

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## Definition

\begin{definition}
\label{TheDefinition}
Let $M,N$ be a [[pair]] of [[group|groups]] (not necessarily [[abelian group|abelian]]) endowed with [[group action|actions]]  by [[group homomorphism|group]] [[automorphisms]] on each other: $\rho$ an action of $M$ on $N$, and $\sigma$ of $N$ on $M$.

Then the *tensor product* $M \otimes N$ is the group generated by [[elements]] of the form $m\otimes n$ subject to the following [[generators and relations|relations]]:

* $(m m')\otimes n
  \,=\,
  (m m' m^{-1} \otimes \rho(m)(n))(m\otimes n),$

* $m\otimes (n n') \,=\, (m\otimes n)(\sigma(n)(m)\otimes n n' n^{-1} ),$

for $m,m'\in M$ and $n,n'\in N$.
\end{definition}

This definition is due to [Brown & Loday 1987, Section 2](#BL87) (there in the context of the [[van Kampen theorem]]).

+-- {: .num_remark}
###### Remark
Def \ref{TheDefinition} is to be understood as the generalization of the [[tensor product of abelian groups]], since as one can verify, whenever $M,N$ act trivially on each other, then the above definition reduces to the ordinary [[tensor product of abelian groups|tensor product]] of the [[abelianizations]] $(-)^{op}$ of the given groups:

$$
M\otimes N\cong M^{ab} \otimes_{\mathbb{Z}} N^{ab}.
$$
=--

## References

The definition first appears in:

* [[Ronnie Brown]], [[Jean-Louis Loday]]. *Van Kampen theorems for diagrams of spaces*. Topology **26** 3 (1987) 311-335 [doi:10.1016/0040-9383(87)90004-8]


tensor product of infinity-modules

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#Contents#
* table of contents
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## Idea

The generalization of the notion of [[tensor product of modules]] to [[∞-modules]].

## Definition

We start by defining a collection of [[colored operad|colored]] [[symmetric operad]] $Tens^\otimes$ parameterized by the [[simplex category]] $\Delta$ such that for each $k$-[[simplex]] $[k] \in Delta$ the [[algebras over an operad]] over $Tens^\otimes_{[k]}$ are $(n+1)$-tuples of [[associative algebras]] $(A_i)$ together with a consecutive sequence of [[bimodules]] over these (the right algebra of every bimodule being the left algebra of the next one).

The definition is a straightforward generalization of the of the [[operad for modules]] and the [[operad for bimodules]].

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

Write $Tens^\otimes$ for the category (to be thought of as a family of [[categories of operators]] of [[symmetric operads]]) whose

* [[objects]] are triples consisting of

  * an object $\langle n\rangle \in Assoc^\otimes$ of the [[category of operators]] of the [[associative operad]];

  * an object $[k] \in \Delta$ of the [[simplex category]];

  * two [[functions]] $c_-, c_+ \colon \langle n\rangle^\circ \to [k]$ such that for all $i in \langle n\rangle^\circ$ either $c_+(i) = c_-(i)$ or $c_+(i) = c_-(i) + 1$;

* [[morphisms]] consist if

  * a morphism $\alpha \colon \langle n\rangle \to \langle n'\rangle$ in $Assoc^\otimes$

  * a morphism $\lambda \colon [k'] \to [k]$ in $\Delta$

  such that (...)

=--

([Lurie, def. 4.3.4.1](#Lurie))

We disuss how an object of this category is to be thought of as labeled with "algebra labels $\mathfrak{a}_i$" for vertices of a simplex, an "bimodule lables $\mathfrak{n}_{i, j}$" for edges of the simplex.

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

By construction there are [[forgetful functors]]

$$
  \Delta^{op}
  \leftarrow
  Tens^\otimes
  \rightarrow
  \mathcal{Ass}^\otimes
  \,.
$$

=--

([Lurie, 4.3.4.1](#Lurie))

+-- {: .num_defn #NotationForTensS}
###### Definition (Notation)

For $S \to \Delta^{op}$ an [[(∞,1)-functor]] (given as a map of simplicial sets from a [[quasi-category]] $S$ to the [[nerve]] of the [[simplex category]]), write

$$
  Tens^\otimes_{S}
   \coloneqq
  Tens^\otimes \underset{\Delta^{op}}{\times} S
$$

for the [[fiber product]] in [[sSet]].

=--

([Lurie, notation 4.3.4.5, 4.3.4.15](#Lurie))

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

We have

* $Tens^\otimes_{[0]} \simeq Assoc^\otimes$, the [[associative operad]];

* $Tens^\otimes_{[1]} \simeq BM^\otimes$ the [[operad for bimodules]].

* $Tens^\otimes_{[k]}
    \simeq
   Tens^\otimes_{\{0,1\}}
    \underset{Tens^\otimes_{\{1\}}}{\coprod}
   Tens^\otimes_{\{1,2\}}
     \underset{Tens^\otimes_{\{2\}}}{\coprod}
     \cdots
     \underset{Tens^\otimes_{\{k-1\}}}{\coprod}
   Tens^\otimes_{\{k-1,k\}}
  $

  as an [[(∞,1)-colimit]] in the [[(∞,1)-category]] of [[(∞,1)-operads]] (a dual _[[Segal condition]]_)

=--

([Lurie, example 4.3.4.6, 4.3.4.7, prop. 4.3.4.11](#Lurie))

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

Prop. \ref{SegalConditionOfTens} implies that for
$\mathcal{C}^\otimes$ an [[(∞,1)-operad]], the [[(∞,1)-algebras over an (∞,1)-operad]] over the fiber $Tens^\otimes_{[k]}$ in $\mathcal{C}$ form the [[(∞,1)-category]]

$$
  Alg_{Tens^\otimes_{[k]}}(\mathcal{C})
  \simeq
  \underbrace{
  BMod(\mathcal{C})
   \underset{Alg(\mathcal{C})}{\times}
  BMod(\mathcal{C})
   \underset{Alg(\mathcal{C})}{\times}
  \cdots
   \underset{Alg(\mathcal{C})}{\times}
  BMod(\mathcal{C})
  }_{k\;factors}
  \,.
$$

=--

([Lurie, 4.3.5](#Lurie))

+-- {: .num_defn #NotationForAlS}
###### Definition (Notation)

For $\mathcal{C}^\otimes \to Tens^\otimes_S$ a [[fibration]] in the [[model structure for quasi-categories]] which exhibits $\mathcal{C}^\otimes$ as an $S$-[[family of (∞,1)-operads]], write

$$
  Alg_S(\mathcal{C}) \hookrightarrow Fun_{Tens^\otimes_S}(Step_S, \mathcal{C}^\otimes)
$$

for the full [[sub-(∞,1)-category]] on those [[(∞,1)-functors]] which send inert morphisms to inert morphisms.

=--

([Lurie, notation 4.3.4.15](#Lurie))

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

For an [[(∞,1)-functor]] $S \to \Delta^{op}$ and a [[fibration]] in the [[model structure for quasicategories]] $q \colon \mathcal{C}^\otimes \to Tens_S^\otimes$ exhibiting $\mathcal{C}^\otimes$ as an $S$-[[family of (∞,1)-operads]], then there is an [[equivalence of (∞,1)-categories]]

$$
  Alg_{/Tens_S}(\mathcal{C})
  \to
  Alg_S(\mathcal{C})
  \,.
$$

=--

([Lurie, prop. 4.3.4.17](#Lurie)).

+-- {: .num_defn #NotationForTensGt}
###### Definition (Notation)

Let $\Delta^1 \to \Delta^{op}$ be the map that picks the morphism $\{0,2\} \hookrightarrow \Delta^2$ in the [[simplex category]]. With def. \ref{NotationForTensS} write

$$
  Tens^\otimes_{\gt}
  \coloneqq
  Tens_{\Delta^1}^\otimes
  \coloneqq
  Tens^\otimes \underset{\Delta^{op}}{\times} \Delta^1
  \,.
$$

=--

([Lurie, notation 4.3.5.1](#Lurie))

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

The $Tens^\otimes_{\gt}$ of def. \ref{NotationForTensGt} is a [[correspondence]] of [[(∞,1)-operads]] which exhibits bilinear maps as follows:

An [[∞-algebra over an (∞,1)-operad]] $\gamma_1 \colon Tens^\otimes_{\gt} \times_{\Delta^1} \{1\} \to \mathcal{C}^\otimes$ is equivalently a bimodule

$$
  X \in {}_{A'} Mod(\mathcal{C})_{C'}
  \,,
$$

while an $\infty$-algebra
$\gamma_0 \colon Tens^\otimes_{\gt} \times_{\Delta^1} \{0\} \to \mathcal{C}^\otimes$ is equivalently a pair of bimodules

$$
  N_1 \in {}_A Mod(\mathcal{C})_B
  \;\;, \;\;
  N_2 \in {}_B Mod(\mathcal{C})_C
 \,.
$$

An extension of $(\gamma_0, \gamma_1)$ through the correspondence hence to a map of [[generalized (∞,1)-operads]] $Tens^\otimes_{\gt} \to \mathcal{C}^\otimes$ is equivalently a pair of [[A-∞ algebra]] maps $A \to A'$ and $B \to B'$ together with a bilinear map $N_1 \otimes N_2 \to X$.

=--

([Lurie, beginning of 4.3.4]).

+-- {: .num_defn #RelativeTensorProduct}
###### Definition
(**relative tensor product of $\infty$-bimodules**)

For $q \colon \mathcal{C}^\otimes \to \mathcal{O}^\otimes$
a [[fibration of (∞,1)-operads]], consider a morphism of
[[generalized (∞,1)-operads]]

$$
  F \colon Tens_{\gt}^\otimes \to \mathcal{C}^{\otimes}
  \,.
$$

This exhibits three [[A-∞ algebras]] $A_i \coloneqq F|_{\{i\}}$, a pair of bimodule objects

$$
  (N_1, N_2) = F|_{[2]}
$$

over $A_0$-$A_1$ and over $A_1$-$A_2$, respectively,
and a bimodule object $N = F|_{[1]}$ over $A_0$-$A_2$. We say that $N$ exhibits the **relative [[tensor product of ∞-modules]]** of $N_1$ with $N_2$ over $A_1$

$$
  N \simeq N_1 \otimes_{A_1} N_2
$$

if $F$ is an operadic $q$-[[(∞,1)-colimit]]-[[diagram]].

=--

([Lurie, def. 4.3.5.3](#Lurie)).

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

Let $\mathcal{C}^\otimes \to Assoc^\otimes$ exhibit a [[monoidal (∞,1)-category]] such that $\mathcal{C}$ has [[geometric realization]] of [[simplicial objects in an (∞,1)-category|simplicial objects]] and the tensor product preserves these separately in each argument.

Then the tensor product of $\infty$-modules def. \ref{RelativeTensorProduct} extends to an [[(∞,1)-functor]]

$$
  BMod(\mathcal{C})
  \underset{Alg(\mathcal{C})}{\times}
  BMod(\mathcal{C})
  \to
  BMod(\mathcal{C})
  \,.
$$

=--

([Lurie, example 4.3.5.11](#Lurie))

## Related concepts

* [[(∞,1)-bimodules]], [[(∞,2)-category of (∞,1)-bimodules]]

## References

Section 4.3.5 of

* [[Jacob Lurie]], _[[Higher Algebra]]_
 {#Lurie}

[[!redirects tensor product of ∞-modules]]

[[!redirects tensor product of (∞,1)-modules]]
[[!redirects tensor product of (infinity,1)-modules]]


tensor product of Lie algebras

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* table of contents
{: toc}

## Definition

We first recall the definition of an [[action]] of a [[Lie algebra]].

