triangle identities

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

# Contents
* table of contents
{: toc}

## Idea
 {#Idea}

The __triangle identities__ or __zigzag identities__ are identities characterized by the [[unit of an adjunction|unit and counit]] of an [[adjunction]], such as a [[adjoint pair|pair]] of [[adjoint functors]]. These identities _define_, equivalently, the nature of adjunction ([this prop. ](adjoint+functor#GeneralAdjunctsInTermsOfAdjunctionUnitCounit)).

## Statement
 {#Statement}

Consider:

1. $\mathcal{C}, \mathcal{D}$ a [[pair]] of [[categories]], or, generally, of [[objects]] in a given [[2-category]];

1. $L \colon \mathcal{C} \to \mathcal{D}$ and $R \colon \mathcal{D} \to \mathcal{C}$ a pair of [[functors]] between these, or generally [[1-morphisms]] in the ambient [[2-category]];

1. $\eta \colon id_{\mathcal{C}} \Rightarrow R \circ L$ and $\epsilon \colon L \circ R \Rightarrow id_{\mathcal{D}}$ two [[natural transformations]] or, generally [[2-morphisms]].

This data is called a _[[adjoint pair|pair]] of [[adjoint functors]]_ (generally: an _[[adjunction]]_) if the _triangle identities_ are satisfied, which may be expressed in any of the following equivalent ways:

1. _[As equations](#AsEquations)_

1. _[As diagrams](#AsDiagrams)_

1. _[As string diagrams](#AsStringDiagrams)_

$\,$

### As equations
  {#AsEquations}

As [[equations]], the triangle identities read

$$
  \big(
    \epsilon L
  \big)
  \circ
  \big(
    L \eta
  \big)
  \;=\;
  id_L
$$

$$
  \big(
    R \epsilon
  \big)
  \circ
  \big(
    \eta R
  \big)
  \;=\;
  id_R
$$

Here juxtaposition denotes the [[whiskering]] operation of [[1-morphisms]] on [[2-morphisms]], as made more manifest in the diagrammatic unravelling of these expressions:

### As diagrams
 {#AsDiagrams}

In terms of [[diagrams]] in the [[functor categories]] this means

$$
  L
   \overset{\;\;L\eta\;\;}{\Rightarrow}
  L R L
   \overset{\;\;\epsilon L\;\;}{\Rightarrow}
  L
  \;\;
  =
  \;\;
  L \overset{\;\;id_L\;\;}{\Rightarrow} L
$$

and

$$
  R
    \overset{\;\;\eta R\;\;}{\Rightarrow}
  R L R
    \overset{\;\;R\epsilon\;\;}{\Rightarrow}
  R
  \;\;
  =
  \;\;
  R \overset{\;\;id_R\;\;}{\Rightarrow} R
$$

In terms of diagrams of [[2-morphisms]] in the ambient [[2-category]], this looks as follows:

\begin{xymatrix}
  \mathcal{C}
  \ar[r]|-{ \;L\; }
  \ar@/^3pc/[rr]|-{ \;\mathrm{id}_{\mathcal{C}}\; }_-{\ }="s1"
  &
  \mathcal{D}
  \ar[r]|-{ \;R\; }
  \ar@/_3pc/[rr]|-{ \;\mathrm{id}_{\mathcal{D}}\; }^-{\ }="t2"
  &
  \mathcal{C}
  \ar[r]|-{ \;L\; }
  &
  \mathcal{D}
  &
    =
  &
  \mathcal{C}
  \ar[r]|-{ \;L\; }
  &
  \mathcal{D}
  %
  \ar@{=>}^\eta "s1"+(0,-2); "s1"+(0,-8)
  \ar@{=>}^\epsilon "t2"+(0,8); "t2"+(0,2)
\end{xymatrix}

\begin{xymatrix}
  \mathcal{D}
  \ar[r]|-{ \;R\; }
  \ar@/_3pc/[rr]|-{ \;\mathrm{id}_{\mathcal{D}}\; }^-{\ }="s1"
  &
  \mathcal{C}
  \ar[r]|-{ \;L\; }
  \ar@/^3pc/[rr]|-{ \;\mathrm{id}_{\mathcal{C}}\; }_-{\ }="t2"
  &
  \mathcal{D}
  \ar[r]|-{ \;R\; }
  &
  \mathcal{C}
  &
    =
  &
  \mathcal{D}
  \ar[r]|-{ \;R\; }
  &
  \mathcal{C}
  %
  \ar@{=>}^\eta "t2"+(0,-2); "t2"+(0,-8)
  \ar@{=>}^\epsilon "s1"+(0,8); "s1"+(0,2)
\end{xymatrix}

where on the right the [[identity morphism|identity]] [[2-morphisms]] are left notationally implicit.

If we leave the [[identity morphism|identity]] [[1-morphisms]] on the left notationally implicit, then we get the following suggestive form of the triangle identities:

(taken from _[[geometry of physics -- categories and toposes]]_).

### As string diagrams
 {#AsStringDiagrams}

As [[string diagrams]], the triangle identities appear as the action of "pulling zigzags straight" (hence the name):

+-- {: style='text-align:center'}
[[adjunction-up-string.png:pic]][[adjunction-down-string.png:pic]]
=--

With labels left implicit, this notation becomes very economical:

+-- {: style='text-align:center'}
[[adjunction-up-string-minimal.png:pic]] [[adjunction-down-string-minimal.png:pic]]
=--

## Related concepts

* [[adjunction]]

* [[unit of an adjunction]]

* [[adjunct]]

## References

Textbook accounts include

* {#Borceux94} [[Francis Borceux]], Theorem 3.1.5 and Diagram 3.3 in: _Basic Category Theory_, Vol. 1 of _[[Handbook of Categorical Algebra]]_, Cambridge University Press (1994)

See the references at _[[category theory]]_ for more.

[[!redirects zigzag identities]]
[[!redirects zigzag identity]]
[[!redirects zig-zag identities]]
[[!redirects zig-zag identity]]
[[!redirects zig-zag law]]
[[!redirects triangle identity]]

[[!redirects zigzag-identity]]
[[!redirects zigzag-identities]]
[[!redirects zigzag-identity]]
[[!redirects zig-zag-identities]]
[[!redirects zig-zag-identity]]


triangle inequality

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

#Contents#
* table of contents
{:toc}

## Idea

The key conditions on a [[distance]] function for _[[metric spaces]]_ and on _[[norms]]_ on [[normed vector spaces]] is that the distance $d(x,z)$ between any two points $x,z$ is no larger than the sum of the distances $d(x,y) + d(y,z)$ via any third point $y$

$$
  d(x,z) \leq d(x,y) + d(y,z)
  \,.
$$

For the usual metric/norm on [[Euclidean space]] and $f$ and $g$ two edges of a [[triangle]]

$$
  \array{
    && y
    \\
    & {}^{\mathllap{f}}\nearrow && \searrow^{\mathrlap{g}}
    \\
    x && \underset{f + g}{\longrightarrow} && z
  }
$$

then this [[inequality]] expresses that the straight path from $x$ to $z$ is always shorter or at worst as long as the path from $x$ to $z$ via $y$.

However, [[normed fields]] and hence [[normed vector spaces]] for which this example gives the correct intuition are rare among all normed fields and vector spaces (they are the _[[archimedean]]_ ones).  Most norms instead satisfy the stronger [[ultrametric]] triangle inequality which says that

$$
  {\vert f + g \vert} \leq max({\vert f\vert}, {\vert g\vert })
  \,.
$$

A norm with this property is called _[[non-archimedean]].

## Interpretation in enriched category theory
 {#InterpretationInEnrichedCategoryTheory}

One may equivalently regard the triangle equality in [[metric spaces]] as the [[composition]] operation in a certain incarnation of the metric space as an [[enriched category]] ([Lawvere 73](#Lawvere73)). From this perspective some concepts from [[analysis]] usefully generalize to other [[enriched categories]], notably the concept of _[[Cauchy complete categories]]_. For more on this see at _[Lawvere metric space](metric+space#LawvereMetricSpace)_.

Namely regard the [[half-open interval]] $[0,\infty) \subset \mathbb{R}$ as a [[poset]] under the [[relation]] $\geq$, and regard this poset as a [[category]] (see at [[(0,1)-category]]). The operation of addition of real numbers makes this a [[monoidal category]].

This means that a category [[enriched category|enriched over]] this monoidal category $([0,\infty){\geq}, +)$ is

1. ([[objects]]) a set $X$;

1. ([[hom objects]]) for every [[pair]] of points $(x,y) \in X \times X$ a real number $d(x,y) \in [0,\infty)$

1. ([[composition]]) for all $x,y,z \in X$ a morphism in $[0,\infty)_{\geq}$ of the form

   $$
     \circ_{x,y,z} \;\colon\; d(x,y) + d(y,z) \longrightarrow d(x,z)
   $$

such that

1. ([[unitality]]) $d(x,x) = 0$;

1. ([[associativity]]) ...

Now since the [[category]] $[0,\infty)_{\geq}$ is a [[poset]], there is at most one morphism between any given pair of objects, and hence the choice of [[composition]] morphism $\circ_{x,y,z}$ above is really a condition on $d(-,-)$. Moreover, since a morphism $x \to y$ in $[0,\infty)$ exists precisely if $x \geq y$, then this condition is exactly the triangle identity

$$
  d(x,y) + d(y,z) \geq d(x,z)
  \,.
$$

Moreover, the [[unitality]] condition is part of the non-degeneracy condition on a metric, $d(x, y) = 0$ iff $x = y$, and the [[associativity]] condition is automatically satisfied once [[composition]] is defined.

## References

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

* {#Lawvere73}  [[Bill Lawvere]] (1973).  _Metric spaces, generalized logic and closed categories_.  Reprinted in [[TAC]], 1986.  [Web](http://www.tac.mta.ca/tac/reprints/articles/1/tr1abs.html).

[[!redirects triangle inequalities]]


triangular Hopf algebra

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

#Contents#
* table of contents
{:toc}

## Idea

A _(quasi-)triangular Hopf algebra_ is a [[Hopf algebra]] whose underlying [[bialgebra]] is a [[quasitriangular bialgebra|(quasi-)triangular]].  This means that its [[category of modules]] is not just a [[rigid monoidal category]] but a [[braided monoidal category]]/[[symmetric monoidal category]].

If this rigid symmetric monoidal category in addition satisfies a certain regularity condition (see [here](Deligne%27s+theorem+on+tensor+categories#Regularity)) then the corresponding triangular Hopf algebra is equivalent to a  [[superalgebra|supercommutative]] Hopf algebra, hence to the [[formal dual]] of a affine algebraic [[supergroup]]. See at _[[Deligne's theorem on tensor categories]]_ for more on this.

## Definition

(...) e.g. ([Gelaki, 2.1](#Gelaki)) (...)

## Properties

### Tannaka duality

[[!include structure on algebras and their module categories - table]]

## Examples

* A [[quantum group]] (in the sense of Drinfeld-Jimbo) is a quasitriangular Hopf algebra.

* The [[Drinfeld double]] of a [[Hopf algebra]] is a [[quasi-triangulated Hopf algebra]].

## Related concepts

* [[Yang-Baxter equation]]

* [[quasitriangulated quasi-Hopf algebra]]

## References

* {#Gelaki} [[Shlomo Gelaki]], _On the classification of finite-dimensional triangular Hopf algebras_, New directions in Hopf algebras, MSRI publications volume 43 (2002) ([pdf](http://library.msri.org/books/Book43/files/gelaki.pdf))

* Wikipedia, _[Quasitriangular Hopf algebra](http://en.wikipedia.org/wiki/Quasitriangular_Hopf_algebra)_

* Nicolas Andruskiewitsch, [[Pavel Etingof]], Shlomo Gelaki, _Triangular Hopf algebras with the Chevalley property_ ([arXiv:math/0008232](http://arxiv.org/abs/math/0008232))

* [[Pavel Etingof]], [[Shlomo Gelaki]], _On families of triangular Hopf algebras_ ([arXiv:math/0110043](http://arxiv.org/abs/math/0110043))

Discussion in view of [[Deligne's theorem on tensor categories]] is in

* [[Pavel Etingof]], [[Shlomo Gelaki]], _The classification of finite-dimensional triangular Hopf algebras over an algebraically closed field of characteristic 0_ ([arXiv:0202258](http://de.arxiv.org/abs/math/0202258))

[[!redirects triangular Hopf algebras]]

[[!redirects quasitriangular Hopf algebra]]
[[!redirects quasitriangular Hopf algebras]]
[[!redirects quasi-triangular Hopf algebra]]
[[!redirects quasi-triangular Hopf algebras]]


triangular matrix

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

#Contents#
* table of contents
{:toc}

## Idea

In the context of square [[matrices]] an _upper triangular matrix_ is one all whose entries below the diagonal vanish. Analogously for a _lower triangular matrix_.

## Examples

* The [[table of marks]] of a [[finite group]] is a lower triangular matrix over the [[integers]] ([this Prop.](table+of+marks#TableOfMarksIsInvertibleUpperTriangular)).