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

Let $\mathfrak{m},\mathfrak{n}$ be a pair of Lie algebras over some [[field]] $k$. An **action** of $\mathfrak{m}$ on $\mathfrak{n}$ is a $k$-bilinear map of $k$-vector spaces $\rho:\mathfrak{m}\otimes \mathfrak{n}\to \mathfrak{n}$ satisfying

* $\rho([m,m'])(n) = \rho(m)(\rho(m')(n))-\rho(m')(\rho(m)(n))$

* $\rho(m)([n,n']) = [\rho(m)(n),n']+[n,\rho(m)(n')]$

for $m\in \mathfrak{m}$ and $n,n'\in \mathfrak{n}$.
=--

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

Let $\mathfrak{m},\mathfrak{n}$ be a pair of Lie algebras, endowed with an action $\rho$ of $\mathfrak{m}$ on $\mathfrak{n}$, and an action $\sigma$ of $\mathfrak{n}$ on $\mathfrak{m}$. Their **tensor product** $\mathfrak{m}\otimes \mathfrak{n}$ is generated by elements $m\otimes n$ such that

* $\lambda(m\otimes n) = (\lambda m)\otimes n = m\otimes (\lambda n),$

* $(m+m')\otimes n = m\otimes n + m'\otimes n,$

* $(m\otimes (n+n') = m\otimes n + m\otimes n',$

* $[m,m']\otimes n = m\otimes \rho(m')(n) - m'\otimes \rho(m)(n)$

* $m\otimes [n,n'] = \sigma(n')(m)\otimes n - \sigma(n)(m)\otimes n',$

* $[m\otimes n, m'\otimes n'] = - (\sigma(n)(m))\otimes (\rho(m')(n')),$

for $\lambda\in k$, $m,m'\in \mathfrak{m}$, and $n,n'\in \mathfrak{n}$.
=--

This definition, due to [Ellis (1989)](#El89), is to be thought of as the [[infinitesimal object|infinitesimal]] version of the [[tensor product of groups]]. It was further extended in [Garcia-Martinez et al. (2015)](#GM15) to [[super Lie algebra|super Lie algebras]].

## References

This definition first appears in

* {#El89} [[Graham Ellis]]. *A non-abelian tensor product of Lie algebras*. Glasgow Mathematical Journal, 33(1), pp.101-120. ([doi](https://doi.org/10.1017/S0017089500008107))

The corresponding definition for [[super Lie algebra|super Lie algebras]] is in

* {#GM15} Xabier García-Martínez, Emzar Khmaladze, Manuel Ladra. *Non-abelian tensor product and homology of Lie superalgebras*. Journal of Algebra Volume 440, 15 October 2015, Pages 464-488. ([doi](https://doi.org/10.1016/j.jalgebra.2015.05.027))


tensor product of modules

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

#Contents#
* table of contents
{:toc}

## Idea

The [[tensor product]] of [[modules]].

## Definition

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

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 theory|category-theoretically]]:

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

The tensor product $A \otimes_R B$
is 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 $B$.

=--

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

If the ring $R$ happens to be a [[field]], then $R$-modules are [[vector spaces]] and the tensor product of $R$-modules becomes the [[tensor product of vector spaces]].

=--

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

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.

=--

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

The tensor product of modules can be generalized to the [[tensor product of functors]].

=--

## Properties

### Monoidal category structure
 {#MonoidalCategoryStructure}

The category $R$[[Mod]] equipped with the tensor product of modules $\otimes_R$ becomes a [[monoidal category]], in fact a [[distributive monoidal category]].

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

A [[monoid]] in $(R Mod, \otimes)$ is equivalently an $R$-[[associative algebra|algebra]].

=--

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

The tensor product of modules [[distributivity|distributes]] over the [[direct sum]] of modules:

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

=--

### Characterization by exact additive functors

See _[[Eilenberg-Watts theorem]]_.

### Exactness properties
 {#ExactnessProperties}

Let $R$ be a [[commutative ring]].

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

For $N \in R Mod$ a module, the [[functor]] of tensoring with this module

$$
  (-) \otimes_R N \colon R Mod \to R Mod
$$

is an [[additive functor|additve]] [[right exact functor]].

=--

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

The functor is [[additive functor|additive]] by the [[distributivity]] of tensor products over [[direct sums]], prop. \ref{DistributivityOverDirectSum}.

A [[category theory|general abstract]] way of seeing that the functor is right exact is to notice that $(-)\otimes_R N$ is a [[left adjoint]] functor, its [[right adjoint]] being the [[internal hom]] $[N,-]$ (see at [[Mod]]). By the discussion at _[[adjoint functor]]_ this means that $(-) \otimes_R N$ even preserves all [[colimits]], in particular the [[finite colimits]].

=--

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

The functor $(-)\otimes_R N$ is **not** a [[left exact functor]] (hence not an [[exact functor]]) for all choices of $R$ and $N$.

=--

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

Let $R \coloneqq \mathbb{Z}$, hence $R Mod \simeq$ [[Ab]] and let $N \coloneqq \mathbb{Z}/2\mathbb{Z}$ the [[cyclic group]] or [[order]] 2.
Moreover, consider the inclusion $\mathbb{Z} \stackrel{\cdot 2}{\hookrightarrow} \mathbb{T}$ sitting in the [[short exact sequence]]

$$
  0 \to \mathbb{Z} \stackrel{\cdot 2}{\to} \mathbb{Z} \to \mathbb{Z}/2\mathbb{Z} \to 0
  \,.
$$

The functor $(-) \otimes \mathbb{Z}/2\mathbb{Z}$ sends this to

$$
  0 \to \mathbb{Z}/2\mathbb{Z}
    \stackrel{0}{\to}
  \mathbb{Z}/2\mathbb{Z}
   \stackrel{id}{\to}
  \mathbb{Z}/2\mathbb{Z}
  \to 0
  \,.
$$

Here the morphism on the left is the 0-morphism: in components it is given for all $n_1, n_2 \in \mathbb{Z}$ by

$$
  \begin{aligned}
    (n_1, n_2 mod 2)
    & \mapsto
    (2 n_1, n_2 mod 2)
    \\
    & \simeq 2 (n_1, n_2 mod 2)
    \\
    & \simeq (n_1, 2 n_2 mod 2)
    \\
    & \simeq (n_1, 0)
    \\
    & \simeq 0
  \end{aligned}
  \,.
$$

Hence this is not a [[short exact sequence]] anymore.

=--

One kind of module $N$ for which $(-)\otimes_R N$ is always exact are [[free modules]].

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

Let $i \colon N_1 \hookrightarrow N_2$ be an inclusion of a [[submodule]]. For $S \in $ [[Set]] write $R^{\oplus {\vert S\vert}} = R[S]$ for the [[free module]] on $S$. Then

$$
  i \otimes_R N \colon N_1 \otimes_R R^{\oplus {\vert S\vert}} \to N_2 \otimes_R R^{\oplus {\vert S\vert}}
$$

is again a [[monomorphism]]. Indeed, due to the [[distributivity]] of the tensor product over the [[direct sum]] and using that $R \in R Mod$ is the tensor unit, this is

$$
  i^{\oplus {\vert S\vert}} \colon N_1^{\oplus {\vert S\vert}} \hookrightarrow N_2^{\oplus {\vert S\vert}}
  \,.
$$

=--

There are more modules $N$ than the free ones for which $(-)\otimes_R N$ is exact. One says

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

If $N \in R Mod$ is such that $(-)\otimes_R N \colon R Mod \to R Mod$ is a [[left exact functor]] (hence an [[exact functor]]), $N$ is called a **[[flat module]]**.

=--

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

For a general module, a measure of the failure of $(-)\otimes_R N$ to be exact is given by the [[Tor]]-functor $Tor^1(-,N)$. See there for more details.

=--

## Related concepts

* [[tensor product of abelian groups]]

* [[tensor product of bimodules]]

* [[tensor product of vector bundles]]

* [[tensor product of algebras]]

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

* [[tensor product of chain complexes]]

* [[tensor product of ∞-modules]]

* [[tensor product of functors]]

## References

Textbook accounts:

* [[Frank W. Anderson]], [[Kent R. Fuller]], §19 in: _Rings and Categories of Modules_, Graduate Texts in Mathematics, **13** Springer (1992) [[doi:10.1007/978-1-4612-4418-9](https://doi.org/10.1007/978-1-4612-4418-9)]

* [[Paul Edwin Bland]], §2.3 in: *Rings and Their Modules*, De Gruyter (2011) [[doi:10.1515/9783110250237](https://doi.org/10.1515/9783110250237), [pdf](http://site.iugaza.edu.ps/mashker/files/%D9%85%D8%AD%D8%A7%D8%B6%D8%B1%D8%A7%D8%AA-%D8%AC%D8%A8%D8%B1-%D8%AF%D9%83%D8%AA%D9%88%D8%B1%D8%A7%D8%A9.pdf)]

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]]]

See also:

* Wikipedia, *[Tensor product of modules](https://en.wikipedia.org/wiki/Tensor_product_of_modules)*

[[!redirects tensor products of modules]]


tensor product of monads

## Idea

There is not in general a tensor product on [[monads]] analogous to the [[Boardman-Vogt tensor product]] on [[operads]]. But in suitable special cases it can be defined (e.g. [BGLS 13](#BGLS13)).

## References

* {#BGLS13} Nathan Bowler, Sergey Goncharov, Paul Blain Levy, Lutz Schröder, _Exploring the Boundaries of Monad Tensorability on Set_, Logical Methods in Computer Science, Volume 9, Issue 3 (September 18, 2013) lmcs:740 ([arXiv:1309.2128](https://arxiv.org/abs/1309.2128))

[[!redirects tensor products of monads]]


tensor product of presentable (infinity,1)-categories

#Contents#
* tabe of contents
{:toc}

## Idea

The [[tensor product]] of [[presentable (infinity,1)-category|presentable]] $(\infty,1)$-[[(infinity,1)-category|categories]] is the product in the [[symmetric monoidal (infinity,1)-category of presentable (infinity,1)-categories]]. See there for more details.

This is the $(\infty,1)$-category $C \otimes D$ which is 'the universal recipient of a bilinear functor' from $C \times D$. Here, we think of coproducts in $C$ and $D$ as addition, and then if a functor $C \times D \to E$ preserves colimits in each variable, in particular it preserves coproducts and so is 'bilinear'. Such a bilinear functor will factor uniquely (in a homotopic sense) through a universal bilinear functor $C \times D \to C \otimes D$, just like for bilinear maps and tensor products of abelian groups.

## Related concepts

* [[tensor product of abelian categories]]

[[!redirects tensor product of presentable (∞,1)-categories]]


tensor product of representations

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

#Contents#
* table of contents
{:toc}

## Idea

The [[tensor product]] of [[linear representations]].

## Definition

Let $G$ be a [[group]] and let

$$
  \rho_i \;\colon\; G \times V_i \longrightarrow V_i
$$

be two [[linear representations]] of $G$ on [[vector spaces]] $V_i$, for $i \in \{1,2\}$. Then the _tensor product of representations_ of these is the linear representation whose underlying vector space is the [[tensor product of vector spaces]] $V_1 \otimes_k V_2$ equipped with the $G$-action induced by the [[diagonal action]]

$$
  G \times V_1 \times V_2
     \overset{\Delta_G \times id}{\longrightarrow}
  G \times G \times V_1 \times V_2
     \simeq
   G \times V_1 \times G \times V_1
     \overset{\rho_1 \times \rho_2}{\longrightarrow}
   V_1 \times V_2
  \,.
$$

## Related concepts

* [[category of representations]]

* [[Clebsch-Gordan coefficient]]

* [[Wigner 3j symbol]], [[Wigner 6j symbol]], [[Wigner 9j symbol]]

* [[McKay quiver]]

[[!redirects tensor product of representations]]


tensor product of vector bundles

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

#Contents#
* table of contents
{:toc}

## Idea

Given two [[vector bundles]] $V_1 \to X$ and $V_2 \to X$ over the same base space $X$, their _tensor product of vector bundles_ $V_1 \otimes V_2 \to X$ is the vector bundle over $X$ whose [[fiber]] over any point is the [[tensor product of vector spaces]] (i.e. the [[tensor product of modules]]) of the respective fibers of $V_1$ and $V_2$ (the [fiber-wise](topological+vector+bundle#FiberwiseOperations) tensor product).

The tensor product of vector bundles makes the category [[Vect(X)]] into a [[symmetric monoidal category|symmetric]] [[monoidal category]], in fact a [[distributive monoidal category]].

## Definition

+-- {: .num_defn }
###### Definition
**(tensor product of [[topological vector bundles]])**

Let $X$ be a [[topological space]], and let $E_1 \to X$ and $E_2 \to X$ be two [[topological vector bundles]] over $X$.

Let $\{U_i \subset X\}_{i \in I}$ be an [[open cover]] with respect to which both vector bundles locally trivialize (this always exists: pick a [[local trivialization]] of either bundle and form the joint [[refinement]] of the respective [[open covers]] by [[intersection]] of their patches). Let

$$
  \left\{
    (g_1)_{i j}
    \colon U_i \cap U_j \to GL(n_1)
  \right\}
  \phantom{AAA}
  \text{and}
  \phantom{AAA}
  \left\{
    (g_2)_{i j}
    \colon
     U_i \cap U_j \longrightarrow GL(n_2)
  \right\}
$$

be the [[transition functions]] of these two bundles with respect to this cover.

For $i, j \in I$ write

$$
  \array{
    (g_1)_{i j} \otimes (g_2)_{i j}
    &\colon&
    U_i \cap U_j
      &\longrightarrow&
    GL(n_1 \cdot n_2)
  }
$$

be the pointwise [[tensor product of vector spaces]] of these transition functions

Then the _tensor product bundle_ $E_1 \otimes E_2$ is the one glued from this tensor product of the transition functions (by [this construction](topological+vector+bundle#TopologicalVectorBundleFromCechCocycle)):

$$
  E_1 \otimes E_2
  \;\coloneqq\;
  \left(
    \left( \underset{i}{\sqcup} U_i \right)
    \times
    \left(
      \mathbb{R}^{n_1 \cdot n_2}
    \right)
  \right)/ \left( \left\{ (g_1)_{i j} \otimes (g_2)_{i j} \right\}_{i,j \in I} \right)
  \,.
$$

=--

## Examples

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

Let $X$ be a [[topological space]] and let $E_i \overset{p_i}{\to} X$ be a two [[topological vector bundles]] over $X$, of [[finite number|finite]] [[rank of a vector bundle]]. Then a [[homomorphism]] of vector bundles

$$
  f \;\colon\; E_1 \rightarrow E_2
$$

is equivalently a [[section]] of the tensor product of $E_2$ with the [[dual vector bundle]] of $E_1$:

$$
  Hom_{Vect(X)}(E_1, E_2)
   \;\simeq\;
  \Gamma_X( E_1^\ast \otimes_X E_2)
  \,.
$$

Moreover, this section is a trivializing section ([this example](topological+vector+bundle#FiberwiseLinearlyIndependentSectionsTrivialize)) precisely if the corresponding morphism is an [[isomorphism]].

=--

## Related concepts

* [[external tensor product of vector bundles]]

* [[direct sum of vector bundles]]

* [[inner product on vector bundles]]

* [[dual vector bundle]]

* [[tensor product]]

* [[tensor category]]

## References

* [[Max Karoubi]], §4.8 in: _K-Theory -- An introduction_, Grundlehren der mathematischen Wissenschaften **226** Springer (1978) [[pdf](https://webusers.imj-prg.fr/~max.karoubi/K.book/MK.book.pdf), [doi:10.1007%2F978-3-540-79890-3](https://link.springer.com/book/10.1007%2F978-3-540-79890-3)]

[[!redirects tensor products of vector bundles]]

[[!redirects tensor product of line bundles]]
[[!redirects tensor products of line bundles]]


tensor product of vector spaces

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

#Contents#
* table of contents
{:toc}

## Idea

The _tensor product_ of two vector spaces is a new vector space with the property that [[bilinear maps]] out of the [[Cartesian product]] of the two spaces are equivalently [[linear maps]] out of the tensor product.

The tensor product of vector spaces is just the special case of the [[tensor product of modules]] over some [[ring]] $R$ for the case that this ring happens to be a [[field]].

The tensor product of vector spaces makes the [[category]] [[Vect]] of all [[vector spaces]] into a [[monoidal category]], in fact a [[distributive monoidal category]].

## Definition

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

Given two [[vector spaces]] over some [[field]] $k$,  $V_1, V_2 \in Vect_k$, their [[tensor product of vector spaces]] is the vector space denoted

$$
  V_1 \otimes_k V_2 \in Vect
$$

whose elements are [[equivalence classes]] of formal linear combinations of [[tuples]] $(v_1,v_2)$ with $v_i \in V_i$, for the [[equivalence relation]] given by

$$
  (k v_1 , v_2) \;\sim\; k( v_1 , v_2) \;\sim\; (v_1, k v_2)
$$

$$
  (v_1 + v'_1 , v_2) \; \sim \; (v_1,v_2) + (v'_1, v_2)
$$

$$
  (v_1 , v_2 + v'_2) \; \sim \; (v_1,v_2) + (v_1, v'_2)
$$

More abstractly this means that the [[tensor product of vector spaces]] is the vector space characterized by the fact that

1. it receives a [[bilinear map]]

   $$
     V_1 \times V_2 \longrightarrow V_1 \otimes V_2
   $$

   (out of the [[Cartesian product]] of the underlying sets)

1. any other [[bilinear map]] of the form

   $$
     V_1 \times V_2 \longrightarrow V_3
   $$

   factors through the above bilinear map via a unique [[linear map]]

   $$
     \array{
        V_1 \times V_2 &\overset{bilinear}{\longrightarrow}& V_3
        \\
        \downarrow & \nearrow_{\mathrlap{\exists ! \, linear}}
         \\
         V_1 \otimes_k V_2
     }
   $$

=--

## References

* *Tensor Products*, §III.89 in: *The Princeton Companion to Mathematics* (2009) [[doi:10.1515/9781400830398.301a](https://doi.org/10.1515/9781400830398.301a)]

## Related concepts

* [[tensor product of modules]]

* [[tensor product of chain complexes]]

* [[tensor product of vector bundles]]

* [[graded vector space]]

* [[inductive tensor product]]

[[!redirects tensor products of vector spaces]]


tensor product theory

If $A$ and $B$ are [[algebraic theory|algebraic theories]], the algebraic theory $A\otimes B$ is characterized by the fact that its models can be identified with $A$-models in $B Mod$, or equivalently as $B$-models in $A Mod$. There are maps of theories $A\to A\otimes B$ and $B\to A\otimes B$ which are universal for maps of theories $A\to C$ and $B\to C$ whose images commute, for any theory $C$.


tensor projection

## Idea

How do you type check [[linear logic|MILL]], but with projections rather than matching to eliminate $\otimes$?  [Mike Shulman](#Shulman) gave one such system, but its type-checking algorithm requires two passes (if this algorithm were incorporated into the rules, there would be an extra intermediate judgment with "activeness" annotations on the types), and doesn't handle additional type constructors such as [[linear implication]].

This page is intended to give algorithmic rules for a system with tensor projections and linear implication, but at present it is unverified and its interpretation into monoidal categories is unclear.

## Judgment form, Notations

$U \subseteq \Gamma \vdash t\,:\,T\,\boxtimes\,U' \subseteq \Gamma'$

$U$, $\Gamma$, and $t$ are inputs; $T$, $U'$, and $\Gamma'$ are outputs.

The observation mentioned in [Allais](#Allais) motivating the use of "$\boxtimes$" was that a term generally doesn't use all the resources it's given, so the rest are passed back out, tensored with the result type.

$\Gamma$ and $\Gamma'$ are [[contexts]] with some natural number of semicolon "scope" delimiters. $U$ and $U'$ are "usage" sets, which are subsets of the (domains of the) respective contexts. In [Allais](#Allais), the usage sets are represented as bit vectors (which is a fine way to implement them). Unlike [Allais](#Allais), the usage before and after are for *different* contexts, because type checking a term generally adds projections to the context as potentially-consumed resources.

The purpose of delimited scopes in the context is to remember by when we must've consumed tensor projections. Currently, only lambda adds a scope delimiter. The current algorithm puts projections in the outermost scope it can get away with. This approach will not work when adding additive connectives.

In the rules, I implicitly coerce a context to the usage set of its entire domain. The abbreviation $(\Gamma \mapsto \Gamma')U$ "weakens" $U$ to include all entries in $\Gamma'$ but not $\Gamma$. Set theoretically speaking: $(\Gamma \mapsto \Gamma')U \coloneqq U \cup (\Gamma' \setminus \Gamma)$. "$\Delta$" and variants are used as metavariables for contexts *without any delimiters*.

Some important (intended) properties of the judgment form are that if $U \subseteq \Gamma \vdash t\,:\,T\,\boxtimes\,U' \subseteq \Gamma'$, then:

* $U \subseteq \Gamma$ and $U' \subseteq \Gamma'$
* $\Gamma$ and $\Gamma'$ have the same number of delimiters.
* $\Gamma'$ includes all the entries of $\Gamma$, in the same scopes. (So in particular, $\Gamma \subseteq \Gamma'$.)
* $U' \subseteq (\Gamma \mapsto \Gamma')U$

Note that the context grows while the usage shrinks. Erm, I guess these usage sets are the sets of resources *not* used (and thus available). $(\Gamma \mapsto \Gamma')U$ should be understood as the *actual* usage indicated by $U \subseteq \Gamma$, in light of the additional resources in $\Gamma'$. So as the type checker proceeds, more resources are known of, but fewer remain available.

## Generators and Functions

There are generator morphisms, which can be used any number of times. $f \in \mathcal{G}(A;\overrightarrow{B})$ means you can get $f\,:\,A \multimap \overrightarrow{B}$ whenever you want.

Functions take exactly one argument, and have one or more results. Semantically, the results are tensored ($\otimes$) together. Following [Shulman](#Shulman), multiple results are accessed using "generalized Sweedler notation". For example, if $f(a)$ has two results, they are typically written as $f_{(1)}(a)$ and $f_{(2)}(a)$. Using *both* results is a *single* use of linear implication elimination. This is actually the only way to use tensor projections, currently. Combining it with function application was purely a matter of convenience, and is probably a bad idea, for adding more connectives.