## Related concepts

* [[Borel subgroup]]

* [[symmetric matrix]]

## References

See also

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

[[!redirects triangular matrices]]

[[!redirects upper triangular form]]
[[!redirects upper triangular forms]]

[[!redirects lower triangular form]]
[[!redirects lower triangular forms]]

[[!redirects lower triangular matrix]]
[[!redirects lower triangular matrices]]

[[!redirects upper triangular matrix]]
[[!redirects upper triangular matrices]]

[[!redirects upper-triangular form]]
[[!redirects upper-triangular forms]]

[[!redirects lower-triangular form]]
[[!redirects lower-triangular forms]]

[[!redirects lower-triangular matrix]]
[[!redirects lower-triangular matrices]]

[[!redirects upper-triangular matrix]]
[[!redirects upper-triangular matrices]]


triangulated categories of sheaves

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

#Contents#
* table of contents
{:toc}

## Idea

Let $(X, O_X)$ be a [[ringed space]].  In analogy with [[modules]] over a [[commutative ring]], one may study $(X, O_X)$ by studying the category $Mod(O_X)$ of [[sheaf of modules|sheaves of modules]] over $O_X$.  There is a [[symmetric monoidal model category|symmetric monoidal]] [[cofibrantly generated model category|cofibrantly generated]] [[model structure]] on $Mod(O_X)$.  The [[homotopy category]] with respect to this model structure, i.e. the [[derived category]] of $Mod(O_X)$, is a [[triangulated category]] $D(Mod(O_X))$ associated to $X$.  Here we discuss various interesting subcategories of $D(Mod(O_X))$.  In particular, the bounded derived category of coherent sheaves is an important invariant studied by the school of [[Bondal]], [[Orlov]] and others.  The triangulated category of perfect complexes is a very significant construction in [[noncommutative algebraic geometry]].

## Definitions

### Derived category of quasi-coherent sheaves

Recall that one has the [[full subcategory|full]] [[abelian category|abelian]] [[subcategory]] $QCoh(O_X) \subset Mod(O_X)$ of [[quasi-coherent sheaves]], which are the ones corresponding locally to [[modules]] over a [[ring]].  One gets a triangulated subcategory $D(QCoh(O_X)) \subset D(Mod(O_X))$.

### Derived category of coherent sheaves

When $X$ is a [[noetherian scheme]], one has the full abelian subcategory $Coh(O_X) \subset QCoh(O_X)$ of [[coherent sheaves]]; these are the ones corresponding locally to [[finitely generated modules]].  Hence one gets the triangulated subcategory $D(Coh(O_X)) \subset D(QCoh(O_X))$.

### Bounded derived category of coherent sheaves

Let $X$ be a [[noetherian scheme]] and let $D^b(Coh(O_X))$ denote the [[bounded derived category]] of $Coh(O_X)$.

+-- {: .num_prop}
###### Proposition
The canonical [[fully faithful]] [[functor]]
  $$ D^b(Coh(O_X)) \hookrightarrow D(Mod(O_X)) $$
identifies $D^b(Coh(O_X))$ with the [[full subcategory]] of $D(Mod(O_X))$ of [[bounded complexes]] whose [[cohomology]] objects are [[coherent sheaves]].
=--

See [(SGA 6, Exp. II, Corollaire 2.2.2.1)](#SGA6).

### Triangulated category of perfect complexes

Let $X$ be a [[ringed space]] and let $Pf(X) \subset D(Mod(O_X))$ denote the [[full subcategory]] of [[perfect complexes]] of $O_X$-[[modules]].  This is a [[triangulated subcategory]] that is contained in $D_{coh}(Mod(O_X)) \subset D(Mod(O_X))$, the full subcategory of complexes with [[coherent cohomology]].  It is stable under [[derived tensor product]], [[derived inverse image]], and [[derived direct image]] of [[proper morphisms]].

+-- {: .num_prop}
###### Proposition
If $X$ is a [[smooth scheme]], there is a canonical [[equivalence]]
  $$ Pf(X) \stackrel{\sim}{\to} D^b(Coh(O_X)) $$
of [[triangulated categories]].
=--

This follows from the previous proposition combined with [(SGA 6, Exp. I, Exemples 5.11)](#SGA6).

## Derived equivalence

The bounded derived category of coherent sheaves $D^b(Coh(O_X))$ is usually called simply the **derived category of $X$** and denoted $D(X)$.  It is an important invariant of $X$, which has been studied extensively by [[Mukai]], [[Bondal]]-[[Orlov]], and others.

There are several geometrically interesting examples of non-isomorphic [[varieties]] $X$ and $Y$ with $D(X) \simeq D(Y)$; for example, the derived category of an [[abelian variety]] $X$ is equivalent to the derived category of its dual $\hat{X}$.  In such cases one often says that $X$ and $Y$ are **derived equivalent**.  However, the derived category is not a weak invariant of $X$, indeed in some cases it is as strong as isomorphism; see [[Bondal-Orlov reconstruction theorem]].  In fact, the construction $X \mapsto D(X)$ seems to lose just enough information so that derived equivalence becomes a geometrically interesting invariant.

### Fourier-Mukai functors

+-- {: .num_thm}
###### Theorem ([[Orlov]])
Let $X$ and $Y$ be smooth projective [[varieties]] over a [[field]] $K$.  Let $F : D(X) \to D(Y)$ be a [[triangulated functor|triangulated]] [[fully faithful functor]].  There exists an object $E \in D(X \times Y)$ which is unique up to isomorphism and is such that $F$ is isomorphic to the [[Fourier-Mukai transform]] of $E$, that is the functor
  $$ F \mapsto Rq_*(Lp^*(F) \otimes^L E)) $$
where $p : X \times Y \to X$ and $q : X \times Y \to Y$ are the [[projections]].
=--

See [[Fourier-Mukai functor]] for details.

### Derived invariants

Let us say that a functor on the category of smooth projective [[varieties]] is a **derived invariant** if it maps derived equivalent varieties to isomorphic objects.

* The [[canonical ring]] is a derived invariant [(Orlov 2003)](#OrlovSurvey).

* [[Orlov]] has proved that for [[abelian varieties]], the functor $X \mapsto X \times \hat{X}$ is a derived invariant [(Orlov 2003)](#OrlovSurvey).

* [[Rouquier]] has proved that the functor $X \mapsto Pic^0(X) \rtimes Aut^0(X)$ valued in [[algebraic groups]] is a derived invariant [(Rouquier 2010)](#Rouquier2010).  (Note that this a generalization of Orlov's result on abelian varieties above.)

* [[Orlov]] has proved that the derived category $D(X)$ determines the rational [[Chow motive]] $M(X)$ up to [[Tate twists]].  Under a certain condition on the [[Fourier-Mukai functor|corresponding]] object $E \in D(X \times Y)$, this may be refined to derived invariance of the integral [[Chow motive]].  In particular the corresponding statements are true for any [[Weil cohomology theory]] (e.g. [[etale cohomology]], [[Betti cohomology]], [[de Rham cohomology]], etc.).  See [(Orlov, 2005)](#Orlov2005).

* The derived category $D(X)$ determines the [[noncommutative Chow motive]] $NM(X)$ up to isomorphism.  In particular, it follows that all [[additive invariants]], like [[K-theory]] or [[Hochschild homology]], are derived invariants.  See [(Yusufzai)](#Yusufzai).

## Related concepts

* [[abelian sheaf cohomology]]

* [[derived category of coherent sheaves]]

* [[perfect complex]]

* [[deformation theory of sheaves]]

## References

* [[SGA 6]]
{#SGA6}

For the [[model structure]] on $Mod(O_X)$, see [[model structure on chain complexes]] and in particular

* [[Denis-Charles Cisinski]], F. Déglise, _Local and stable homologial algebra in Grothendieck abelian categories_, Homology, Homotopy and Applications, vol. 11 (1) (2009)  ([url](http://www.intlpress.com/HHA/v11/n1/a11/))
{#CisinskiDeglise}

### Derived categories of quasi-coherent sheaves

* [[R. Thomason]], T. Trobaugh. _Higher algebraic K-theory of schemes and of derived categories_. The Grothendieck Festschrift, Vol. III Birkh¨auser, Boston (1990), 247-436.

* [[Amnon Neeman]], _The Grothendieck duality theorem via Bousfield's techniques and Brown representability_, J. Amer. Math. Soc., vol. 9, no. 1, 1996, pp. 205-236, [doi](http://dx.doi.org/10.1090/S0894-0347-96-00174-9).

### Bounded derived category of coherent sheaves

For a summary of the results of [[Bondal]]-[[Orlov]], see

* [[Aleksei Bondal]], [[Dmitri Orlov]], _Derived categories of coherent sheaves_, [arXiv](http://arxiv.org/abs/math/0206295).

For a detailed survey, see

* [[Dmitri Orlov]], _Derived categories of coherent sheaves and equivalences between them_, Russian Math. Surveys, 58 (2003), 3, 89-172, [translation](http://www.mi.ras.ru/~orlov/papers/Uspekhi2003.pdf).
 {#OrlovSurvey}

On the relationship between derived categories and Chow motives, see

* [[Dmitri Orlov]], _Derived categories of coherent sheaves and motives_.
[math.AG/0512620](http://arxiv.org/abs/math/0512620)

For some discussion of the above result and the relationship between derived categories and [[additive invariants]] see

* [[Adeel Khan Yusufzai]], _Perfect correspondences and Chow motives_, master's thesis, 2013.  [arXiv](http://arxiv.org/abs/1310.0249).

For the derived invariance of $Pic^0 \rtimes Aut^0$, see

* [[Rouquier]], _Automorphismes, graduations et categories triangulees_, [arXiv](http://arxiv.org/abs/1008.1971).

### Triangulated category of perfect complexes

* [[Luc Illusie]]. _Le complexe cotangent I et II_, Lecture Notes in Math. 239, 283, Springer-Verlag, Berlin (1971,1972).

* [[Andre Hirschowitz]], [[Carlos Simpson]].  _Descente pour les n-champs_, [arXiv:math/9807049](http://arxiv.org/abs/math/9807049)

* [[Stacks Project]], [tag 08CL](http://stacks.math.columbia.edu/tag/08CL)

[[!redirects triangulated category of sheaves]]
[[!redirects triangulated category of perfect complexes]]
[[!redirects bounded derived category of coherent sheaves]]


triangulated category

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Stable homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

Any [[(infinity,1)-category]] $C$ can be flattened, by ignoring higher morphisms, into a [[1-category]] $ho(C)$ called its [[homotopy category of an (∞,1)-category|homotopy category]].  The notion of a _triangulated structure_ is designed to capture the additional structure canonically existing on $ho(C)$ when $C$ has the property of being [[stable (infinity,1)-category|stable]].  This structure can be described roughly as the data of an invertible [[suspension functor]], together with a collection of sequences called _distinguished triangles_, which behave like [[decategorification|shadows]] of [[homotopy cofibre sequence|homotopy (co)fibre sequences]] in [[stable (infinity,1)-categories]], subject to various axioms.

A central class of examples of triangulated categories are the [[derived categories]] $D(\mathcal{A})$ of [[abelian categories]] $\mathcal{A}$. These are the [[homotopy category of an (∞,1)-category|homotopy categories]] of the stable [[(∞,1)-categories of chain complexes]] in $\mathcal{A}$.  However the notion also encompasses important examples coming from nonabelian contexts, like the [[stable homotopy category]], which is the [[homotopy category]] of the [[stable (infinity,1)-category|stable]] [[(infinity,1)-category of spectra]].  Generally, it seems that all triangulated categories appearing in nature arise as [[homotopy categories]] of [[stable (infinity,1)-categories]] (though examples of "exotic" triangulated categories probably exist).

By construction, passing from a [[stable (infinity,1)-category]] to its [[homotopy category]] represents a serious loss of information.  In practice, endowing the homotopy category with a triangulated structure is often sufficient for many purposes.  However, as soon as one needs to remember the [[homotopy colimits]] and [[homotopy limits]] that existed in the [[stable (infinity,1)-category]], a triangulated structure is not enough.  For example, even the [[mapping cone]] in a triangulated category is not functorial.  Hence it is often necessary to work with some [[enhanced triangulated category|enhanced]] notion of triangulated category, like [[stable derivators]], [[pretriangulated dg-categories]], [[stable model categories]] or [[stable (infinity,1)-categories]].  See [[enhanced triangulated category]] for more details.

## History

The notion of triangulated category was developed by [[Jean-Louis Verdier]] in his 1963 thesis under [[Alexandre Grothendieck]].  His motivation was to axiomatize the structure existing on the [[derived category]] of an [[abelian category]].  Axioms similar to Verdier's were given by [[Albrecht Dold]] and [[Dieter Puppe]] in a 1961 paper.  A notable difference is that Dold-Puppe did not impose the octahedral axiom (TR4).

## Definition

The traditional definition of _triangulated category_ is the following. But see remark \ref{RedundancyInDefinition} below.