## Rules

$$\frac{t:T \in U}{U \subseteq \Gamma \vdash t\,:\,T\,\boxtimes\,U \setminus \{t\} \subseteq \Gamma}$$

Yes, you get *terms* out of the context, not just variables. For now, the only terms that end up in the context are variables and tensor projections. $(t:T) \in U$ really means $(t:T) \in \Gamma$ and $t \in U$, since the type is stored in $\Gamma$.

$$\begin{gathered}
\frac{f \in \mathcal{G}(A;\overrightarrow{B})}
{U \subseteq \Gamma \vdash f\,:\,A \multimap \overrightarrow{B}\,\boxtimes\,U \subseteq \Gamma} \\
\\
\frac{}{U \subseteq \Gamma \vdash ()\,:\,\mathbf{1}\,\boxtimes\,U \subseteq \Gamma} \\
\\
\frac{U \subseteq \Gamma \vdash c\,:\,\mathbf{1}\,\boxtimes\,U_1 \subseteq \Gamma_1 \qquad
U_1 \subseteq \Gamma_1 \vdash t\,:\,T\,\boxtimes\,U_2 \subseteq \Gamma_2}
{U \subseteq \Gamma \vdash c;\,t\,:\,T\,\boxtimes\,U_2 \subseteq \Gamma_2} \\
\\
\frac{U \subseteq \Gamma \vdash a\,:\,A\,\boxtimes\,U_1 \subseteq \Gamma_1 \qquad
U_1 \subseteq \Gamma_1 \vdash b\,:\,B\,\boxtimes\,U_2 \subseteq \Gamma_2}
{U \subseteq \Gamma \vdash (a,b)\,:\,A \otimes B\,\boxtimes\,U_2 \subseteq \Gamma_2} \\
\\
\frac{}{U \subseteq \Gamma \vdash \pi\,:\,(A \otimes B) \multimap (A,B)\,\boxtimes\,U \subseteq \Gamma}
\end{gathered}$$

Yeah, that's tensor elim now. Kind of funky. It turns out this rule is redundant: you can also use $(\lambda x:A \otimes B.x)$.

$$\frac{U \cup \{x\} \subseteq \Gamma;x:A \vdash b\,:\,B_1 \otimes \dots \otimes B_n\,\boxtimes\,U' \subseteq \Gamma';\Delta \qquad
U' \cap \Delta = \emptyset}
{U \subseteq \Gamma \vdash \lambda x:A.b\,:\,A \multimap (B_1,\dots,B_n)\,\boxtimes\,U' \subseteq \Gamma'}$$

There is a new scope for checking the body, which starts out with just the bound variable. When done checking, there's generally more stuff in it. Currently, that would only be tensor projections. In any case, everything in the new scope must've been consumed.

$$\frac{U \subseteq \Gamma \vdash f\,:\,A \multimap B\,\boxtimes\,U_1 \subseteq \Gamma_1 \qquad
U_1 \subseteq \Gamma_1 \vdash a\,:\,A\,\boxtimes\,U_2 \subseteq \Gamma_2}
{U \subseteq \Gamma \vdash f(a)\,:\,B\,\boxtimes\,U_2 \subseteq \Gamma_2}$$

That's the *nice* application rule, for when there's one result. Now for the *nasty* ones...

$$\frac{\begin{array}{l}f_{(i)}(a) \notin \Gamma \\
U \subseteq \Gamma \vdash f\,:\,A \multimap \overrightarrow{B}\,\boxtimes\,U_1 \subseteq \Gamma_1 \\
U_1 \subseteq \Gamma_1 \vdash a\,:\,A\,\boxtimes\,U_2 \subseteq \Gamma_2 \\
|\overrightarrow{B}| \gt 1 \qquad 0 \lt i \leq |\overrightarrow{B}| \\
U_d = (\Gamma \mapsto \Gamma_2)U \setminus U_2 \\
\Gamma_2 = \Gamma_l;\Delta;\Gamma_r \\
U_d \cap \Gamma_r = \emptyset \qquad U_d \cap \Delta \neq \emptyset \\
\Gamma_3 = \Gamma_l;\Delta,\overrightarrow{f(a):B};\Gamma_r \\
U_3 = (\Gamma_2 \mapsto \Gamma_3)U_2 \setminus \{f_{(i)}(a)\}\end{array}}
{U \subseteq \Gamma \vdash f_{(i)}(a)\,:\,B_i\,\boxtimes\,U_3 \subseteq \Gamma_3}$$

$$\;$$

$$\frac{\begin{array}{l}f^u_{(i)}(a) \notin \Gamma \\
U \subseteq \Gamma \vdash f\,:\,A \multimap \overrightarrow{B}\,\boxtimes\,U_1 \subseteq \Gamma_1 \\
U_1 \subseteq \Gamma_1 \vdash a\,:\,A\,\boxtimes\,U_2 \subseteq \Gamma_2 \\
|\overrightarrow{B}| \gt 1 \qquad 0 \lt i \leq |\overrightarrow{B}| \\
(\Gamma \mapsto \Gamma_2)U \setminus U_2 = \emptyset \\
\Gamma_2 = \Delta_l;\Gamma_r \\
\Gamma_3 = \Delta_l,\overrightarrow{f^u(a):B};\Gamma_r \\
U_3 = (\Gamma_2 \mapsto \Gamma_3)U_2 \setminus \{f^u_{(i)}(a)\}\end{array}}
{U \subseteq \Gamma \vdash f^u_{(i)}(a)\,:\,B_i\,\boxtimes\,U_3 \subseteq \Gamma_3}$$

Whew. These rules add tensor projections of a function application to the context, the first time the application is encountered in a given scope. There are two rules because if the application doesn't consume any resources, there's generally an ambiguity between multiple instances of the application, so we add a "label" to disambiguate. $u$ is the label metavariable. This labeling idea is from [Shulman](#Shulman). See there for a discussion of the problem.

In the rule mentioning $U_d$, $U_d$ is the set of resources consumed by the application. That is, both the function and argument. The premise $U_d \cap \Delta \neq \emptyset$ implies that $U_d$ is nonempty. Meanwhile, the other rule checks that the same computed usage *is* empty.

In both rules, $\Gamma_2$ is split to find the scope to which the projections are added. If the application uses resources, that scope is the innermost scope with resources that were used. Otherwise it's the outermost scope.

The projections cannot go in a scope outside one with resources used, because then there would be no way to get the variables out there in ordinary MILL using a match.

On the other hand, it can be trouble to put the projections in a scope more inside than necessary, since the lambda rule checks that everything in its scope was consumed. The projections only need to be consumed before one of their dependencies goes out of scope. For example, ($\lambda x:A.((\lambda y:B.(y,f_{(1)}(x))),f_{(2)}(x))$) means ($\lambda x:A.let\;(f_1,f_2) = f(x)\;in\;((\lambda y:B.(y,f_1)),f_2)$), but would be rejected if we naively put $f(x)$ into the scope in which it's first encountered.

## References

These algorithmic rules are loosely based on:

* {#Allais} Guillaume Allais, _Typing with Leftovers – A mechanization of Intuitionistic Multiplicative-Additive Linear Logic_, TYPES 2017 ([pdf](http://drops.dagstuhl.de/opus/volltexte/2018/10049/pdf/LIPIcs-TYPES-2017-1.pdf))

The idea to allow tensor projections in some sound way, and a lot of the notation, are from:

* {#Shulman} [[Michael Shulman]], _A practical type theory for symmetric monoidal categories_, 2019 ([arXiv](https://arxiv.org/abs/1911.00818))


tensor ring

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

# Contents
* table of contents
{: toc}

## Idea

A [[free monoid]] object in [[Ab]]

## Definition

Given an [[abelian group]] $G$, the **tensor ring** is a [[ring]] $T(G)$ with an abelian group homomorphism $g:G \to T(G)$, such that for every other ring $R$ with abelian group homomorphism $h:G \to R$, there is a unique ring homomorphism $i:T(G) \to R$ such that $i \circ g = h$.

## See also

* [[ring]]

* [[symmetric ring]]

* [[exterior ring]]

* [[Clifford ring]]

* [[Weyl ring]]

* [[tensor algebra]]

[[!redirects tensor rings]]


tensor triangulated category

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###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Stable homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A _tensor triangulated category_ is a [[category]] that carries the structure of a [[symmetric monoidal category]] and of a [[triangulated category]] in a compatible way.

## Definition

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

There are different variants of the definition in the literature, asking for successively more structure.

To start with, a **tensor triangulated category** must be at least a [[category]] $Ho$ equipped with

1. the structure of a [[symmetric monoidal category]] $(Ho, \otimes, 1, \tau)$ ("[[tensor category]]");

1. the structure of a [[triangulated category]] $(Ho, \Sigma, CofSeq)$

1. for all objects $X,Y\in Ho$ [[natural isomorphisms]]

   $$
     e_{X,Y}
       \;\colon\;
     (\Sigma X) \otimes Y
       \overset{\simeq}{\longrightarrow}
     \Sigma(X \otimes Y)
   $$

such that

1. (tensor product is additive) for each object $V$ the functor $V \otimes (-) \simeq (-) \otimes V$ is an [[additive functor]];

1. (tensor product is exact) for each object $V \in Ho$ the functor $V \otimes (-) \simeq (-)\otimes V$ preserves distinguished triangles in that for

   $$
     X
       \overset{f}{\longrightarrow}
     Y
       \overset{g}{\longrightarrow}
     Y/X
       \overset{h}{\longrightarrow}
     \Sigma X
   $$

   in $CofSeq$, then also

   $$
     V \otimes X
       \overset{id_V \otimes f}{\longrightarrow}
     V\otimes Y
       \overset{id_V \otimes g}{\longrightarrow}
     V \otimes Y/X
       \overset{id_V \otimes h}{\longrightarrow}
     V \otimes (\Sigma X)
       \simeq
     \Sigma(V \otimes X)
   $$

   in $CofSeq$, where the equivalence at the end is $e_{X,V}\circ \tau_{V, \Sigma X}$.

Jointly this says that the isomorphisms $e$ give $V \otimes (-)$ the structure of a [[triangulated functor]], for all $V$.

=--

([Balmer 05, def. 1.1](#Balmer05))

In addition one may ask that

1. (coherence) for all $X, Y, Z \in Ho$ the following [[commuting diagram|diagram commutes]]

   $$
     \array{
       ( \Sigma(X) \otimes Y) \otimes Z
         &\overset{e_{X,Y} \otimes id}{\longrightarrow}&
       (\Sigma (X \otimes Y)) \otimes Z
         &\overset{e_{X \otimes Y, Z}}{\longrightarrow}&
       \Sigma( (X \otimes Y) \otimes Z )
       \\
       {}^{\mathllap{\alpha_{\Sigma X, Y, Z}}}\downarrow
         &&
         &&
       \downarrow^{\mathrlap{\Sigma \alpha_{X,Y,Z}}}
       \\
       \Sigma (X) \otimes (Y \otimes Z)
         &&
         \underset{e_{X, Y \otimes Z }}{\longrightarrow}
         &&
      \Sigma( X \otimes (Y \otimes Z) )
     }
   $$

   is in $CofSeq$, where $\alpha$ is the [[associator]] of $(Ho, \otimes, 1)$.

1. (graded commutativity) for all $n_1, n_2 \in \mathbb{Z}$ the following [[commuting diagram|diagram commutes]]

   $$
     \array{
       (\Sigma^{n_1} 1) \otimes (\Sigma^{n_2} 1)
       &\overset{\simeq}{\longrightarrow}&
       \Sigma^{n_1 + n_2} 1
       \\
       {}^{\mathllap{\tau_{\Sigma^{n_1}1, \Sigma^{n_2}1}}}\downarrow
         &&
       \downarrow^{\mathrlap{(-1)^{n_1 \cdot n_2}}}
       \\
       (\Sigma^{n_2} 1) \otimes (\Sigma^{n_1} 1)
       &\underset{\simeq}{\longrightarrow}&
       \Sigma^{n_1 + n_2} 1
     }
     \,,
   $$

   where the horizontal isomorphisms are composites of the $e_{\cdot,\cdot}$ and the braidings.

This is ([Hovey-Palmieri-Strickland 97, def. A.2.1](#HoveyPalmieriStrickland97)) except for statements concerning possible further [[closed monoidal category]] structure. There this is called "symmetric monoidal structure compatible with the triangulation".

Finally, one can ask for the existence of additional compatibility commutative diagrams, for instance representing a "derived shadow" of the [[pushout product]] axiom of a [[monoidal model category]].  These can be found as (TC3), (TC4), and (TC5) in ([May](#May01)).

## Examples

* The archetypical example is the [[stable homotopy category]] equipped with the [[smash product of spectra]]. For details see at _[[model structure on orthogonal spectra]]_ the section _[The monoidal stable homotopy category](model+structure+on+orthogonal+spectra#TheMonoidalStableHomotopyCategory)_.

## Related concepts

* [[spectrum of a tensor triangulated category]]

* [[tensor category]], [[triangulated category]]

* [[tensor (∞,1)-category]]

## References

* {#HoveyPalmieriStrickland97} [[Mark Hovey]], [[John Palmieri]], [[Neil Strickland]], _Axiomatic stable homotopy theory_, Memoirs of the AMS 610 (1997) ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/axiomatic.pdf), [doi:10.1090/memo/0610](http://dx.doi.org/10.1090/memo/0610))

* {#Balmer05} [[Paul Balmer]], _The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math., 588:149–168, 2005 ([arXiv:0409360](http://arxiv.org/abs/math/0409360), [doi:10.1515/crll.2005.2005.588.149](https://doi.org/10.1515/crll.2005.2005.588.149))

* {#May01} [[Peter May]], _The Additivity of Traces in Triangulated Categories_, Advances in Mathematics, Volume 163, Issue 1, 2001, Pages 34-73 ([doi:10.1006/aima.2001.1995](https://doi.org/10.1006/aima.2001.1995))

Review:

* {#Stevenson11} [[Greg Stevenson]], _Tensor actions and locally complete intersection_ PhD thesis 2011 ([pdf](http://www.math.uni-bielefeld.de/~gstevens/Stevenson_thesis.pdf))

* [[Paul Balmer]], _A guide to tensor-triangular classification_, in: [[Haynes Miller]] (ed.) _[[Handbook of Homotopy Theory]]_ ([arXiv:1912.08963](https://arxiv.org/abs/1912.08963))

[[!redirects tensor triangulated categories]]


tensor unit

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#### Monoidal categories
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=--
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#Contents#
* table of contents
{:toc}

## Idea

The [[unit object]] of a [[tensor product]].

## Related concepts

* [[unit]]

* [[unitality]]

* [[unitor]]

[[!redirects tensor units]]


tensorial costrength

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#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A __tensorial costrength__ (or simply __costrength__) for a [[functor]] $F \colon V \to V$ on a [[monoidal category]] $V$ is, in essence, the [[opposite category|dual]] of a [[tensorial strength]]. Whereas a strength provides a coherent [[natural transformation]] of the form:
$$X \otimes F(Y) \to F(X \otimes Y)$$
or
$$F(X) \otimes Y \to F(X \otimes Y)$$
(a __left-strength__ or __right-strength__ respectively),
a costrength provides a coherent [[natural transformation]] of the form:
$$F(X \otimes Y) \to X \otimes F(Y)$$
or
$$F(X \otimes Y) \to F(X) \otimes Y$$
(a __left-costrength__ or __right-costrength__ respectively).

When $V$ is [[symmetric monoidal category|symmetric]], the [[braiding]] $b$ allows to obtain a left-costrength from a right-costrength, and in this setting, one usually drops “left-” or “right-” and simply talks about __tensorial costrengths__.

More generally, a __costrength__ (on a monoidal category which is not necessarily symmetric) is a left-costrength and a coright-strength such that the two induced maps $T(X\otimes (Y\otimes Z)) \to (X \otimes T Y) \otimes Z$ agree.

## Definition

For now, see *[[tensorial strength]]*, whose definition may be appropriately dualised.

## Terminology

The terminology "costrength" was introduced by [Blute, Cockett & Seely (1997)](#BCS97) for the concept described here. This terminology is consistent with the usual convention in category theory of prefixing a term by "co-" when it is the same concept in the [[opposite category]] $V^{op}$.

Unfortunately, in the same year, [Power & Robinson](#PR97) also used the terminology "costrength" to describe a [[tensorial strength|right-strength]]. A right-strength is a left-strength with respect to the reversal of the tensor product, i.e. a left-strength in the [[reverse monoidal category]] $V^{rev}$. While both usages can be found in recent literature, we prefer the terminology that is consistent with prior categorical terminology.

(Note that the notion of "cotensorial strength" from [Kock (1972)](#kock72) is different from both concepts, and involves a costrength-like transformation involving the [[internal hom]] of a monoidal category, rather than the tensor.)

## Related concepts

* [[tensorial strength]]
* [[strong monad]]
* [[commutative monad]]

## References

* {#BCS97} [[R. F. Blute]], [[J. R. B. Cockett]], and [[R. A. G. Seely]]. _Categories for computation in context and unified logic._ Journal of Pure and Applied Algebra 116.1-3 (1997): 49-98.

The following paper introduces a conflicting definition of "costrength", which we call a [[tensorial strength|right-strength]]:

* {#PR97} [[John Power]], and [[Edmund Robinson]]. _Premonoidal categories and notions of computation_. Mathematical structures in computer science 7.5 (1997): 453-468.

The notion of cotensorial strength is mentioned briefly in:

* {#kock72} [[Anders Kock]], *Strong functors and monoidal monads*, Arch. Math **23** (1972) 113–120 [[doi:10.1007/BF01304852](https://doi.org/10.1007/BF01304852), [pdf](http://home.imf.au.dk/kock/SFMM.pdf)]

[[!redirects tensorial costrength]]
[[!redirects tensorial costrengths]]
[[!redirects costrong functor]]
[[!redirects costrong functors]]
[[!redirects costrength]]
[[!redirects costrengths]]

[[!redirects cotensorial strength]]
[[!redirects cotensorial strengths]]


tensorial strength

+-- {: .rightHandSide}
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###Context###
#### Enriched category theory
+--{: .hide}
[[!include enriched category theory contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In the simplest case, _tensorial strength_ is a property of an [[endofunctor]] $T \colon V \to V$ on a [[monoidal category]] which characterizes it as a  $V$-[[enriched functor]] when $V$ is canonically regarded as an [[enriched category]] over itself.

More generally, tensorial strength on a functor
$$
  F \,\colon\, \mathbf{C}_0 \to \mathbf{C}_1
$$

between categories that are [[tensoring|tensored]] ([[copowering|copowered]]) over $V$

$$
  \textstyle{\otimes}
    \,\colon\,
  V \times C_i \longrightarrow C_i
$$

is a [[natural transformation]]

$$
  v \otimes F(c) \to F(v \otimes c)
$$

exhibiting compatibility of $F$ with that [[tensoring]].

In the case that $V$ is also [[closed monoidal category|closed monoidal]], there is a close relation between strong functors and [[enriched functors]] (see [below](RelationToEnrichedFunctors)).

{#WarningOnTerminology} In fact, beware that [Kock 1972](#Kock72) originally said *tensorial strength* in constrast to the *strength* of *strong functors* by which he referred to $V$-[[enriched functors]]. Beware that today "strong functor" (certainly as in "[[strong monad]]") is usually understood in the sense of tensorial strength, not in the sense of "enriched functor"  (to the extent that the two notions differ at all, see again [below](RelationToEnrichedFunctors)). Hence historically it would be more accurate to say "tensorially strong functors" for the notion discussed here, but this term is not commonly used at all.

## Definition

### Standard definition

Given a [[monoidal category]] $(V, \otimes, I)$ with [[tensor product]] $\otimes$, a **tensorial left-strength** for a functor

$$F \colon V \to V$$

is usually defined as a [[natural transformation]]

$$
  \beta_{v, w}
  \;\colon\;
  v \otimes F(w)
    \longrightarrow
  F(v \otimes w)
$$

making the following [[commuting diagram|diagrams commute]]:

$$
\array{
  (u \otimes v) \otimes F(w) & & \overset{\beta_{u\otimes v,w}}{\to} & & F((u\otimes v)\otimes w)\\
  ^\mathllap{\alpha_{u,v,F(w)}}\downarrow & & & & \downarrow^{\mathrlap{F(\alpha_{u,v,w})}}\\
  u\otimes (v\otimes F(w)) & \underset{1_u\otimes\beta_{v,w}}{\to} & u\otimes F(v\otimes w) & \underset{\beta_{u,v\otimes w}}{\to} & F(u\otimes (v\otimes w))
}
$$
and
$$
\array{
I\otimes F(v) & \overset{\beta_{I,v}}{\to} & F(I\otimes v) \\
& _\mathllap{\lambda_{F(v)}}\searrow & \downarrow^\mathrlap{F(\lambda_v)}\\
& & F(v)
}
$$
where $\alpha$ is the [[associator]] and $\lambda$ is the [[unitor]] of the [[monoidal category]] $V$.

A **tensorial right-strength** (not to be confused with a [[tensorial costrength]]) is defined symmetrically, as a natural transformation
$$\gamma_{v, w}: F(w) \otimes v \to F(w \otimes v)$$

When $V$ is [[symmetric monoidal category|symmetric]], tensorial left strengths are equivalently tensorial right strengths, and in this setting, one usually drops "left-" or "right-" and simply talks about **tensorial strengths**.

A functor equipped with a tensorial strength is called a **strong functor** (not to be confused with a *strong 2-functor*, which is another name for a [[pseudofunctor]], i.e. a [[lax 2-functor]] whose coherence cells are invertible).

More generally, the notion makes sense not just for endofunctors of $V$, but for functors between any categories that are "tensored over $V$."

### Tentative reformulation

+-- {: .standout}

_Tentative_ -- The following reformulation [[Baez 2009](&Baez09)] of the standard definition is tentative.

=--

It seems we may phrase this definition more conceptually as follows. There is a bicategory $V-Act$ where:

* the objects are categories $C$ equipped with a lax left $V$-action, i.e. a [[lax monoidal functor]] $\alpha : V \to End(C)$,
* the morphisms are functors laxly preserving this action, and
* the 2-morphisms are natural transformations compatible with this action.

$V$ has a canonical left action on itself, which allows us to think of it as an object of $V-Act$.  So, given a functor $F: V \to V$, a **tensorial strength** for $F$ is defined to be a way of equipping it with the structure of a morphism in $V-Act$.