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

A __triangulated category__ is

* an [[additive category]];

* [[category with translation|with (additive) translation]];

* equipped with a collection of [[category with translation|triangles]] called __[[distinguished triangles]]__ (dts)

* such that the following axioms hold

**TR0:** every triangle isomorphic to a distinguished triangle is itself a distinguished triangle;

**TR1:** the triangle

$$
  X \stackrel{Id_X}{\to} X \to 0 \to T X
$$

is a distinguished triangle;

**TR2:** for all $f : X \to Y$, there exists a distinguished triangle

$$
  X \stackrel{f}{\to} Y \to Z \to T X
  \,;
$$

**TR3:** a triangle

$$
  X \stackrel{f}{\to} Y \stackrel{g}{\to} Z \stackrel{h}{\to} T X
$$

is a distinguished triangle precisely if

$$
  Y \stackrel{g}{\to} Z \stackrel{h}{\to} T X \stackrel{-T(f)}{\to} T Y
$$

is a distinguished triangle;

**TR4:** given two distinguished triangles

$$
  X \stackrel{f}{\to} Y \stackrel{g}{\to} Z \stackrel{h}{\to} T X
$$

and

$$X' \stackrel{f'}{\to} Y' \stackrel{g'}{\to} Z' \stackrel{h'}{\to} T X'$$

and given morphisms $\alpha$ and $\beta$ in

$$\array{
  X &\stackrel{f}{\to}& Y
  \\
  \downarrow^\alpha && \downarrow^\beta
  \\
  X' &\stackrel{f'}{\to}& Y'
}$$

there exists a morphism $\gamma : Z \to Z'$
extending this to a morphism of distinguished triangles in that
the diagram

$$
  \array{
    X &\stackrel{f}{\to}& Y
    &\stackrel{g}{\to}& Z
    &\stackrel{h}{\to}& T X
    \\
    \downarrow^\alpha && \downarrow^{\mathrlap{\beta}}
    && \downarrow^{\mathrlap{\exists \gamma}}
    && \downarrow^{\mathrlap{T(\alpha)}}
    \\
    X' &\stackrel{f'}{\to}& Y'
    &\stackrel{g'}{\to}& Z'
    &\stackrel{h'}{\to}& T X'
}
$$

commutes;

**TR5 (octahedral axiom):** given three distinguished triangles of the form

$$
\begin{aligned}
& X \stackrel{f}{\to} Y
    \stackrel{h}{\to} Y/X
    \stackrel{}{\to} T X
\\
& Y \stackrel{g}{\to} Z
    \stackrel{k}{\to} Z/Y
    \stackrel{}{\to} T Y
\\
& X \stackrel{g \circ f}{\to} Z
    \stackrel{l}{\to} Z/X
    \stackrel{}{\to} T X
\end{aligned}
$$

there exists a distinguished triangle

$$
Y/X \stackrel{u}{\to} Z/X
    \stackrel{v}{\to} Z/Y
    \stackrel{w}{\to} T (Y/X)
$$

such that the following big diagram commutes

$$
  \array{
    X
    &&\stackrel{g \circ f}{\to}&&
    Z
    &&\stackrel{k}{\to}&&
    Z/Y
    &&\stackrel{w}{\to}&&
    T (Y/X)
    \\
    &
    {}_{f}\searrow
    &&
    \nearrow_{g}
    &&
    \searrow^{l}
    &&
    \nearrow_{v}
    &&
    \searrow^{}
    &&
    \nearrow_{T(h)}
    \\
    &&
    Y
    &&&&
    Z/X
    &&&&
    T Y
    \\
    &&&
    \searrow^{h}
    &&
    \nearrow_{u}
    &&
    \searrow^{}
    &&
    \nearrow_{T(f)}
    \\
    &&&&
    Y/X
    &&\stackrel{}{\to}&&
    T X
  }
$$

If TR5 is not required, one speaks of a **pretriangulated category**.

=--

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

This classical definition is redundant; TR4 and one direction of TR3 follow from the remaining axioms.  See ([May](#May)).

The octahedral axiom has many equivalent formulations, a concise account is in ([Hubery](#Hubery)). Notice that one of the equivalent axioms, called axiom B, there, is essentially just an axiomatization of the existence of [[homotopy pushouts]].

=--

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

In the context of triangulated categories the translation functor $T : C \to C$ is often called the **suspension functor** and denoted $(-)[1] : X \mapsto X[1]$ (in an algebraic context) or $S$ or $\Sigma$ (in a topological context).  However unlike "translation functor", suspension functor is also the term used when the invertibility is not assumed, cf. [[suspended category]].

=--

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

If $(f,g,h)$ is a [[distinguished triangle]], then $(f,g,-h)$ is not generally distinguished, although it is "exact" (induces [[long exact sequences]] in [[homology]] and [[cohomology]]).  However, $(f,-g,-h)$ *is* always distinguished, since it is isomorphic to $(f,g,h)$:
  $$ \array{ X & \xrightarrow{f} & Y & \xrightarrow{g} & Z & \xrightarrow{h} & T X\\
    ^{id}\downarrow && ^{id} \downarrow && ^{-1} \downarrow && \downarrow^{id}\\
    X & \xrightarrow{f} & Y & \xrightarrow{-g} & Z & \xrightarrow{-h} & T X}$$

=--

## Properties

### Long fiber-cofiber sequences
 {#LongExactSequences}

The following prop. \ref{LongExactSequencesFromDistinguishedTriangle} is the incarnation in the axiomatics of triangulated categories of the [[long exact sequences of homotopy groups]] induced by [[homotopy fiber sequences]] in [[homotopy theory]].

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

Given a [[triangulated category]], def. \ref{TriangulatedCategory}, and

$$
  A \overset{f}{\longrightarrow} B \overset{g}{\longrightarrow} B/A \overset{h}{\longrightarrow} \Sigma A
$$

a distinguished triangle, then

$$
  g\circ f = 0
$$

is the [[zero morphism]].

=--

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

Consider the [[commuting diagram]]

$$
  \array{
    A
      &\overset{id}{\longrightarrow}&
    A
      &\overset{}{\longrightarrow}&
    0
      &\overset{}{\longrightarrow}&
    \Sigma A
    \\
    \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{f}}
    \\
    A
      &\overset{f}{\longrightarrow}&
    B
      &\overset{g}{\longrightarrow}&
    B/A
      &\overset{h}{\longrightarrow}&
    \Sigma A
  }
  \,.
$$

Observe that the top part is a distinguished triangle by axioms T1 and T2 in def. \ref{TriangulatedCategory}. Hence by T3 there is an extension to
a commuting diagram of the form

$$
  \array{
    A
      &\overset{id}{\longrightarrow}&
    A
      &\overset{}{\longrightarrow}&
    0
      &\overset{}{\longrightarrow}&
    \Sigma A
    \\
    \downarrow^{\mathrlap{id}} && \downarrow^{\mathrlap{f}}
    && \downarrow
    && \downarrow^{\mathrlap{\Sigma \mathrlap{id}}}
    \\
    A
      &\overset{f}{\longrightarrow}&
    B
      &\overset{g}{\longrightarrow}&
    B/A
      &\overset{h}{\longrightarrow}&
    \Sigma A
  }
  \,.
$$

Now the commutativity of the middle square proves the claim.

=--

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

Consider a [[triangulated category]], def. \ref{TriangulatedCategory}, with shift functor denoted $\Sigma$ and with [[hom-functor]] denoted $[-,-]_\ast \colon Ho^{op}\times Ho \to Ab$. Then for $X$ any object, and for any distinguished triangle

$$
  A \overset{f}{\longrightarrow} B \overset{g}{\longrightarrow} B/A \overset{h}{\longrightarrow} \Sigma A
$$

the sequences of [[abelian groups]]

1. (long cofiber sequence)

   $$
    [\Sigma A, X]_\ast
      \overset{[h,X]_\ast}{\longrightarrow}
    [B/A,X]_\ast
      \overset{[g,X]_\ast}{\longrightarrow}
    [B,X]_\ast
      \overset{[f,X]_\ast}{\longrightarrow}
    [A,X]_\ast
   $$

1. (long fiber sequence)

   $$
     [X,A]_\ast
       \overset{[X,f]_\ast}{\longrightarrow}
     [X,B]_\ast
       \overset{[X,g]_\ast}{\longrightarrow}
     [X,B/A]_\ast
       \overset{[X,h]_\ast}{\longrightarrow}
     [X,\Sigma A]_\ast
   $$

are [[long exact sequences]].

=--

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

Regarding the first case:

Since $g \circ f = 0$ by lemma \ref{CompositesInADistinguishedTriangleAreZero}, we have an inclusion $im([g,X]_\ast) \subset ker([f,X]_\ast)$. Hence it is sufficient to show that if $\psi \colon B \to X$ is in the kernel of $[f,X]_\ast$ in that $\psi \circ f = 0$, then there is $\phi \colon C \to X$ with $\phi \circ g = \psi$. To that end, consider the commuting diagram

$$
  \array{
    A
      &\overset{f}{\longrightarrow}&
    B
      &\overset{g}{\longrightarrow}&
    B/A
      &\overset{h}{\longrightarrow}&
    \Sigma A
    \\
    \downarrow && {}^{\mathllap{\psi}}\downarrow
    \\
    0
      &\overset{}{\longrightarrow}&
    X
      &\overset{id}{\longrightarrow}&
    X
      &\overset{}{\longrightarrow}&
    0
  }
  \,,
$$

where the commutativity of the left square exhibits our assumption.

The top part of this diagram is a distinguished triangle by assumption, and the bottom part is by condition $T1$ in def. \ref{TriangulatedCategory}. Hence by condition T3 there exists $\phi$ fitting into a commuting diagram of the form

$$
  \array{
    A
      &\overset{f}{\longrightarrow}&
    B
      &\overset{g}{\longrightarrow}&
    B/A
      &\overset{h}{\longrightarrow}&
    \Sigma A
    \\
    \downarrow && {}^{\mathllap{\psi}}\downarrow && \downarrow^{\mathrlap{\phi}} && \downarrow
    \\
    0
      &\overset{}{\longrightarrow}&
    X
      &\overset{id}{\longrightarrow}&
    X
      &\overset{}{\longrightarrow}&
    0
  }
  \,.
$$

Here the commutativity of the middle square exhibits the desired conclusion.

This shows that the first sequence in question is exact at $[B,X]_\ast$. Applying the same reasoning to the distinguished traingle $(g,h,-\Sigma f)$ provided by T2 yields exactness at $[C,X]_\ast$.

Regarding the second case:

Again, from lemma \ref{CompositesInADistinguishedTriangleAreZero} it is immediate that

$$
  im([X,f]_\ast) \subset ker([X,g]_\ast)
$$

so that we need to show that for $\psi \colon X \to B$ in the kernel of $[X,g]_\ast$, hence such that $g\circ \psi = 0$, then there exists $\phi \colon X \to A$ with $f \circ \phi = \psi$.

To that end, consider the commuting diagram

$$
  \array{
    X &\longrightarrow& 0 &\longrightarrow& \Sigma X &\overset{- id}{\longrightarrow}& \Sigma X
    \\
    \downarrow^{\mathrlap{\psi}} && \downarrow
    \\
     B
       &\overset{g}{\longrightarrow}&
     B/A
       &\overset{h}{\longrightarrow}&
     \Sigma A
       &\overset{-\Sigma f}{\longrightarrow}&
     \Sigma B
  }
  \,,
$$

where the commutativity of the left square exhibits our assumption.

Now the top part of this diagram is a distinguished triangle by conditions T1 and T2 in def. \ref{TriangulatedCategory}, while the bottom part is a distinguished triangle by applying T2 to the given distinguished triangle. Hence by T3 there exists $\tilde \phi \colon \Sigma X \to \Sigma A$ such as to extend to a commuting diagram of the form

$$
  \array{
    X &\longrightarrow& 0 &\longrightarrow& \Sigma X &\overset{- id}{\longrightarrow}& \Sigma X
    \\
    \downarrow^{\mathrlap{\psi}}
      &&
    \downarrow
      &&
    \downarrow^{\mathrlap{\tilde \phi}}
      &&
    \downarrow^{\mathrlap{\Sigma \psi}}
    \\
     B
       &\overset{g}{\longrightarrow}&
     B/A
       &\overset{h}{\longrightarrow}&
     \Sigma A
       &\overset{-\Sigma f}{\longrightarrow}&
     \Sigma B
  }
  \,,
$$

At this point we appeal to the condition in def. \ref{TriangulatedCategory} that $\Sigma \colon Ho \to Ho$ is an [[equivalence of categories]], so that in particular it is a [[fully faithful functor]]. It being a [[full functor]] implies that there exists $\phi \colon X \to A$ with $\tilde \phi = \Sigma \phi$. It being faithful then implies that the whole commuting square on the right is the image under $\Sigma$ of a commuting square

$$
  \array{
    X &\overset{-id}{\longrightarrow}& X
    \\
    {}^{\mathllap{\phi}}\downarrow && \downarrow^{\mathrlap{\psi}}
    \\
    A &\underset{-f}{\longrightarrow}& B
  }
  \,.
$$

This exhibits the claim to be shown.

=--

### From stable model categories and stable $\infty$-categories
 {#FromStableModelCategories}

A [[pointed category|pointed]] [[model category]] $\mathcal{C}$ is called a _[[stable model category]]_ if the canonically induced [[reduced suspension]]-functor on its [[homotopy category of a model category|homotopy category]]

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

is an [[equivalence of categories]].

In this case $(Ho(\mathcal{C}),\Sigma)$ is a triangulated category. ([Hovey 99, section 7](#Hovey99), for review see also [Schwede, section 2](#Schwede)).