A bit more precisely:

$$V-Act = Lax(B V, Cat)$$

Here $B V$ is $V$ regarded as a 1-object weak 2-category, $Cat$ is regarded as a 2-category, and $Lax$ is the bicategory of

* weak 2-categories
* [[lax functor|lax functors]]
* [[lax natural transformation|lax natural transformations]]

There is a forgetful 2-functor

$$V-Act \to Cat $$

and a **tensorial strength** for a functor $F: V \to V$ is a way of lifting it to a morphism $\tilde{F}: \tilde{V} \to \tilde{V}$ in $V-Act$, where $\tilde{V}$ is $V$ equipped with its canonical left action on itself.

## Properties

### Relation to enriched functors
 {#RelationToEnrichedFunctors}

> (For more references see also at *[enriched monad -- relation to Strong monads](enriched+monad#RelationToStrongMonads)*)

One possible reason why it may be hard to grasp the notion of strength is because in the case $V = Set$, they're sort of invisible. Every functor $T: Set \to Set$ has one, in fact a canonical one! This will make more sense in a moment.

Strengths are easier to understand by considering the case where $V$ is [[closed monoidal category|closed monoidal]]. Here $V$ is [[enriched category|enriched]] in itself, and one of the most important aspects of strengths is this:

* A functor $T: V \to V$ with a strength is the same thing as a $V$-enrichment on $T$.

Here's how that works. An enrichment on $T$ in $V$-[[enriched category theory]] consists of a natural family of morphisms in $V$,

$$hom(a, b) \to hom(T(a), T(b)),$$

satisfying some suitable axioms. Here $hom$ denotes the enrichment of $V$ in itself -- the [[internal hom]]. Starting from a strength on $T$, you can cook up an enrichment on $T$, roughly as follows: the map above is equivalent by hom-tensor [[adjunction]] to a map

$$hom(a, b) \otimes T(a) \to T(b)$$

and now the strength allows you to slide $hom(a, b)$ inside $T$, and from there you just apply $T$ to the evaluation:

$$T(hom(a, b) \otimes a) \overset{T(eval_{a, b})}{\to} T(b)$$

and presto, you're done. The strength axioms ensure that this enrichment structure on $T$ satisfies the axioms you need for a functor to be enriched.

(And going back the other way, from an enrichment to a strength, is also easy -- there you have to dualize. That is, instead of using the evaluation which is the counit of the hom-tensor adjunction, you use the coevaluation which is the unit. )

After some work, one can convince oneself that the notions of strength and enrichment for endofunctors on closed monoidal categories really are equivalent notions.

And now we can understand why strengths in the case $V = Set$ are so "invisible" -- every functor $T: Set \to Set$ is $Set$-enriched; that's what we mean by an ordinary functor!

## Examples

(Plenty of examples of [[strong monads]] are of interest, particularly in the context of [[monads in computer science]]. See there for more.)

\begin{example}
\label{ForHeytingAlgebras}
Suppose the theory of [[Boolean algebras]] -- say we're studying the structure of a free Boolean algebra on a set of generators, $B[X]$. Then again we can think of this, or of any [[Heyting algebra]], as enriched in itself. Further, we have definable unary operators

$$T \colon B[X] \to B[X]$$

in the theory, such as $T = p \wedge (-)$, or $T = p \Rightarrow (-)$, etc. The great discovery of [[Charles S. Peirce]] is that any definable unary operator in the theory of Boolean algebras carries a strength, or if you prefer is enriched. That may be taken to be the essence of Peirce's iteration rule for Alpha, and it [[vertical categorification|categorifies]] right over to a similar statement for the theory of [[closed monoidal category|closed categories]], [[star-autonomous category|star-autonomous categories]], what have you: all definable unary covariant functors in such theories carry canonical strengths.

(Peirce went a little further, and incorporated notions of contravariant strength as well.)
\end{example}

## Related concepts

* [[strong natural transformation]]

* [[cartesian closed functor]]

* [[strong monad]]

* [[tensorial costrength]]

* [[commutative monad]]

## References

The notion of tensorial strength originates in

* {#Kock72} [[Anders Kock]], below Thm. 1.3 of: *Strong functors and monoidal monads*, Arch. Math **23** (1972) 113–120 [[doi:10.1007/BF01304852](https://doi.org/10.1007/BF01304852), [pdf](http://home.imf.au.dk/kock/SFMM.pdf)]

Comprehensive review:

* {#Ratkovic12} [[Kruna S. Ratkovic]], *Tensorial Strength*, Section 3 of: *Morita theory in enriched context* (2012) [[arXiv:1302.2774](https://arxiv.org/abs/1302.2774), [hal:tel-00785301](https://theses.hal.science/tel-00785301/document)]

Review in the context of [[monads in computer science]]:

* [[Tarmo Uustalu]], *Strong functors, Strong monads*, in: *[Containers for effects and contexts](https://www.ioc.ee/~tarmo/ssgep15/)*, Lecture notes (2015) [[pdf](https://www.ioc.ee/~tarmo/ssgep15/ssgep-1a.pdf), [[Uustalu-StrongFunctors.pdf:file]]]

* {#McDermottUustalu22} [[Dylan McDermott]], [[Tarmo Uustalu]], Section 3 of: *What Makes a Strong Monad?*, EPTCS **360** (2022) 113-133 [[arXiv:2207.00851](https://arxiv.org/abs/2207.00851), [doi:10.4204/EPTCS.360.6](https://doi.org/10.4204/EPTCS.360.6)]

The tentative 'more conceptual' definition of tensorial strength, as well as the 'description' above, arose in this discussion:

* {#Baez09} [[John Baez]], _The Monads Hurt My Head -- But Not Anymore_ ([blog](http://golem.ph.utexas.edu/category/2009/07/the_monads_hurt_my_head_but_no.html#c025476))

[[!redirects tensorial strengths]]
[[!redirects strong functor]]
[[!redirects strong functors]]
[[!redirects strength]]


tensr product of chain complexes

+-- {: .rightHandSide}
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#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Monoidal categories
+--{: .hide}
[[!include monoidal categories - contents]]
=--
=--
=--

## Idea

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

#


Terence Tao

[[!redirects Terrence Tao]]

**Terence Tao** is an Australian-American mathematician, Fields medalist, and a professor at UCLA. Some of his major interests are in analysis (especially PDE-s, [[functional analysis]] and [[harmonic analysis]]), dynamical systems, [[number theory]] (especially the study of *integer sequences*), probability theory, and ergodic theory.

* [wikipedia](http://en.wikipedia.org/wiki/Terence_Tao)
* Tao's [homepage](http://www.math.ucla.edu/~tao) at UCLA
* Terence Tao's blog: [What's new](http://terrytao.wordpress.com/)

## Selected writings

* [[Vitaly Bergelson]], [[Terence Tao]], *Multiple recurrence in quasirandom groups*. ([arXiv:1211.6372](https://arxiv.org/abs/1211.6372))

On [[measure theory]] via (secretly) [[Boolean toposes]]:

* [[Asgar Jamneshan]], [[Terence Tao]], _Foundational aspects of uncountable measure theory: Gelfand duality, Riesz representation, canonical models, and canonical disintegration_ ([arXiv:2010.00681](arXiv:2010.00681))

category: people

[[!redirects Terry Tao]]


Terence Wall

Charles Terence Clegg Wall is a British mathematician, who was particularly involved with the development of [[surgery theory]].  He was professor of mathematics at Liverpool University.

* [Wikipedia entry](http://en.wikipedia.org/wiki/C._T._C._Wall)

## Selected writings

On [[surgery]] and introducing [[L-theory]] ([[Hermitian K-theory]]):

* {#Wall69} [[C. T. C. Wall]] (ed. [[Andrew Ranicki]]), *Surgery on compact Manifolds*, Math. Surveys and Monographs **69** (1999) [[pdf](https://www.maths.ed.ac.uk/~v1ranick/books/scm.pdf)]

On [[free group actions on n-spheres]]:

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

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

* {#Wall13} [[C. T. C. Wall]], *On the Structure of Finite Groups with Periodic Cohomology*, 381-413 in: *Lie Groups: Structure, Actions, and Representations*, Progress in Mathematics **306** Birkhäuser 2013 ([doi:10.1007/978-1-4614-7193-6_16](https://doi.org/10.1007/978-1-4614-7193-6_16))

## Related entries

* [[Brauer group]], [[superalgebra]]

* [[Wall's finiteness obstruction]]

* [[Hermitian form]]

category: people

[[!redirects Charles Terence Clegg Wall]]
[[!redirects Charles Wall]]
[[!redirects Terry Wall]]

[[!redirects C. T. C. Wall]]
[[!redirects C.T.C. Wall]]


Teresa Rivas

Maria Teresa Rivas Rodriguez  is a Spanish topologist, who has worked on [[proper homotopy theory]], [[shape theory
]] and on adapting ideas from those areas to give new insights into [[dynamical systems]].

She is based at the University of La Rioja, Spain.

* [University home page](https://aps.unirioja.es/ldapweb/buscar?dn=uid%3Dmarivas%2Cou%3DPersonas%2C+o%3Dunirioja.es%2Cdc%3Dunirioja%2Cdc%3Des&accion=verEntrada)

category: people

[[!redirects Maria Teresa Rivas Rodriguez]]


term

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

#Contents#
* table of contents
{:toc}

## Idea

In [[formal logic]] such as [[type theory]] a _term_ $z$ is an entity/expression of the formal language which is of some [[type]] $Z$. One writes $z : Z$ to express this (a _typing [[judgement]]_). The [[semantics]] of terms in [[Set]] is:  _[[elements]]_ of a [[set]], where one writes $z \in Z$. One also says $z$ is an _inhabitant_ of the type $Z$ and that $Z$ is an _[[inhabited type]]_ if it has a term.

A term $z : Z$ may _depend_ on [[free variables]] $x$ that are themselves terms $x : X$ of some other [[type]] $X$. For instance $z \coloneqq x + 3$ may be a term of type $Z \coloneqq \mathbb{Z}$ (the [[integers]]) which depends on a [[variable]] term $x$ also of type $X \coloneqq \mathbb{Z}$ the integers. The notation for this in the [[metalanguage]] is

$$
  x : X \vdash z : Z
  \,.
$$

Generally here also the type $Z$ itself may depend on the variable $x$ (hence the term $z$ may be of different type depending on its free variables) in which case one says that $z$ is a term of $X$-[[dependent type]].

## Definition

In the [[metalanguage]] of [[type theory]] called [[natural deduction]], terms are what the [[term introduction rules]] produce.

## Categorical semantics

Here are comments on the interpretation of types in the  [[categorical semantics]] of [[dependent type theory]].

In the [[internal language]] of any [[category]] $C$, a [[morphism]]

$$
  f : B \to A
$$

is a _term_ $f(x)$ of _[[type]]_ $A$ where $x$ is a _free [[variable]]_ of _[[type]]_ $B$, which in [[type theory|type-theoretic]] symbols is given by

$$
  x\colon B \vdash f(x)\colon A
  \,.
$$

## Related concepts

* [[element]], [[generalized element]]

* [[dependent term]]

* [[term introduction rule]], [[term elimination rule]]

* [[telescope]]

[[!redirects terms]]
[[!redirects inhabitant]]
[[!redirects inhabitants]]


term elimination

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

\tableofcontents

## Idea

In [[type theory]], **term elimination** are the [[natural deduction]] rules for how to use [[terms]] of a given [[type]].

For example, the term elimination rules for the [[sum type]] are as follows:

$$\frac{\Gamma, z:A + B \vdash C \; \mathrm{type} \quad \Gamma, x:A \vdash c:C[\mathrm{inl}(x)/z] \quad \Gamma, y:B \vdash d:C[\mathrm{inr}(y)/z] \quad \Gamma \vdash e:A + B}{\Gamma \vdash \mathrm{ind}_{A + B}(z.C, x.c, y.d, e):C[e/z]}$$

## Contextual term elimination

Similar to the [[conversion rules]] for types, there are also **contextual term elimination** rules for types. These differ from the usual term elimination rules in that in that there is an additional context member $\Delta$ attached to the end of the context $\Gamma, x:A$ so that the full context becomes $\Gamma, x:A, \Delta$. By definition, $\Delta$ is dependent upon $x:A$, and the conclusion usually involves substituting $x:A$ by some given term $a:A$ in the context, becoming $\Gamma, \Delta[a/x]$. For example, the contextual term elimination rules for the [[sum type]] are given by:

$$\frac{\Gamma, z:A + B, \Delta \vdash C \; \mathrm{type} \quad \Gamma, x:A, \Delta[\mathrm{inl}(x)/z] \vdash c:C[\mathrm{inl}(x)/z] \quad \Gamma, y:B, \Delta[\mathrm{inr}(y)/z] \vdash d:C[\mathrm{inr}(y)/z] \quad \Gamma \vdash e:A + B}{\Gamma, \Delta[e/z] \vdash \mathrm{ind}_{A + B}(z.C, x.c, y.d, e):C[e/z]}$$

And in [[first order logic]] over [[type theory]], the contextual term elimination rules for the sum types are given by:

$$\frac{\Gamma, z:A + B, \Delta \vert \Phi \vdash C \; \mathrm{type} \quad \Gamma, x:A, \Delta[\mathrm{inl}(x)/z] \vert \Phi[\mathrm{inl}(x)/z] \vdash c:C[\mathrm{inl}(x)/z] \quad \Gamma, y:B, \Delta[\mathrm{inr}(y)/z] \vert \Phi[\mathrm{inr}(y)/z] \vdash d:C[\mathrm{inr}(y)/z] \quad \Gamma \vdash e:A + B}{\Gamma, \Delta[e/z] \vert \Phi[e/z] \vdash \mathrm{ind}_{A + B}(z.C, x.c, y.d, e):C[e/z]}$$

## Related concepts

* [[formation rule]], [[introduction rule]], [[conversion rule]]

* [[beta-conversion]], [[eta-conversion]]

* [[definition rule]]

* [[polarity in type theory]]

* [[equality]]

## References

Contextual [[dependent product types]] and contextual [[identity types]] are defined in the appendix of:

* [[Benno van den Berg]], [[Martijn den Besten]], *Quadratic type checking for objective type theory* ([arXiv:2102.00905](https://arxiv.org/abs/2102.00905))

where the term elimination rules are contextual term elimination rules.

[[!redirects term elimination]]
[[!redirects term elimination rule]]

[[!redirects term eliminator]]
[[!redirects term eliminators]]

[[!redirects term elimination rules]]
[[!redirects elimination rule]]
[[!redirects elimination rules]]

[[!redirects contextual term elimination]]
[[!redirects contextual term elimination rule]]
[[!redirects contextual term elimination rules]]
[[!redirects contextual elimination rule]]
[[!redirects contextual elimination rules]]


term in context

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### Context
#### Type theory
+-- {: .hide}
[[!include type theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

In [[type theory]], a _term in context_ is a [[term]] such as

$$
  a \colon A \; \vdash \; b(a) \colon B(a)
$$

which may involve [[free variables]] from some [[context]] (here, $a:A$).  In [[dependent type theory]], the [[type]] of a term in context may also depend on the same context, an $A$-[[dependent type]], such as

$$
  a \colon A \;\vdash \; B(a) \colon Type
  \,.
$$

## Related concepts

* [[hypothetical judgment]], [[generic judgment]]

* [[telescope]]

[[!redirects terms in context]]


term model

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

#Contents#
* table of contents
{:toc}

## Idea

Given a ([[dependent type theory|dependent]]) [[type theory]], then its _term model_ is a [[model]] which consists of exactly those [[types]] and [[terms]] which may be constructed (via the relevant [[natural deduction]] operations) in the type theory.

The term model is supposed to be the [[initial object]] in the [[category]] of all [[models]] of the given type theory; see [[initiality conjecture]].

e.g. [Hofmann, example 1](#Hofmann)

Under the [[relation between type theory and category theory]] the term model is a [[category]], essentially the _[[syntactic category]]_ of the type theory.

## References

* {#Hofmann} [[Martin Hofmann]], *On the interpretation of type theory in locally cartesian closed categories*, in *Computer Science Logic. CSL 1994*, Lecture Notes in Computer Science **933** (1994) 427–441 [[doi:10.1007/BFb0022273](https://doi.org/10.1007/BFb0022273)]

* {#BuisseDybjer07} Alexandre Buisse, [[Peter Dybjer]], _Towards Formalizing Categorical Models of Type Theory in Type Theory_, 2007 ([pdf](http://www.cse.chalmers.se/~peterd/papers/Bremen2007.pdf))

* {#Aczel12} [[Peter Aczel]], _What is a type theory and what should a type theory be?_, 2012  ([pdf](http://4wft.fmf.uni-lj.si/wp-content/uploads/2012/04/Aczel.pdf), [pdf](http://4wft.fmf.uni-lj.si/wp-content/uploads/2012/08/slides_pap.pdf))

[[!redirects term models]]


terminal > history

< [[terminal]]
[[!redirects terminal -- history]]


terminal category

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

#Contents#
* table of contents
{:toc}

## Definition

The __terminal category__ or __trivial category__ or __final category__ is [[generalized the|the]] [[terminal object]] in [[Cat]].  It is the unique (up to isomorphism) [[category]] with a single [[object]] and a single [[morphism]], necessarily the [[identity morphism]] on that object. It is often denoted $1$ or $\mathbf{1}$ or $\ast$.

In [[enriched category theory]], often instead of the terminal category one is interested in the [[unit enriched category]].

## Properties

A [[functor]] from the terminal category $1$ to any [[category]] $C$ is [[equivalence|equivalently]] an [[object]] of $C$.  More generally, the [[functor category]] $[1, C] \simeq C$ from the terminal category to $C$ is canonically [[equivalence of categories|equivalent]] (in fact, isomorphic) to the category $C$ itself.

The terminal category is a [[discrete category]] that, as a [[set]], may be called the _singleton_.  As a [[subset]] of the singleton, it is in fact a [[truth value]], true ($\top$).  In general, all of these (and their analogues in [[higher category theory]] and [[homotopy theory]]) may be called the [[point]].

So far we have interpreted "terminal" as referring to the 1-category $Cat$.  If instead we interpret "terminal" in the [[2-limit|2-categorical]] sense, then any category [[equivalence of categories|equivalent]] to the one-object-one-morphism category described above is also terminal.  A category is terminal in this sense precisely when it is [[inhabited set|inhabited]] and [[indiscrete category|indiscrete]].  For such a category $1$, the functor category $[1,C]$ is equivalent, but not isomorphic, to $C$.

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

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

1. The following are equivalent:

   1. $\mathcal{C}$ has a [[terminal object]];

   1. the unique [[functor]] $\mathcal{C} \to \ast$ to the [[terminal category]] has a [[right adjoint]]

      $$
        \ast
          \underoverset
             {\underset{}{\longrightarrow}}
             {\overset{}{\longleftarrow}}
             {\bot}
        \mathcal{C}
      $$

    Under this equivalence, the [[terminal object]] is identified with the image under the right adjoint of the unique object of the [[terminal category]].

1. Dually, the following are equivalent:

   1. $\mathcal{C}$ has an [[initial object]];

   1. the unique [[functor]] $\mathcal{C} \to \ast$ to the [[terminal category]] has a [[left adjoint]]

      $$
        \mathcal{C}
          \underoverset
             {\underset{}{\longrightarrow}}
             {\overset{}{\longleftarrow}}
             {\bot}
        \ast
      $$

    Under this equivalence, the [[initial object]] is identified with the image under the left adjoint of the unique object of the [[terminal category]].

=--

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

Since the unique [[hom-set]] in the [[terminal category]] is [[generalized the|the]] [[singleton]], the hom-isomorphism characterizing the [[adjoint functors]] is directly the [[universal property]] of an [[initial object]] in $\mathcal{C}$

$$
  Hom_{\mathcal{C}}( L(\ast) , X )
  \;\simeq\;
  Hom_{\ast}( \ast, R(X) )
  =
  \ast
$$

or of a [[terminal object]]

$$
  Hom_{\mathcal{C}}( X , R(\ast) )
  \;\simeq\;
  Hom_{\ast}( L(X), \ast ) = \ast
  \,,
$$

respectively.

=--

## Related concepts

* [[initial category]]

* [[point]]

[[!redirects terminal categories]]

[[!redirects trivial category]]
[[!redirects trivial categories]]

[[!redirects final category]]
[[!redirects final categories]]

[[!redirects terminal groupoid]]
[[!redirects terminal groupoids]]


terminal coalgebra for an endofunctor

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###Context###
#### Algebra
+--{: .hide}
[[!include higher algebra - contents]]
=--
#### Induction
+-- {: .hide}
[[!include induction - contents]]
=--
=--
=--

# Terminal coalgebras
* table of contents
{: toc}

## Introduction

A __terminal coalgebra__, also called __final coalgebra__, for an [[endofunctor]] $F$ on a category $C$ is a [[terminal object]] in the category of [[coalgebra for an endofunctor|coalgebras]] of $F$.

If $F$ has a terminal coalgebra $\alpha\colon X \to F(X)$, then $X$ is isomorphic to $F(X)$ (see below); in this sense, $X$ is a fixed point of $F$.  Being terminal, $X$ is the largest fixed point of $F$ in that there is a map to $X$ from any other fixed point (indeed, any other coalgebra), and this map is an [[injection]] if $C$ is [[Set]].

The dual concept is [[initial algebra]]. Just as initial algebras allow for [[induction]] and [[recursion]], so terminal coalgebras allow for [[coinduction]] and [[corecursion]].

## Details

Given two coalgebras $(x, \eta: x \to F x)$, $(y, \theta: y \to F y)$, a coalgebra map is a morphism $f: x \to y$ which respects the coalgebra structures:

$$\theta \circ f = F(f) \circ \eta$$

A **terminal coalgebra** (usually called a _final coalgebra_ in the literature) is of course a terminal object in the category of coalgebras. Many data structures can be expressed as terminal coalgebras of suitable endofunctors; a simple but useful theorem says that terminal coalgebras $x$ are "fixed points" of their endofunctors, in that $F x \cong x$. This is the dual form of a theorem discovered long ago by Lambek:

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

If $(x, \theta: x \to F x)$ is a terminal object in the category of $F$-coalgebras, then $\theta$ is an isomorphism.
=--

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

Define a coalgebra structure on $F x$ by $F\theta: F x \to F F x$. By terminality of $x$, there is a unique coalgebra map $f: F x \to x$. We claim this is inverse to $\theta$. Indeed, by how we defined the coalgebra structure on $F x$, it is tautological that $\theta$ is a coalgebra map. By terminality of $x$ again, this gives an equation of coalgebra maps:

$$f \circ \theta = 1_x.$$

On the other hand,

$$\theta \circ f = F(f) \circ F(\theta) = F(f \circ \theta) = F(1_x) = 1_{F x}$$

where the first equation holds because $f$ is a coalgebra map. This completes the proof.
=--

To construct terminal coalgebras, the following result is useful and practical. See [[Adámek's theorem on terminal coalgebras]] for an extension of this result.