Similarly, the [[homotopy category of an (infinity,1)-category|homotopy category]] of a [[stable (∞,1)-category]] is a triangulated category, see [there](stable+infinity-category#TheTriangulatedHomotopyCategory).

## Examples

* The [[homotopy category of chain complexes]] $K(\mathcal{A})$ in an [[abelian category]] (the category of chain complexes modulo [[chain homotopy]]) is a triangulated category: the translation functor is the [[suspension of chain complexes]] and the distinguished triangles are those coming from the [[mapping cone]] construction $X \stackrel{f}{\to}Y \to Cone(f) \to T X$ in $Ch_\bullet(\mathcal{A})$.

* The [[stable homotopy category]] (the [[homotopy category of an (∞,1)-category|homotopy category]] of the [[stable (∞,1)-category of spectra]]) is a triangulated category.  This is also true for [[parameterized spectra|parametrized]], [[equivariant spectra|equivariant]], etc. spectra. Also the [[full subcategory]] called the [[Spanier-Whitehead category]] is triangulated.

* The stable category of a [[Quillen exact category]] is [[suspended category]] as exhibited by [[Bernhard Keller]]. If the exact category is Frobenius, i.e. has enough injectives and enough projective and these two classes coincide, then the suspension of the stable category is in fact invertible, hence the stable category is triangulated. A triangulated category equivalent to a triangulated categories is said to be an [[algebraic triangulated category]].

* As mentioned before, the [[homotopy category of an (infinity,1)-category|homotopy category]] of a [[stable (∞,1)-category]] is a triangulated category.  Slightly more generally, this applies also to a [[stable derivator]], and slightly less generally, it applies to a [[stable model category]].  This includes both the preceding examples.

* The [[localization]] $C/N$ of any triangulated category $C$ at a [[null system]] $N \hookrightarrow C$, i.e. the localization using the [[calculus of fractions]] given by the morphisms $f : X \to Y$ such that there exists distinguished triangles $X \to Y \to Z \to T X$ with $Z$ an object of a [[null system]], is still naturally a triangulated category, with the distinguished triangles being the triangles isomorphic to an image of a distinguished triangle under $Q : C \to C/N$.

  * In particular, therefore, the [[derived category]] of any [[abelian category]] is a triangulated category, since it is the localization of the homotopy category at the null system of acyclic complexes.  This example is also the homotopy category of a stable $(\infty,1)$-category, and usually of a stable model category.

* In [[algebraic geometry]], important examples are given by the various [[triangulated categories of sheaves]] associated to a [[variety]] (e.g. bounded derived category of [[coherent sheaves]], triangulated category of [[perfect complexes]]).

## Related concepts

* [[exact triangle]], [[distinguished triangle]]

* [[enhanced triangulated category]], [[pretriangulated dg-category]], [[stable (∞,1)-category]]

* [[tensor triangulated category]]

* [[derived category]]

* [[t-structure]], [[Bridgeland stability condition]]

* [[suspended category]], [[n-angulated category]]

* [[Bousfield lattice]], [[thick subcategory]], [[additive category]],

* [[exceptional collection]]

* [[well-generated triangulated category]], [[compactly generated triangulated category]]

* [[spectrum of a triangulated category]]

* [[triangulated categories of sheaves]], [[Bondal-Orlov reconstruction theorem]]

* [[triangulated subcategory]], [[full triangulated subcategory]], [[periodic triangulated category]]

## References

The concept orignates in the thesis

* [[Verdier, Jean-Louis]], _Des Catégories Dérivées des Catégories Abéliennes_, Astérisque **239** (1996) [[doi:10.24033/ast.364](https://smf.emath.fr/publications/des-categories-derivees-des-categories-abeliennes), [numdam:AST_1996__239__R1_0](http://www.numdam.org/issues/AST_1996__239__R1_0), [pdf](http://www.numdam.org/item/AST_1996__239__R1_0.pdf)]

Similar axioms were already given in

* [[Albrecht Dold]], [[Dieter Puppe]], _Homologie nicht-additiver Funktoren_, Annales de l'Institut Fourier (Université de Grenoble) 11: 201–312, 1961, [eudml](https://eudml.org/doc/73776).

Monographs:

* {#GelfandManin96} [[Sergei Gelfand]], [[Yuri Manin]], Section IV of: *[[Methods of homological algebra]]*,  transl. from the 1988 Russian (Nauka Publ.) original, Springer (1996, 2002) [[doi:10.1007/978-3-662-12492-5](https://doi.org/10.1007/978-3-662-12492-5)]

* {#Neeman01} [[Amnon Neeman]], *Triangulated Categories*, Annals of Mathematics Studies **213**, Princeton University Press (2001) [[ISBN:9780691086866](https://press.princeton.edu/titles/7102.html), [pdf](http://www.mi-ras.ru/~akuznet/homalg/Neeman%20Triangulated%20categories.pdf)]

Discussion of the relation to [[stable model categories]] originates in

* {#Hovey99} [[Mark Hovey]], section 7 of _Model Categories_ Mathematical Surveys and Monographs, Volume 63, AMS (1999) ([pdf](https://www.math.rochester.edu/people/faculty/doug/otherpapers/hovey-model-cats.pdf), [Google books](http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover))

Other surveys:

* [[Masaki Kashiwara]], [[Pierre Schapira]], Section 10 of: *[[Categories and Sheaves]]*, Grundlehren der Mathematischen Wissenschaften **332**, Springer (2006) [[doi:10.1007/3-540-27950-4](https://link.springer.com/book/10.1007/3-540-27950-4), [pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf)]

* {#Hubery} [[Andrew Hubery]], _Notes on the octahedral axiom_, ([pdf](http://math-www.uni-paderborn.de/user/hubery/static/Octahedral.pdf))

* [[Masaki Kashiwara]], [[Pierre Schapira]], section 10 _[[Categories and Sheaves]]_,

* [[Jacob Lurie]], section 3 of _[[Stable Infinity-Categories]]_,

* {#Noohi08} [[Behrang Noohi]], _Lectures on derived and triangulated categories_, pp. 383-418 in [[Masoud Khalkhali]], [[Matilde Marcolli]] (eds.), _An Invitation to Noncommutative Geometry_, World Scientific  (2008) ([doi:10.1142/9789812814333_0006](https://doi.org/10.1142/9789812814333_0006) [arXiv:0704.1009](http://arxiv.org/abs/0704.1009)).

* [[Fernando Muro]] ([pdf](http://personal.us.es/fmuro/htag.pdf))

* {#Schwede} [[Stefan Schwede]], _Triangulated categories_ ([pdf](https://math.berkeley.edu/~aaron/atf/triangulated-categories.pdf))

* _[[Introduction to Stable homotopy theory]] --  [Triangulated structure](Introduction+to+Stable+homotopy+theory+--+1-1#TriangulatedStructure)_

A survey of formalisms used in [[stable homotopy theory]] to present the triangulated [[homotopy category]] of a [[stable (∞,1)-category]] is in

* [[Neil Strickland]], _Axiomatic stable homotopy - a survey_ ([arXiv:math.AT/0307143](http://front.math.ucdavis.edu/0307.5143))

Discussion of the redundancy in the traditional definition of triangulated category is in

* {#May} [[Peter May]], _The additivity of traces in triangulated categories_, ([pdf](http://www.math.uchicago.edu/~may/PAPERS/AddJan01.pdf))

There was also some discussion at the [nForum](http://nforum.mathforge.org/discussion/3638/triangulated-category).

On [[localization]] of triangulated categories:

* {#Krause08} [[Henning Krause]], *Localization theory for triangulated categories*, in proceedings of *[Workshop on Triangulated Categories, Leeds 2006](https://www.commalg.org/2006/08/13/workshop-on-triangulated-categories-leeds/)* [[arXiv:0806.1324](https://arxiv.org/abs/0806.1324)]

category: triangulated categories

[[!redirects triangulated categories]]
[[!redirects distinguished triangle]]
[[!redirects distinguished triangles]]
[[!redirects triangulated structure]]
[[!redirects triangulated structures]]

[[!redirects pretriangulated category]]
[[!redirects pretriangulated categories]]
[[!redirects pre-triangulated category]]
[[!redirects pre-triangulated categories]]

[[!redirects octahedral axiom]]
[[!redirects octahedral axioms]]


triangulated category of singularities

#Contents#
* table of contents
{:toc}

## Idea

A [[derived category]] of [[coherent sheaves]] on [[singularities]] in a [[variety]].

For a [[smooth variety]] $X$ then the full bounded [[derived category]] $D^b(X) = D^b(Coh X)$ coincides with that of [[perfect complexes]], while for a singular variety there are objects not represented by perfect complexes. Hence these may be attributed to be due to the contribution of the singularities. The quotient $D^b(X)/Perf(X)$ hence serves as the derived category of the singularities themselves.

$\mathbb{C}^\times$-equivariant singularity category is in good cases equivalent to the category of matrix
factorizations, which is the category of B-branes
in a [[Landau-Ginzburg theory]].

One sometimes considers a dg-[[enhanced triangulated category|enhancement]] (which is unique by Orlov-Lunts) of the derived category instead, the dg-category of singularities.

## References

The concept is due to

* [[Dmitri Orlov]], _Triangulated categories of singularities and D-branes in Landau-Ginzburg models_, _Proc. Steklov Inst. Math. 2004, no. 3 (246), 227--248_ ([arXiv:math/0302304](http://arxiv.org/abs/math/0302304))

* [[Dmitri Orlov]], _Derived categories of coherent sheaves and triangulated categories of singularities_, Algebra, Arithmetic, and Geometry: In Honor of Yu.I. Manin, vol. II, Progr. Math. __270__, Birkhäuser 2009, pp. 503-531 ([arXiv:0503632](https://arxiv.org/abs/math/0503632))

* [[Dmitri Orlov|Dmitri O. Orlov]], _Triangulated categories of singularities and equivalences between Landau-Ginzburg models_, Sbornik: Mathematics __197__:12 (2006) 1827 [doi](https://doi.org/10.1070/SM2006v197n12ABEH003824)

Idempotent completions of triangulated categories of singularities of two schemes are equivalent if the formal completions of these schemes along singularities are isomorphic:

* [[Dmitri Orlov]], _Formal completions and idempotent completions of triangulated categories of singularities_, Adv. Math. __226__:1 (2011) 206-217 [doi](https://doi.org/10.1016/j.aim.2010.06.016)

The [[algebraic K-theory]] of these categories of singularities is considered in

* [[Goncalo Tabuada]],  _Algebraic K-theory with coefficients of cyclic quotient singularities_, Comptes Rendus Mathematique __354__:5 (2016) 449-452 ([arXiv:1512.01205](https://arxiv.org/abs/1512.01205) [doi](https://doi.org/10.1016/j.crma.2016.01.017))

If a variety $Y$ over a field $k$ is the zero scheme of a section of a vector bundle on a smooth $k$-variety, then there is a construction of a singular variety $Z$ so that  $k^\times$-equivariant/graded singularity category of $Z$
is equivalent (as an [[enhanced triangulated category]]) to the bounded derived category of coherent sheaves on $Y$:

* M. Umut Isik, _Equivalence of the derived category of a variety with a singularity category_, International Mathematics Research Notices 2013, no. 12, pp. 2787-2808, [doi](https://doi.org/10.1093/imrn/rns125) [arXiv:1011.1484](https://arxiv.org/abs/1011.1484)

There is also a relative singularity category (usually defined in terms of some ring and module theoretic data) related to [[matrix factorization]]s

* [[Alexander I. Efimov]], [[Leonid Positselski]], _Coherent analogues of matrix factorizations and relative singularity categories_, Algebra and Number Theory, 9(5):1159-1292 (2015) [arXiv:1102.0261](https://arxiv.org/abs/1102.0261)
* Massimo Pippi, _On the structure of dg categories of relative singularities_, High. Struct. __6__:1 (2022) 375--402 [arXiv:1911.01332](https://arxiv.org/abs/1911.01332)

[[!redirects triangulated categories of singularities]]
[[!redirects derived category of singularities]]
[[!redirects derived categories of singularities]]
[[!redirects dg-categories of singularities]]
[[!redirects dg-category of singularities]]
[[!redirects category of singularities]]


triangulated functor

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Homological algebra
+--{: .hide}
[[!include homological algebra - contents]]
=--
#### Stable homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A **triangulated functor** $F : T \to T'$ between two [[triangulated categories]] $T$ and $T'$ is an [[additive functor]] that commutes with translation and preserves the distinguished triangles.

([Neeman 01, def. 2.1.1](#Neeman01))

Triangulated functors are also called _exact functors_ (not to be confused with the _other_ meaning of _[[exact functor]]_).

## Related concepts

* [[tensor triangulated category]]

* [[Fourier-Mukai transform]]

## References

* {#Neeman01} [[Amnon Neeman]], _Triangulated Categories_, Princeton University Press (2001)

[[!redirects triangulated functors]]


triangulated subcategory

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Stable homotopy theory
+--{: .hide}
[[!include stable homotopy theory - contents]]
=--
#### Notions of subcategory
+-- {: .hide}
[[!include notions of subcategory]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A [[subcategory]] of a [[triangulated category]] which is itself triangulated.