+-- {: .num_theorem #Adam}
###### Theorem
**(Adámek)**
If $C$ has a terminal object $1$ and the limit $L$ of the diagram

$$\ldots F^3 1 \stackrel{F^2 !}{\to} F^2 1 \stackrel{F !}{\to} F 1 \stackrel{!}{\to} 1 \qquad (1)$$

exists in $C$ and $F$ preserves this limit, then the limit
carries a structure of terminal coalgebra.
=--

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

Let $\pi_n: L \to F^n 1$ be the $n^{th}$ projection of the limiting cone. Then we have a cone from $F L$ to the diagram (1) whose components are

$$F L \stackrel{F\pi_n}{\to} F^{n+1} 1 \stackrel{F^n !}{\to} F^n 1$$

and the induced map $F L \to L$ to the limit is invertible by hypothesis; let $\theta: L \to F L$ be the inverse. We claim the coalgebra $(L, \theta)$ is terminal.

Indeed, suppose $(x, \eta: x \to F x)$ is any coalgebra. We recursively define maps $f_n: x \to F^n 1$: let $f_0: x \to 1$ be the unique map, and

$$f_{n+1} = F(f_n) \circ \eta$$

It is easily checked by induction that we have a commutative diagram

$$\array{
F^{n+1}1 & \stackrel{F^n !}{\to} & F^n 1 \\
 ^\mathllap{F f_n} \uparrow & \nwarrow  ^\mathrlap{f_{n+1}} & \uparrow ^\mathrlap{f_n} \\
F x & \underset{\eta}{\leftarrow} & x
}$$

defining a cone from $x$ to the diagram (1), and inducing a map $f: x \to L$ such that the following diagram commutes:

$$\array{
F L & \stackrel{\theta^{-1}}{\to} & L \\
 ^\mathllap{F f} \uparrow & & \uparrow  ^\mathrlap{f} \\
F x & \underset{\eta}{\leftarrow} & x
}$$

This diagram gives the fact that $f$ is a coalgebra map. Moreover, any coalgebra map $f: x \to L$ leads to a sequence $f_n = \pi_n \circ f$ that satisfies $f_{n+1} = F(f_n) \circ \eta$, by gluing the second diagram to the commutative diagram

$$\array{
F^{n+1}1 & \stackrel{F^n!}{\to} & F^n 1 \\
 ^\mathllap{F \pi_n} \uparrow & \nwarrow  ^\mathrlap{\pi_{n+1}} & \uparrow  ^\mathrlap{\pi_n} \\
F L & \underset{\theta^{-1}}{\to} & L
}$$

so that we were forced to define the $f_n$ by recursion as we did, and the coalgebra map $f: x \to L$ is therefore uniquely determined.
=--

## Examples

### Unit interval in the real line
 {#UnitInterval}

As first observed by [[Peter Freyd]], the [[unit interval]] $[0, 1] \hookrightarrow \mathbb{R}$ inside the [[real line]] can be characterized as a suitable terminal coalgebra. There are various ways of realizing this; we give one (but see remarks below).

Consider the [[category]] of [[intervals]] $Int$, i.e., linearly ordered sets with separate [[top]] and [[bottom]] elements $1$ and $0$, and let

$$F: Int \to Int$$

be the endofunctor which takes an interval $X$ to $X \vee X$, the linear order obtained by taking two copies of $X$ and gluing the top element of the first copy to the bottom element of the second. The real interval $[0, 1]$ becomes a coalgebra if we identify $[0, 1] \vee [0, 1]$ with $[0, 2]$ and consider the multiplication-by-2 map $[0, 1] \to [0, 2]$ as giving a coalgebra structure.

+-- {: .num_theorem}
######Theorem
The interval $[0, 1]$ is terminal in the category of coalgebras.
=--

+-- {: .proof}
######Proof (sketch)
Given any coalgebra structure $f: X \to X \vee X$, any value $f(x)$ lands either in the "lower" half (the first $X$ in $X \vee X$), the "upper" half (the second $X$ in $X \vee X$), or at the precise spot between them, where the top element in the first copy is glued to the bottom element of the second. Intuitively, one could think of a coalgebra structure $\theta: X \to X \vee X$ as giving an automaton where on input $x_0$ there is output of the form $(x_1, h_1)$, where $h_1$ is either "upper", "lower", or "between". By iteration, this generates a behavior stream $(x_n, h_n)$. Interpreting upper as 1 and lower as 0, the $h_n$ form a binary expansion to give a number between 0 and 1, and therefore we have an interval map $X \to [0, 1]$ which sends $x_0$ to that number. Of course, should we ever hit $(x_n, between)$, we have a choice to resolve it as either $(bottom_X, upper)$ or $(top_X, lower)$ and continue the stream, but these streams are identified, and this corresponds to the identification of binary expansions

$$.h_1... h_{n-1} 100000... = .h_1... h_{n-1}011111...$$

as real numbers. In this way, we produce a unique well-defined interval map $X \to [0, 1]$, so that $[0, 1]$ is the terminal coalgebra.
=--

+-- {: .num_remark}
###### Remarks
(More material can be found at [[coalgebra of the real interval]].)

1. The same proof shows that we could have considered instead the category of [[poset|posets]] with separate top and bottom, or even the category of sets with separate top and bottom, with an analogous endofunctor. The reason we chose the category of intervals is (besides the availability of the succinct term 'interval') to indicate that choice of interval $[0, 1]$,
as the model which classifies the [[geometric realization]] functor, can be justified on the grounds of a universal property, as shown by this theorem.

1. Freyd, in his original [post](#Freyd) on this result, was inspired by a similar theorem due to [Pavlovic and Pratt](#PP), that the half-open interval $[0, \infty)$ can be described as the terminal coalgebra for the endofunctor that sends a linearly ordered set $X$ to $\omega \times X$ with the dictionary order.

1. The theorem holds in an arbitrary topos (with $[0, 1]$ being the interval of [[real number object|Dedekind reals]]), provided that the word "separate" is interpreted correctly:
$$\forall p\colon P\; (\neg(0 = p) \vee \neg(1 = p))$$
and provided that the process of gluing endpoints is given correctly. See Johnstone's [[Elephant]], section D.4.7, for an extended discussion.

=--

### Categorified example: Trees

The notion of terminal coalgebra may be categorified. For example, given a 2-category $C$ and a (pseudo) functor $F: C \to C$, one may speak of a 2-terminal (pseudo) coalgebra.

A theoretically important example is the [[tree|category of trees]], seen as a 2-terminal coalgebra for the endofunctor on $Cat$ which takes a locally small category $C$ to its small-coproduct cocompletion. Further discussion of this point is given at [[pure set]].

The small-coproduct cocompletion of $C$ is given by a comma category construction: objects are pairs $(X, F: X_d \to C)$ where $X$ is a set and $F$ is a functor whose domain is the discrete category on $X$, denoted $X_d$. A morphism from $(X, F)$ to $(Y, G)$ is a pair $(f, \Phi)$ where $f: X \to Y$ is a function and $\Phi: F \to G \circ f_d$ is a natural transformation. This category is denoted $Set \wr C$; it is called a "[[categorical wreath product]]" (see also the discussion at [[club]]).

Adámek's theorem may be adapted to this 2-categorical situation. The iterated [[categorical wreath product|wreath product]] $(Set \wr)^n 1$ may be identified with the category of $n$-stage trees:

$$(Set \wr)^n 1 = Set^{[n]^{op}}$$

where $[n]$ is the linear order $1 \leq 2 \leq \ldots \leq n$. Or, what is the same, the category of presheaves $T: [n+1]^{op} \to Set$ with the condition that $T(0) = 1$ is terminal; the element of $T(0)$ is considered to be the root of the tree.

Indeed, we realize an explicit equivalence

$$\Sigma: Set \wr Set^{[n]^{op}} \simeq Set^{[n+1]^{op}}$$

by defining $\Sigma(X, F: X \to Set^{[n]^{op}})$ to be the functor $T: [n+1]^{op} \to Set$ that on the object level takes $1$ to $X$, and $i+1$ to

$$\sum_{x \in X} F(x)(i).$$

On the morphism level, $T(i+1 \to i)$ is the coproduct of morphisms

$$\sum_{x \in X} F(x)(i \to i-1)$$

(and this makes sense for all $1 \leq i \leq n$ under the convention $F(x)(0) = 1$).

The morphism

$$(Set \wr)^n !: (Set \wr)^n Set \to (Set \wr)^n 1$$

used in Adámek's theorem is identified with the restriction functor

$$Set^{[n+1]^{op}} \to Set^{[n]^{op}}$$

which restricts presheaves along the inclusion $[n] \hookrightarrow [n+1]$.

The 2-limit of the diagram in Adámek's theorem is then

$$Set^{\omega^{op}},$$

aka the [[tree|category of trees]], where $\omega$ is the colimit of the finite ordinals $[n]$. The statement that the category of trees is equivalent to its small-coproduct cocompletion says that the category of trees is equivalent to the category of forests.

### Fractals

There is a [[category theory|category theoretic]] treatments of the self-similarity found in [[fractals]] in terms of terminal coalgebras, see [Leinster 10](#Leinster10), [Bhattacharya-Moss-Ratnayake-Rose](#BhattacharyaMossRatnayakeRose).

### Further examples

[[!include list of notable initial algebras and terminal coalgebras]]

## Related concepts

Terminal coalgebras are the [[categorical semantics]] of [[coinductive types]], for instance [[M-types]].

* [[list of notable initial algebras and terminal coalgebras]]

## References

* {#Freyd} [[Peter Freyd]], _Real coalgebra_ Mailing to the categories list, Dec. 22, 1f999. ([link](http://www.mta.ca/~cat-dist/catlist/1999/realcoalg))

* {#PP} [[Dusko Pavlovic]], [[Vaughan Pratt]], _On coalgebra of real numbers_, 1999. ([web](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.47.5204))

Cross-relations between algebraic and coalgebraic aspects of real numbers may be found in this article:

* [[Peter Freyd]], _Algebraic Real Analysis, Theor._ Appl. Cat., vol. 20, no. 10 (2008), 215-308. ([link](http://www.tac.mta.ca/tac/volumes/20/10/20-10abs.html))

For [[category theory|category theoretic]] treatments of the self-similarity found in [[fractals]] in terms of terminal coalgebras, see

* {#Leinster10} [[Tom Leinster]], _A general theory of self-similarity_, ([arXiv:1010.4474](https://arxiv.org/abs/1010.4474))

* {#BhattacharyaMossRatnayakeRose} Prasit Bhattacharya, Lawrence S. Moss, Jayampathy Ratnayake, and Robert Rose, _Fractal Sets as Final Coalgebras Obtained by Completing an Initial Algebra_, ([pdf](http://www3.nd.edu/~pbhattac/papers/Sierpinski.pdf))

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[[!redirects terminal coalgebras for an endofunctor]]

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[[!redirects terminal coalgebras of an endofunctor]]

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[[!redirects terminal coalgebras of the endofunctor]]