## Related concepts

* [[thick subcategory]]
* [[full triangulated subcategory]]

[[!redirects triangulated subcategories]]


triangulation

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

#Contents#
* table of contents
{:toc}

## Idea

A **triangulation** of a [[topological space]] $Y$ is a [[simplicial set]] $X$ together with a [[homeomorphism]] $h: R X \to Y$, where $R$ denotes the [[geometric realization]] [[functor]].

(Usually, in classical algebraic and geometric topology, the $X$ here is taken to be a [[simplicial complex]], but the difference does not really matter if one is considering uses in homotopy theoretic contexts. For the reason, see the discussion at [[subdivision]]. When considering polyhedral structure as such, for instance for PL manifolds, the simplical complex version would be needed. In such a case we may refer to a [[classical triangulation]].)

Explicitly, $R X$ is given by a [[coend]] formula

$$\int^{n \in \Delta} X(n) \cdot \sigma(n)$$

where $\sigma: \Delta \to Top$ is the standard affine [[simplex]] functor. Provided that $Top$ is interpreted as a [[nice category of spaces]] (such as $CGHaus$ -- see the discussion at [[geometric realization]]), the functor $R$ is left exact, and in particular preserves products.

## Triangulation
 {#StandardAffineSimplexFunctor}

There are various ways of understanding the affine simplex functor $\sigma: \Delta \to Top$ from a categorical perspective. (Note: in this article we will be working with the algebraist's version of the simplex category $\Delta$, namely the category of finite ordinals and order-preserving maps, including the initial or empty object which represents a (-1)-dimensional simplex. The $n$-element ordinal is conventionally, but perhaps unfortunately, denoted $[n-1]$, to indicate the dimension.)

### First definition

One way begins by regarding $\Delta^{op}$ as isomorphic to the category of nonempty ordinals, for which maps are functions that preserve order and the top and bottom elements. In other words, as the category of finite intervals, where an [[interval]] is a totally ordered set with a top and bottom element.
Indeed, for each ordinal $[n-1]$, the hom-set $\hom([n-1], [1])$ inherits from $[1]$ an interval structure under the pointwise definitions, and

$$\hom(-, [1]) \colon \Delta^{op} \to FinInt$$

is an equivalence (this can also be seen as a restriction of the Stone duality between finite posets and finite distributive lattices).

Under this contravariant equivalence, the $n$-element object $[n-1]$ of $\Delta$ corresponds to the $(n+1)$-element finite interval (again denoted $[n-1]$). Consider the functor that arises by homming into the standard unit interval $I$:

$$\Delta \simeq (FinInt)^{op} \hookrightarrow Int^{op} \stackrel{\hom(-, I)}{\to} Set$$

This functor lifts to a functor $hom_{Int}(-, I): \Delta \to Top$ that takes $[n-1]$ to the space of interval maps $[n-1] \to I$,

$$\{(x_0, x_1, \ldots, x_n): 0 = x_0 \leq x_1 \leq \ldots \leq x_n = 1\},$$

topologized as a subspace $\{0 \leq x_1 \leq \ldots \leq x_{n-1} \leq 1\}$ of $I^{n-1}$. This gives the affine simplex functor $\sigma: \Delta \to Top$.

The category of intervals is an $\omega$-accessible category where the finitely presentable objects are the finite intervals. It follows that each representable on $Int$, in particular $\hom(-, I): FinInt^{op} \to Set$, is a filtered colimit of representable presheaves on $FinInt$.

### Second definition

A second way of understanding $\sigma$ is by taking advantage of the fact that the algebraist's $\Delta$ is the [[walking structure|walking]] [[monoid]]. This means that given a monoidal structure on $Top$ and a monoid $M$ therein, there is a unique monoidal functor $\sigma: \Delta \to Top$ which sends the generic monoid $[0]$ to the monoid $M$. To this end, take the monoidal product on $Top$ to be "topological simplicial join": the join $X \star Y$ of two spaces $X$, $Y$ may be defined to be the pushout of the diagram

$$X \stackrel{\pi_1}{\leftarrow} X \times Y \stackrel{1_X \times \{0\} \times 1_Y}{\to} X \times I \times Y \stackrel{1_X \times \{1\} \times 1_Y}{\leftarrow} X \times Y \stackrel{\pi_2}{\to} Y$$

and now take the monoid in $Top$ to be the 1-point space $1$ with its unique monoid structure.

The induced monoidal functor is the affine simplex functor $\sigma: \Delta \to Top$. In effect, it identifies the $n$-dimensional simplex with an iterated simplicial join of $n+1$ copies of $1$:

$$\sigma(n) = 1 \star \ldots \star 1$$

because $[n]$ is itself the $(n+1)^{st}$ monoidal power of the 1-element ordinal $[0]$. Equivalently, it can be regarded as the result of applying the cone functor $C X = 1 \star X$ $n$ times to $1$.

## Cubulation

A **[[cubulation]]** of a [[topological space]] $Y$ is a [[cubical set]] $C$ together with a [[homeomorphism]] $h: R_{cub}C \to Y$ where $R_{cub}$ denotes the realization functor for [[cubical set]]s $Set^{\Box^{op}}$. Explicitly, $R_{cub}C$ is given by a [[coend]] formula

$$R_{cub}C = \int^{m \in Cube} C(m) \cdot \Box(m)$$

where $\Box: Cube \to Top$ is the standard geometric cube functor.

### Standard geometric cube functor

The category $Cube$ may be regarded as a "walking interval" in a sense slightly different to the sense of interval above: it is initial among monoidal categories that are equipped with an object $I$, two maps $i_0, i_1: 1 \to I$ (where $1$ is the monoidal unit) and a map $p: I \to 1$ such that $p \circ i_0 = id_1 = p \circ i_1$. The monoidal unit $1$ in $Cube$ is terminal, hence there is a unique map $!: X \to 1$ for any object $X$. The interval $I$ of $Cube$ monoidally generates $Cube$ in the sense of [[PROP|PROS]].

It follows that if $Top$ is considered as a cartesian monoidal category equipped with $I = [0, 1]$ in this sense of interval, we get an induced monoidal functor

$$\Box: Cube \to Top$$

The monoidal product on $Cube$ induces a monoidal product $\otimes$ on $Set^{Cube^{op}}$ by [[Day convolution]]. The cubical realization functor $R_{cub}: Set^{Cube^{op}} \to Top$ is, up to isomorphism, the unique cocontinuous monoidal functor which extends the monoidal functor $\Box$ along the Yoneda embedding; therefore $R_{cub}$ takes $\otimes$-products of cubical sets to the corresponding cartesian products of spaces.

## Properties

### General properties

(...)

### Relation between triangulation and cubulation

As explained below, there is a "cubulation" functor for standard simplices, $\Sigma: \Delta \to Set^{Cube^{op}}$, such that the affine simplex functor $\sigma: \Delta \to Top$ is naturally isomorphic to the composite

$$\Delta \stackrel{\Sigma}{\to} Set^{\Box^{op}} \stackrel{R_{cub}}{\to} Top$$

Given a triangulation $(X, h: R X \to Y)$ of a space $Y$, we have [[isomorphism]]s

$$\array{
Y & \cong & \int^n X(n) \cdot \sigma(n) \\
& \cong & \int^n X(n) \cdot (\int^m \Sigma_n(m) \cdot \Box(m)) \\
& \cong & \int^m (\int^n X(n) \cdot \Sigma_n(m)) \cdot \Box(m)
}$$

where in the last line we used the [[coend Fubini theorem]] for interchange of coends. Thus, defining the cubical set $C$ by

$$C(m) = \int^n X(n) \cdot \Sigma_n(m)$$

we have a homeomorphism $Y \cong \int^m C(m) \cdot \Box(m) = R_{cub} C$, i.e., we obtain a cubulation of $Y$.

There is also a triangulation functor for standard cubes, $\Box: Cube \to Set^{\Delta^{op}}$, which can be used to triangulate the realizations of cubical sets.

### Cubulating simplices and triangulating cubes

The functor $\Sigma$ effectively regards an $n$-simplex as an iterated [[join of simplicial sets]] and then produces the analogous join in the category of cubical sets. This for instance regards the 2-simplex as a square with one degenerate edge.

In other words, to define $\Sigma: \Delta \to Set^{Cube^{op}}$, we mimic the second construction of the affine simplex functor given above, replacing $Top$ by cubical sets and the topological simplicial join by a suitable "cubical simplicial join". Formally, we define a monoidal structure on cubical sets by taking $X \star Y$ to be the pushout of the diagram

$$X \stackrel{\pi_1}{\leftarrow} X \otimes Y \stackrel{1_X \otimes i_0 \otimes 1_Y}{\to} X \otimes I \otimes Y \stackrel{1_X \otimes i_1 \otimes 1_Y}{\leftarrow} X \otimes Y \stackrel{\pi_2}{\to} Y$$

where the projection maps $\pi_1$, $\pi_2$ are defined by taking advantage of the fact that the monoidal unit of $\otimes$ is terminal:

$$\pi_1 = (X \otimes Y \stackrel{1_X \otimes !}{\to} X \otimes 1 \cong X)$$

$$\pi_2 = (X \otimes Y \stackrel{! \otimes 1_Y}{\to} 1 \otimes Y \cong Y)$$

The terminal cubical set is of course a monoid with respect to this monoidal product, so by the walking monoid property we obtain a monoidal functor

$$\Sigma: \Delta \to Set^{Cube^{op}}$$

which plays a role analogous to the affine simplex functor into $Top$.

Observe that geometric realization $R_{cub}: Set^{Cube^{op}} \to Top$ takes cubical simplicial joins to topological simplicial joins, because $R_{cub}$ sends $\otimes$-products to cartesian products, and preserves pushouts because it is cocontinuous. We conclude that both $\sigma: \Delta \to Top$ and $R_{cub} \circ \Sigma: \Delta \to Top$ take monoidal products in $\Delta$ to topological simplicial joins, and both take the walking monoid of $\Delta$ to the one-point space. By the universal property of $\Delta$, it follows that there is a natural isomorphism

$$\sigma \cong R_{cub} \circ \Sigma$$

(as monoidal functors), which is what we want.

Similarly, we can easily define a monoidal functor $\Box_{\delta}: Cube \to Set^{\Delta^{op}}$ such that

$$(\Box: Cube \to Top) \cong (Cube \stackrel{\Box_\delta}{\to} Set^{\Delta^{op}} \stackrel{R}{\to} Top) \qquad (2)$$

In detail, regard the category of simplicial sets as a cartesian monoidal category equipped with the representable $hom(-, [1])$ as an interval (with two face maps from and a projection to the terminal object $hom(-, [0])$). By the walking interval property of $Cube$, there is an induced functor

$$\Box_{\delta}: Cube \to Set^{\Delta^{op}}$$

Finally, because $R: Set^{\Delta^{op}} \to Top$ is product-preserving and preserves the interval objects, the isomorphism (2) obtains by the universal property of $Cube$.

## Related concepts

* [[triangulation theorem]]

* [[equivariant triangulation]]

* [[equivariant triangulation theorem]]

* [[cubulation]]

## References

### General

Texbook accounts

in a context of [[mathematical physics]]:

* [[Mikio Nakahara]], p. 120 onwards in: _[[Geometry, Topology and Physics]]_, IOP 2003 ([doi:10.1201/9781315275826](https://doi.org/10.1201/9781315275826), pdf)

See also

* Wikipedia, _Triangulation (geometry)_

[[!include triangulation theorems -- references]]

[[!redirects triangulations]]


triangulation theorem

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Manifolds and cobordisms
+-- {: .hide}
[[!include manifolds and cobordisms - contents]]
=--
#### Homotopy theory
+--{: .hide}
[[!include homotopy - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

For $X$ a [[manifold]],

* a  *simplicial [[triangulation]]* is a [[simplicial complex]] $Tr(X)$ and a [[homeomorphism]] $\left\vert Tr(X)\right\vert \xrightarrow{homeo} X$ from its [[geometric realization]] to the [[underlying]] [[topological space]] of $X$.

* a *combinatorial triangulation* is a simplicial triangulation such that for each simplex $\sigma$ the *link* of $\sigma$ (the [[union]] of all simplices $\tau$ that intersect $\sigma$ such that both $\sigma$ and $\tau$ are faces of a simplex) is [[homeomorphism|homeomorphic]] to a [[sphere]].

A *triangulation conjecture* [[conjecture|conjectures]] and *triangulation theorem* [[proof|proves]] that a given kind of triangulation exists for a given kind of [[manifold]]. Conversely, deep theorems assert that a given kind of triangulation does *not* generally exists for a given class of manifolds.

## Statements

### Existence of triangulations

It is easy to show that every [[piecewise-linear manifold]] admits a combinatorial triangulation, so that combinatorial triangulability is often understood to mean existence of a piecewise-linear structure.

A compatible piecewise-linear structure, hence a combinatorial triangulation, hence a simplicial triangulation, does exist for

* every [[smooth manifold]]

  ([Cairns 1935](#Cairns35), [Whitehead 1940](#Whitehead40))

* every [[topological manifold]] of [[dimension]] 2, hence every [[surface]]

  ([Rado 1924](#Rado25))

* every [[topological manifold]] of [[dimension]] 3, hence every [[3-manifold]]

  ([Moise 1952](#Moise52))

### Non-existence of triangulations

In each dimension $\geq$ 4 there exist [[topological manifolds]] *not* admitting a combinatorial triangulation ([Kirby & Siebenmann 1969](#KirbySiebenmann69)).

In each dimension $\geq$ 5 there exist [[topological manifolds]] *not* even admitting a simplicial triangulation ([Manolescu 2016](#Manolescu16))

For [[topological manifolds]] $X$ of [[dimension of a manifold|dimension]] $dim(X) \leq 3$ triangulations still exist in general, but for every dimension $\geq 4$ there exist topological manifolds which do not admit a triangulation. For $dim(X) \geq 5$ there is an [[obstruction]] class $\Delta(X) \in H^4(X; \mathbb{Z}/2)$ in the [[ordinary cohomology]]  with [[coefficients]] in [[finite group of order 2|$\mathbb{Z}/2$]], in that $X$ admits a triangulation if and only if $\Delta(X) = 0$.

## Related concepts

* [[equivariant triangulation theorem]]

## References

[[!include triangulation theorems -- references]]

[[!redirects triangulation theorems]]

[[!redirects triangulation conjecture]]
[[!redirects triangulation conjectures]]


triangulation theorems -- references

### Triangulation theorems for manifolds

On [[triangulation conjectures]] and [[triangulation theorems]] on existence of [[triangulations]] of [[manifolds]].

Review:

* [[Ciprian Manolescu]], *Triangulations of manifolds*, Notices of the International Congress of Chinese Mathematicians **2** 2 (2014) ([pdf](https://web.stanford.edu/~cm5/tm.pdf), [[Manolescu_TriangulationsOfManifolds.pdf:file]], [doi:10.4310/ICCM.2014.v2.n2.a2](https://dx.doi.org/10.4310/ICCM.2014.v2.n2.a2))

The question of triangulability of [[smooth manifolds]] was first raised in

* [[Henri Poincaré]], *Complément à l'analysis situs*, Rendiconti del Circolo Matematico di Palermo (1884-1940) volume 13, pages 285–343 (1899) ([doi:10.1007/BF03024461](https://doi.org/10.1007/BF03024461))

and for general [[topological manifolds]] in

* [[Hellmuth Kneser]], *Die Topologie der Mannigfaltigkeiten*, Jahresbericht der Deutschen Mathematiker-Vereinigung (1926), Volume: 34, page 1-13 ([eudml:145701](https://eudml.org/doc/145701))

Proof that every [[surface]] admits a combinatorial triangulation:

* {#Rado25} [[Tibor Radó]], *Über den Begriff der Riemannschen Fläche*, Acta Litt. Sci. Szeged, 2 (101-121), 10 ([pdf](https://www.maths.ed.ac.uk/~v1ranick/papers/rado.pdf), [pdf](http://acta.bibl.u-szeged.hu/13312/1/math_002_101-121.pdf))
Proof that every [[smooth manifold]] admits a combinatorial triangulation:

Proof that every [[smooth manifold]] admits a combinatorial triangulation is due to

* {#Cairns35} [[Stewart S. Cairns]], *Triangulation of the manifold of class one*, Bull. Amer. Math. Soc. 41(8): 549-552 ([euclid:1183498332](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-41/issue-8/Triangulation-of-the-manifold-of-class-one/bams/1183498332.full))

* {#Whitehead40} [[J. H. C. Whitehead]], *On $C^1$-complexes*, Ann. of Math. (2) 41 (1940), 809–824 ([doi:10.2307/1968861](https://doi.org/10.2307/1968861), [jstor:1968861](https://www.jstor.org/stable/1968861))

with further accounts in:

* [[Hassler Whitney]], Section IV.B of: *Geometric Integration Theory* (1957), Princeton Legacy Library (2016) ([doi:10.1515/9781400877577](https://doi.org/10.1515/9781400877577))

* {#Cairns61} [[Stewart S. Cairns]], *A simple triangulation method for smooth manifolds*, Bull. Amer. Math. Soc. 67 (1961), 389-390 ([doi:10.1090/S0002-9904-1961-10631-9](https://doi.org/10.1090/S0002-9904-1961-10631-9))

    > (but see [MO:a/177199](https://mathoverflow.net/a/177199/381), where this "simpler argument" is claimed to be wrong)

* [[Jacob Lurie]], *Existence of Triangulations* ([pdf](https://www.math.ias.edu/~lurie/937notes/937Lecture4.pdf)), Lecture 4 in: *[Topics in Geometric Topology](https://www.math.ias.edu/~lurie/937.html)*

A detailed exposition is available in Chapter II (see Thm. 10.6) of

* [[James R. Munkres]], _Elementary Differential Topology_, Annals of Mathematics Studies 54 (1966), Princeton University Press ([doi:10.1515/9781400882656](http://dx.doi.org/10.1515/9781400882656)).

Generalization to existence of [[equivariant triangulation]] for smooth [[G-manifolds]] ([[equivariant triangulation theorem]]):

* {#Illman72} [[Sören Illman]], *Equivariant algebraic topology*, Princeton University 1972 ([[Illman_EquivariantAlgebraicTopology1972.pdf:file]])

* {#Illman78} [[Sören Illman]], _Smooth equivariant triangulations of $G$-manifolds for $G$ a finite group_, Math. Ann. (1978) 233: 199 ([doi:10.1007/BF01405351](https://doi.org/10.1007/BF01405351))

* {#Illman83} [[Sören Illman]], _The Equivariant Triangulation Theorem for Actions of Compact Lie Groups_, Mathematische Annalen (1983) Volume: 262, page 487-502 ([dml:163720](https://eudml.org/doc/163720))

Proof that every [[3-manifold]] admits the structure of a [[smooth manifold]] and hence of a combinatorial triangulation:

* {#Moise52} [[Edwin E. Moise]],  *Affine Structures in 3-Manifolds: V. The Triangulation Theorem and Hauptvermutung*, Annals of Mathematics Second Series, Vol. 56, No. 1 (Jul., 1952), pp. 96-114 ([doi:10.2307/1969769](https://doi.org/10.2307/1969769), [jstor:1969769](https://www.jstor.org/stable/1969769))

Proof that in every dimension $dim \geq 4$ there exist topological manifolds without combinatorial triangulation:

* {#KirbySiebenmann69} [[Robion C. Kirby]], [[Laurent C. Siebenmann]], *On the triangulation of manifolds and the Hauptvermutung*, Bull. Amer. Math. Soc. 75(4): 742-749 (July 1969)  ([euclid:1183530633](https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society/volume-75/issue-4/On-the-triangulation-of-manifolds-and-the-Hauptvermutung/bams/1183530633.full))

Proof that in every dimension $dim \geq 5$ there exist topological manifolds without simplicial triangulation:

* {#Manolescu16} [[Ciprian Manolescu]], *$Pin(2)$-equivariant Seiberg-Witten Floer homology and the Triangulation Conjecture*, J. Amer. Math. Soc. 29 (2016), 147-176  ([arXiv:1303.2354](https://arxiv.org/abs/1303.2354), [doi:10.1090/jams829]( https://doi.org/10.1090/jams829))


tribe

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

# Tribes
* table of contents
{: toc}

## Idea

A *tribe* in the sense of [Joyal 2017, Def. 3.1.6](#Joyal17)
is a kind of [[fibration category]] providing some fragment of [[categorical semantics of dependent type theory]] and in particular of [[homotopy type theory]].

## Definitions

...

##Related concepts

* [[clan]]

## References

The particular terminology "tribe" is due to:

* {#Joyal17} [[André Joyal]], _Notes on Clans and Tribes_ [[arxiv:1710.10238](https://arxiv.org/abs/1710.10238)]

following

* [[André Joyal]], _What is categorical type theory_, various talks in 2013 [[pdf](http://www.math.uwaterloo.ca/~asl2013/Slides/Joyal.pdf)]

but the underlying notion of [[categorical semantics of dependent types]] is classical, going back at least to [Seely 1984](relation+between+type+theory+and+category+theory#Seely84b) &  [Hofmann 1997](categorical+semantics+of+dependent+type+theory#Hofmann97), as is the general idea of categorical semantics of [[homotopy type theory]], going back to

* {#AwodeyWarren07} [[Steve Awodey]], [[Michael Warren]], *Homotopy theoretic models of identity type*,  Mathematical Proceedings of the Cambridge Philosophical Society **146** 1 (2009) [[arXiv:0709.0248](http://arxiv.org/abs/0709.0248), [doi:10.1017/S0305004108001783](https://doi.org/10.1017/S0305004108001783)]

reviewed for instance in

* [[Mike Shulman]], _Categorical models of homotopy type theory_ (2012) [[pdf](https://home.sandiego.edu/~shulman/hottminicourse2012/03models.pdf)]

See also:

* {#Shulman15} [[Michael Shulman]], _Univalence for inverse diagrams and homotopy canonicity_, Mathematical Structures in Computer Science, Volume 25, Issue 5 ( _From type theory and homotopy theory to Univalent Foundations of Mathematics_ ) June 2015 ([arXiv:1203.3253](https://arxiv.org/abs/1203.3253), [doi:/10.1017/S0960129514000565](https://doi.org/10.1017/S0960129514000565))

* [[Chris Kapulkin]] and [[Karol Szumilo]], _Internal Language of Finitely Complete (∞,1)-categories_, [arxiv:1709.09519](https://arxiv.org/abs/1709.09519)

The type-theory-literature traditionally refers to such categories generically as *[[display map]]/[[fibration categories]] with such-and-such types*, eg.

[North 2019, Rem. 2.4](#North19):

> The definitions in this section are relatively standard in the literature. A display map category which models [[dependent pair type|$\Sigma$-types]] coincides with Joyal’s notion of *clan*, and a display map category which models [[dependent pair type|$\sum$-types]] and [[dependent function type|$\prod$-types]] coincides with his notion of *$\pi$-clan*. [...]

> A *tribe* in the sense of Joyal is a display map category which models [[dependent pair type|$\Sigma$-types]] and Paulin-Mohring [[identification types|Id-types]].

* {#North19} [[Paige Randall North]], *Identity types and weak factorization systems in Cauchy complete categories*, Mathematical Structures in Computer Science **29** 9 (2019) 1411-1427 [[doi:10.1017/S0960129519000033](https://doi.org/10.1017/S0960129519000033), [arXiv:1901.03567](https://arxiv.org/abs/1901.03567)]

[[!redirects tribe]]
[[!redirects tribes]]
[[!redirects type-theoretic fibration category]]
[[!redirects type-theoretic fibration categories]]
[[!redirects type theoretic fibration category]]
[[!redirects type theoretic fibration categories]]


tricategory

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

#Contents#
* table of contents
{:toc}

## Idea

A **tricategory** is a particular [[algebraic definition of higher category|algebraic]] notion of _weak_ [[3-category]].
The idea is that a tricategory is a category _weakly_ [[enriched category|enriched]] over [[Bicat]]: the [[hom-object|hom-objects]] of a tricategory are [[bicategories]], and the associativity and unity laws of [[enriched category|enriched categories]] hold only up to coherent equivalence.

## Coherence theorems
{#Coherence}

One way to state the coherence theorem for tricategories is that every tricategory is [[equivalence|equivalent]] to a [[Gray-category]], which is a sort of semi-strict 3-category (everything is strict except for the [[interchange law]]).

## Examples {#Examples}

For $R$ a commutative [[ring]], there is a [[symmetric monoidal bicategory]] $Alg(R)$ whose

* [[object]]s are $R$-[[algebra]]s;

* [[morphism]]s are algebra [[bimodule]]s;

* [[2-morphisms]] are bimodule homomorphisms.

The monoidal product is given by [[tensor product]] over $R$.

By [[delooping]] this once, this gives an example of a [[tricategory]] with a single object.

The tricategory statement follows from Theorem 24 in either the journal version, or the arXiv:0711.1761v2 version of:

* [[Richard Garner]], [[Nick Gurski]]: The low-dimensional structures formed by tricategories. Math. Proc. Camb. Phil. Soc. 146 (2009),pp. 551--589, ([arXiv:0711.1761](http://arxiv.org/abs/0711.1761))

This, and that the monoidal bicategory is even symmetric monoidal is given by the main theorem in

* [[Mike Shulman]], _Constructing symmetric monoidal bicategories_ ([arXiv:1004.0993](http://arxiv.org/abs/1004.0993))

## Related entries

*  [[strict 3-category]]
*  [[bicategory]]
*  [[tetracategory]]

## References

The original source:

* [[Robert Gordon]], [[A. John Power]], [[Ross Street]], _Coherence for tricategories_, Mem. Amer. Math Soc. 117 (1995) no 558 ([ISBN:978-1-4704-0137-5](https://bookstore.ams.org/memo-117-558))

refined in:

* [[Nick Gurski]], _An algebraic theory of tricategories_, PhD thesis (2007) [[pdf](https://dokumen.tips/documents/an-algebraic-theory-of-tricategories.html), [[Gurski-AlgebraicTricategories.pdf:file]]]

See also:

Textbook account:

* {#JohnsonYau20} [[Niles Johnson]], [[Donald Yau]], Section 11.2 of: _2-Dimensional Categories_, Oxford University Press 2021 ([arXiv:2002.06055](http://arxiv.org/abs/2002.06055), [doi:10.1093/oso/9780198871378.001.0001](https://oxford.universitypressscholarship.com/view/10.1093/oso/9780198871378.001.0001/oso-9780198871378))

A discussion of [[monoidal category|monoidal]] tricategories, regarded by the discussion at [[k-tuply monoidal (n,r)-category]] as one-object [[tetracategories]], is in section 3 of

* [[Alex Hoffnung]], _Spans in 2-Categories: A monoidal tricategory_ ([arXiv:1112.0560](http://arxiv.org/abs/1112.0560))

See also

* [[Richard Garner]], [[Nick Gurski]], *The low-dimensional structures that tricategories form* ([arXiv:0711.1761](http://arxiv.org/abs/0711.1761))

* Peter Guthmann, _The tricategory of formal composites and its strictification_ ([arXiv:1903.05777](https://arxiv.org/abs/1903.05777))

[[!redirects tricategories]]
[[!redirects coherence theorem for tricategories]]


trichotomous relation

\tableofcontents

## Definition

A **trichotomous relation** on a set $S$ is a [[relation]] $R(x, y)$ which satisfies **trichotomy**:

* for all $x \in S$ and $y \in S$, exactly one of $R(x, y)$, $R(y, x)$, $x = y$ is true.

## Examples

* In [[classical mathematics]], the **law of trichotomy** or **axiom of trichotomy** for the [[real numbers]] state that the [[strict total order]] on the real numbers is trichotomous. In [[constructive mathematics]], this statement is equivalent to the [[analytic LPO]].

* In [[classical mathematics]], every [[strict total order]] is a trichotomous relation, and strict total orders can be defined as a [[transitive relation|transitive]] trichotomous relation. However, in constructive mathematics, only the decidable strict total orders are trichotomous relations.

## Related concepts

* [[strict total order]]

* [[connected relation]]

* [[analytic LPO]]

[[!redirects trichotomous]]
[[!redirects trichotomous relation]]
[[!redirects trichotomous relations]]

[[!redirects trichotomy]]
[[!redirects law of trichotomy]]
[[!redirects axiom of trichotomy]]


trifibration

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

#Contents#
* table of contents
{:toc}

## Idea

$E$ is a **trifibration** if both $E \to B$ and $E^{d} \to B$ are [[bifibrations]], where the category $E^{d}$, fibered over $B$, is the [[dual fibration]] obtained by changing the direction of all the vertical arrows in $E$. The arrows of $E^{d}$ are the equivalence classes of spans with a vertical arrow pointing at the source, and a cartesian arrow pointing at the target ([Pavlovic90, p. 315](#Pavlovic90)).

A fibration $E \to B$ is a bifibration if and only if for every $t: I \to J$ in $B$, the [[inverse image functor]]  $t^{\ast}: E_{J} \to E_{I}$ has a [[left adjoint]] $t_{!}: E_{I} \to E_{J}$. It is a trifibration if and only if there is also a [[right adjoint]] $t_{\ast}: E_{I} \to E_{J}$.

Since a trifibration is _not_ a fibration in three ways as its name suggests, alternative terminology has been used, including $\ast$-bifibration in ([FBMF](#FBMF)).

## References

* {#Pavlovic90} [[Dusko Pavlovic|Duško Pavlović]], _Categorical Interpolation: Descent and the Beck-Chevalley Condition without Direct Images_ , pp.306-325  in _Category theory Como 1990_, LNM **1488** Springer Heidelberg 1991. ([pdf](http://www.isg.rhul.ac.uk/dusko/papers/1990-BCDE-Como.pdf))

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


trigonometric function

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

# Contents
* table of contents
{: toc}

## Idea

A _trigonometric function_ is one derived from the basic [[functions]] of [[trigonometry]], $\cos$, $\sin$ ([[cosine]] and [[sine]]), which themselves are the standard [[coordinate functions]] (equivalently: [[Cartesian product|product projections]] $pr_i$) of the [[Cartesian space]] $\mathbb{R}^2$, [[restriction|restricted]] to the unit [[circle]]:

$$
  \array{
    &
    && \mathbb{R}
    \\
    &
    &
    {}^{\mathllap{cos}}\nearrow
    & \big\uparrow{}^{ \mathrlap{pr_1} }
    \\
    \mathbb{R}
    \overset{exp}{\longrightarrow}
    &
    S^1
    &\hookrightarrow&
    \mathbb{R}^2
    &\simeq&
    \mathbb{R} \times \mathbb{R}
    \\
    &
    & {}_{\mathllap{sin}}\searrow & \big\downarrow{}^{ \mathrlap{pr_2} }
    \\
    &
    && \mathbb{R}
  }
  \,,
$$

or rather the result of [[composition|composing]] these restrictions with an [[arc length]] parametrization $\mathbb{R} \overset{\exp}{\to} S^1$. They are also called _circular functions_.

In elementary [[mathematics]], there are six traditional trigonometric functions;

1. [[sine]] $\;$ $sin$

1. [[cosine]] $\;$ $cos$

1. [[tan|tangent]] $\;$ $\tan = \frac{\sin}{\cos}$,

1. [[cotan|cotangent]] $\;$ $\cot = \frac{\cos}{\sin}$,

1. [[secant]] $\;$ $\sec = \frac1{\cos}$,

1. [[cosecant]] $\;$ $\csc = \frac1{\sin}$.

The early view was that these functions measured the six possible [[ratios]] of side [[lengths]] of right [[triangles]] (as a basic case in terms of which other triangles can be analyzed; "trigonometry" = "triangle measure").

These six functions figure prominently in [[Euclidean geometry]] where the angles of a triangle sum to [[arc length]] $\pi$.

There are more elaborate offshoots such as [[spherical trigonometry]] (see [[elliptic geometry]]) which was historically important for terrestrial navigation. Moreover, there are the related [[hyperbolic functions]] (see [[hyperbolic geometry]]) which result from a projective change of [[conic section]] (from a circle to a [[hyperbola]]).

## Definition

The most useful modern approach to [[cos]] and [[sin]] comes from taking [[Euler's formula]] as a working definition:

Let

$$
  \exp
  \;\colon\;
  \mathbb{C}
    \longrightarrow
   \mathbb{C}^\ast
$$

be the [[complex number|complex]] [[exponential function]], defined by the [[exponential]] [[power series]] formula

$$
  \exp(z)
    \;\coloneqq\;
  \sum_{n \geq 0} \frac{z^n}{n!}
  \,.
$$

These satisfy the following fundamental exponential identities:

* $\exp(z+w) = \exp(z) \cdot \exp(w)$ ([[homomorphism]] from an additive [[group]] to a multiplicative group),

* $\widebar{\exp(z)} = \exp(\widebar{z})$ (because [[complex conjugation]] $\widebar{(-)}$ is a [[continuous map|continuous]] [[field]] [[automorphism]]).

As a result, if $z + \widebar{z} = 0$ (i.e., if $z$ is [[imaginary number|purely imaginary]]: $z = i t$ for some $t \in \mathbb{R}$), we have

$$\exp(z) \cdot \widebar{\exp(z)} = 1$$

so that $w = \exp(z)$ lies on the unit circle defined by $w\widebar{w} = 1$.

+-- {: .num_defn}
###### Definition
The _cosine function_ $\cos: \mathbb{R} \to \mathbb{R}$ is defined by $\cos(t) = Re(\exp(i t))$ ([[real part]]); the _sine function_ $\sin: \mathbb{R} \to \mathbb{R}$ is defined by $\sin(t) = Im(\exp(i t))$ ([[imaginary part]]). In other words: $\exp(i t) = \cos(t) + i\sin(t)$ (Euler).
=--

This implies that $\cos(t) = \frac1{2}(\exp(i t) + \exp(-i t))$ and $\sin(t) = \frac1{2 i}(\exp(i t) - \exp(-i t))$; these equations suggest the simple [[analytic continuation]] of $\cos$ and $\sin$ to functions $\mathbb{C} \to \mathbb{C}$ on the entire complex plane.

## Properties

More or less immediate consequences of the definition include

* $(\cos t)^2 + (\sin t)^2 = 1$ ("[[Pythagorean theorem]]"), since $\exp(i t)$ lies on the unit circle. This is traditionally written as $\cos^2(t) + \sin^2(t) = 1$;

* $\cos(x + y) = \cos(x)\cos(y) - \sin(x)\sin(y)$ and $\sin(x + y) = \sin(x)\cos(y) + \cos(x)\sin(y)$ ("addition formulas"), by taking real and imaginary parts of the identity that says $\exp$ is a homomorphism;

* $(\sin)' = \cos$ and $(\cos)' = -\sin$ (by [[differentiation|differentiating]] $\exp(i t)$); the connection with the arc length parametrization of the unit circle is that the derivative of the position vector $p'(t) = (\exp(i t))'$ is a velocity vector $i\exp(i t)$ of unit length;

* $\cos(x + 2\pi) = \cos(x)$ and $\sin(x+2\pi) = \sin(x)$ ("[[periodicity]]"), according to the modern definition of [[pi]];

* $\cos(x) = \sum_{n \geq 0} (-1)^n \frac{x^{2 n}}{(2 n)!}$ and $\sin(x) = \sum_{n \geq 0} (-1)^n \frac{x^{2 n + 1}}{(2 n + 1)!}$, by exploiting the power series representation of the exponential function.

These name but a few of many [[trigonometric identities]], facility in which can serve as a modern-day shibboleth or barrier of passage in high school or lower-level undergraduate courses in mathematics. They seem also to be popular in mathematics education in India and appear regularly in entrance examinations there. But the ones listed above are the most fundamental.

There is no question that the trigonometric functions are rich in significance; for example; various representations of the tangent, cotangent, secant, etc. appear in [[enumerative combinatorics]] (as in the problem of counting alternating [[permutations]]), representations of [[Bernoulli numbers]], and so on.

## Related concepts

* [[inverse trigonometric function]]

* [[trigonometry]]

* [[trigonometric identity]]

* [[hyperbolic function]]

* [[elliptic function]]

## References

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

* Springer [[eom]]: V.I. Bityutskov, _[Trigonometric functions](http://www.encyclopediaofmath.org/index.php?title=Trigonometric_functions&oldid=14919)_

[[!redirects trigonometric functions]]


trigonometric identities and the irrationality of pi

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

## Goal

This page deals with the question whether it might be possible to prove the [[irrational number|irrationality]] of $\pi$ by using [[trigonometric identities]].

## Details

We begin with an infinite sequence of functions of some variables $\theta_1,\theta_2,\theta_3,\dots$:

$$
\begin{aligned}
f_0(\theta_1,\theta_2,\theta_3,\dots) &= \sum_{\text{even}\;n \ge 0} (-1)^{n/2} \sum_{|A| = n} \prod_{i \in A} \sin\theta_i \prod_{i \notin A} \cos\theta_i \\
f_1(\theta_1,\theta_2,\theta_3,\dots) &= \sum_{\text{odd}\;n \ge 1} (-1)^{(n-1)/2} \sum_{|A| = n} \prod_{i \in A} \sin\theta_i \prod_{i \notin A} \cos\theta_i \\
f_2(\theta_1,\theta_2,\theta_3,\dots) &= \sum_{\text{even}\;n \ge 2} (-1)^{(n-2)/2} n \sum_{|A| = n} \prod_{i \in A} \sin\theta_i \prod_{i \notin A} \cos\theta_i \\
f_3(\theta_1,\theta_2,\theta_3,\dots) &= \sum_{\text{odd}\;n \ge 3} (-1)^{(n-3)/2} (n-1) \sum_{|A| = n} \prod_{i \in A} \sin\theta_i \prod_{i \notin A} \cos\theta_i \\
f_4(\theta_1,\theta_2,\theta_3,\dots) &= \sum_{\text{even}\;n \ge 4} (-1)^{(n-4)/2} n(n-2) \sum_{|A| = n} \prod_{i \in A} \sin\theta_i \prod_{i \notin A} \cos\theta_i \\
f_5(\theta_1,\theta_2,\theta_3,\dots) &= \sum_{\text{odd}\;n \ge 5} (-1)^{(n-5)/2} (n-1)(n-3) \sum_{|A| = n} \prod_{i \in A} \sin\theta_i \prod_{i \notin A} \cos\theta_i \\
f_6(\theta_1,\theta_2,\theta_3,\dots) &= \sum_{\text{even}\;n \ge 6} (-1)^{(n-6)/2} n(n-2)(n-4) \sum_{|A| = n} \prod_{i \in A} \sin\theta_i \prod_{i \notin A} \cos\theta_i \\
f_7(\theta_1,\theta_2,\theta_3,\dots) &= \sum_{\text{odd}\;n \ge 7} (-1)^{(n-7)/2} (n-1)(n-3)(n-5) \sum_{|A| = n} \prod_{i \in A} \sin\theta_i \prod_{i \notin A} \cos\theta_i \\
& \vdots
\end{aligned}
$$

etc.  In each case we could have said $\text{even}\;n \ge 0$ or $\text{odd}\;n \ge 0$, since the coefficients kill off the terms in which $n$ is smaller than the index---the coefficients are the unique simplest expressions that do that.

* Each of these is a symmetric function of $\theta_1,\theta_2,\theta_3,\dots$.

* 0 is an identity element for each of these, i.e. we have $f_k(0,\theta_2,\theta_3,\theta_4,\dots) = f_k(\theta_2,\theta_3,\theta_4,\dots)$.

* Lemma:
$$
f_k(\theta_1,\theta_2,\theta_3,\theta_4,\theta_5,\dots) - f_k(\theta_1+\theta_2,\theta_3,\theta_4,\theta_5,\dots) = \left.\begin{cases} k & \text{if}\;k \ge 2\;\text{is even} \\
k - 1 & \text{if}\;k \ge 2\;\text{is odd}
\end{cases} \right\} \cdot
 \sin\theta_1\sin\theta_2 f_{k-2}(\theta_3,\theta_4,\theta_5,\dots)
$$
and if $k = 0$ or $1$ then the difference is 0.

* Identities:
$$
\begin{aligned}
f_0(\theta_1,\theta_2,\theta_3,\dots) &= \cos(\theta_1 + \theta_2 + \theta_3 + \cdots). \\
f_1(\theta_1,\theta_2,\theta_3,\dots) &= \sin(\theta_1 + \theta_2 + \theta_3 + \cdots). \\
\text{If}\;\sum_{i \ge 1} \theta_i = \pi,\;\text{then}\;f_2(\theta_1,\theta_2,\theta_3,\dots) &= \sum_{i \ge 1} \sin^2 \theta_i. \\
\text{If}\;\sum_{i \ge 1} \theta_i = \pi,\;\text{then}\;f_3(\theta_1,\theta_2,\theta_3,\dots) &= \sum_{i \ge 1} \frac{\sin(2\theta_i)}{2}.
\end{aligned}
$$

There is a simple geometric interpretation of the expression on the right side of the identity involving  $f_3$.  Suppose $\theta_i, i \ge 1$ are the angles between adjacent diagonals of a polygon inscribed in a circle of unit diameter.  For present purposes we consider the sides of the polygon to be "diagonals".  (All but two of the $\theta_i$ appear at each vertex.  For example, in an octagon, at one vertex the angles are $\theta_1,\dots,\theta_6$ at the next vertex they are $\theta_2,\dots,\theta_7$; at the next they are $\theta_3,\dots,\theta_8$; and then at the next they are $\theta_4,\dots,\theta_8,\theta_1$, and so on.  All eight appear at each vertex if one draws a tangent line to the circle and includes the angles between sides of the polygon and the tangent line.)  Then the value of each side of the identity involving $f_3$ is half the area of the polygon.  Each term on the right side of the identity is half the signed area of a triangle having one vertex at the center of the circle.  As one goes counterclockwise around the polygon, the signed area is positive or negative as the ray from the center is then turning counterclockwise or clockwise.  The terms on the left side may perhaps not admit so simple an interpretation when the number of nonzero $\theta_i$ is more than 4.  The special case involving only three nonzero $\theta_i$ is the well-known (?) identity that says that if $\alpha+\beta+\gamma=\pi$ then $4\sin\alpha\sin\beta\sin\gamma = sin(2\alpha) + \sin(2\beta) + \sin(2\gamma)$.

(Is there any simple geometric interpretation of the identity involving $f_2$?)

### A question

Are these four identities the first four terms in a sequence that continues?

+-- {: .query}
[[Michael Hardy]]:  A student in India whom I encountered via the internet told me that the problem of proving the "conditional identity" that $4\sin\alpha\sin\beta\sin\gamma = sin(2\alpha) + \sin(2\beta) + \sin(2\gamma)$ if $\alpha+\beta+\gamma=\pi$ recurs perennially on the joint entrance exam of the Indian Institutes of Technology.  Hence "well known".
=--

##Inversion formulas

$$
\begin{aligned}
\prod_{i=1}^\infty \cos\theta_i & = f_0 + \frac{1}{2} f_2 + \frac{1}{2 \cdot 4} f_4 + \frac{1}{2 \cdot 4 \cdot 6} f_6 + \frac{1}{2 \cdot 4 \cdot 6 \cdot 8} f_8 + \cdots \\
\sum_{j=1}^\infty \sin\theta_j \prod_{i : i \neq j} \cos\theta_i & = f_1 + \frac{1}{2} f_3 + \frac{1}{2\cdot 4} f_5 + \frac{1}{2\cdot 4\cdot 6} f_7 + \frac{1}{2 \cdot 4 \cdot 6 \cdot 8} f_9 + \cdots \\
\sum_{|A| = 2} \prod_{i \in A} \sin\theta_i \prod_{i \notin A} \cos\theta_i & = \frac{1}{2} \left(f_2 + \frac{1}{2} f_4 + \frac{1}{2\cdot 4} f_6 + \frac{1}{2\cdot 4 \cdot 6} f_8 + \frac{1}{2\cdot 4\cdot 6\cdot 8} f_{10} + \cdots\right) \\
\sum_{|A| = 3} \prod_{i \in A} \sin\theta_i \prod_{i \notin A} \cos\theta_i & = \frac{1}{2} \left(f_3 + \frac{1}{2} f_5 + \frac{1}{2\cdot 4} f_7 + \frac{1}{2\cdot 4 \cdot 6} f_9 + \frac{1}{2\cdot 4\cdot 6\cdot 8} f_{11} + \cdots\right) \\
\sum_{|A| = 4} \prod_{i \in A} \sin\theta_i \prod_{i \notin A} \cos\theta_i & = \frac{1}{2\cdot 4} \left(f_4 + \frac{1}{2} f_6 + \frac{1}{2\cdot 4} f_8 + \frac{1}{2\cdot 4 \cdot 6} f_{10} + \frac{1}{2\cdot 4\cdot 6\cdot 8} f_{12} + \cdots\right) \\
\sum_{|A| = 5} \prod_{i \in A} \sin\theta_i \prod_{i \notin A} \cos\theta_i & = \frac{1}{2\cdot 4} \left(f_5 + \frac{1}{2} f_7 + \frac{1}{2\cdot 4} f_9 + \frac{1}{2\cdot 4 \cdot 6} f_{11} + \frac{1}{2\cdot 4\cdot 6\cdot 8} f_{13} + \cdots\right) \\
& \vdots
\end{aligned}
$$

## What Leonhard Euler and Mary Cartwright did

[[Leonhard Euler]] used something similar to the identities for $f_0$ and $f_1$ in order to derive the [[power-series expansion]]s of the [[sine]] and [[cosine]].  The idea is that to find $\sin x$, one lets $x = \theta+\cdots+\theta$, where $\theta$ is infinitely small and the number of terms is an infinitely large integer, and then apply the identities for $f_0$ and $f_1$.  More precisely, Euler wrote $\cos(n\theta)$ and $\sin(n\theta)$ as functions of $\sin\theta$ and $\cos\theta$ and then supposed $n$ is infinitely large and $n\theta$ is finite.  Then $\sin\theta$ becomes $\theta$ and $\cos\theta$ becomes 1.

If one does with $f_k$, for $k \ge 0$, what Euler did with $f_0$ and $f_1$, one gets an infinite sequence of power series in $x$.  The odd-indexed functions of $x$ in this sequence are the functions $J_n$ that appear in Cartwright's proof of the irrationality of $\pi$.  But in defining those functions, Cartwright used neither trigonometric expressions nor power series.  (The proof was published in an appendix to the third edition of Harold Jeffreys' book ''Scientific Inference''.  It is not in later editions.  It appears in [Wikipedia's article](http://en.wikipedia.org/wiki/Proof_that_%CF%80_is_irrational) on the proof of the irrationality of $\pi$.)

## The main question

Could a proof of the irrationality of $\pi$ using only finitary trigonometric identities be lurking somewhere in all this?  ($\left\{f_k(\theta_1,\dots,\theta_\ell)\right\}_{k,\ell}$ is a doubly indexed sequence each term of which is the a sum of finitely many products of finitely many factors.)


trigonometric identity

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
### Context
#### Trigonometry
+-- {: .hide}
[[!include trigonometry -- contents]]
=--
#### Equality and Equivalence
+-- {: .hide}
[[!include equality and equivalence - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

A trigonometric identity is (formally) a [[commutative diagram]] in the [[category]] of [[cartesian spaces]] and [[partial functions]] whose edges are labelled by [[rational functions]] (or sometimes [[algebraic functions]]) and [[trigonometric functions]].

Slightly more precisely: each rational function $p(x_1, \ldots, x_n) \in \mathbb{R}(x_1, \ldots, x_n)$ is interpreted as a [[partial function]] $\mathbb{R}^n \hookleftarrow D \stackrel{f}{\to} \mathbb{R}^n$ where $D$ is the "natural domain" of $p$ (see [[rational function]] for more discussion); these are partial [[analytic functions]]. The basic trigonometric functions [[sine]] ($\sin$) and [[cosine]] ($\cos$) are (total) [[analytic functions]] $\mathbb{R} \to \mathbb{R}$. All of these may be interpreted as partial functions $\mathbb{R}^n \rightharpoonup \mathbb{R}^n$, and generate a class of functions by applying the [[monoidal category]] structure on the category of partial functions that is induced by the [[cartesian product]] on cartesian spaces. A trigonometric identity is then (formally) an [[equality]] of morphisms in the monoidal category thus generated.

Of course, this is complete overkill; category theorists are not oblivious to the fact that this is exactly the kind of description lampooned in Linderholm's _Mathematics Made Difficult_. It's just a formal way of covering bases. So let us add that in practice, a trigonometric identity usually involves functions obtained by substituting trigonometric functions into rational functions, or substituting rational _linear_ (affine) functions into trigonometric functions: the class of functions considered is usually fairly limited in scope.

## Examples

Virtually all trigonometric identities can be seen as arising from suitable [[exponential function]] identities on _[[complex numbers]]_ such as

* $\exp(i t) = \cos(t) + i\sin(t)$ ([[Euler's formula]]);

* $\exp(z + w) = \exp(z)\exp(w)$,

* $\widebar{\exp(z)} = \exp(\widebar{z})$.

These imply relations between the [[sine function]] and the [[cosine function]] as follows:

$$
  \sin\left( \alpha + \beta \right)
  \;=\;
  \sin(\alpha) \cos(\beta) + \cos(\alpha) \sin(\beta)
$$

$$
  \cos(\alpha + \beta)
  \;=\;
  \cos(\alpha) \cos(\beta) - \sin(\alpha) \sin(\beta)
$$

which in turn implies

$$
  \label{ProductOfSinWithCos}
  \sin(\alpha) \cos(\beta)
  \;=\;
  \tfrac{1}{2}
  \left(
    \sin(\alpha + \beta) + \sin(\alpha - \beta)
  \right)
$$

etc.

## Related entries

* See also [[trigonometric function]] for some discussion.

* [[trigonometry]]

## References

See also

* Wikipedia, _[List of trigonometric identities](https://en.wikipedia.org/wiki/List_of_trigonometric_identities)_

[[!redirects trigonometric identity]]
[[!redirects trigonometric identities]]


trigonometry

+-- {: .rightHandSide}
+-- {: .toc .clickDown tabindex="0"}
###Context###
#### Trigonometry
+-- {: .hide}
[[!include trigonometry -- contents]]
=--
#### Geometry
+--{: .hide}
[[!include higher geometry - contents]]
=--
=--
=--

#Contents#
* table of contents
{:toc}

## Idea

__Trigonometry__ is a geometric discipline which comprises the study of metric properties of [[triangles]] (measures of angles, sides and of various auxiliary segments and angles constructed from given triangle), usually on a euclidean or hyperbolic plane or on a [[unit sphere]]. Thus we distinguish the usual euclidean from spherical and hyperbolic trigonometry; there are also generalizations to trigonometry over other fields.

[[trigonometric function|Trigonometric functions]] for real values of argument can be defined in terms of trigonometry of unit triangle (triangle with one of the sides of unit length), where the argument is understood as an angle whose longer side is of unit length. Similarly for [[hyperbolic trigonometric functions]] which can be geometrically defined in terms of hyperbolic unit triangles.

## Related concepts

* [[Euclidean geometry]], [[hyperbolic geometry]], [[elliptic geometry]]

* [[sine]], [[cosine]], [[exponential]], [[Euler's formula]]

* [[trigonometric identity]]

## References

* Norman Wildberger, _Universal hyperbolic geometry I: trigonometry_, Geometriae Dedicata 163(1), 2009 [arxiv/0909.1377](http://arxiv.org/abs/0909.1377) [doi](http://dx.doi.org/10.1007/s10711-012-9746-9)

* wikipedia: [trigonometry](https://en.wikipedia.org/wiki/Trigonometry), [spherical trigonometry](https://en.wikipedia.org/wiki/Spherical_trigonometry)


trigonometry -- contents

**[[trigonometry]]**

* [[Euclidean space]]

* [[triangle]]

* [[angle]], [[arc length]]

* [[trigonometric function]]

  * [[cosine]]

  * [[sine]]

  * [[tangent function|tangent]]

  * [[cotangent]]

  * [[secant]]

  * [[cosecant]]

* [[inverse trigonometric function]]

  * [[arccos]]

  * [[arcsin]]

  * [[arctan]]

  * ...

* [[trigonometric identity]]

  * [[Euler's formula]]

* [[hyperbolic function]]

* [[elliptic function]]