# nLab tensor triangulated category

### Context

#### Homological algebra

homological algebra

Introduction

diagram chasing

### Theorems

#### Stable homotopy theory

stable homotopy theory

Introduction

# Contents

## Idea

A tensor triangulated category is a category that carries the structure of a symmetric monoidal category and of a triangulated category in a compatible way.

## Definition

###### Definition

A tensor triangulated category is a category $Ho$ equipped with

1. the structure of a symmetric monoidal category $(Ho, \otimes, 1, \tau)$ (“tensor category”);

2. the structure of a triangulated category $(Ho, \Sigma, CofSeq)$

3. for all objects $X,Y\in Ho$ natural isomorphisms

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

such that

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

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

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

in $CofSeq$, then also

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

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

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

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

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

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

2. (graded commutativity) for all $n_1, n_2 \in \mathbb{Z}$ the following diagram commutes

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

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

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

## Examples

Review is for instance in

• Greg Stevenson, Tensor actions and locally complete intersection PhD thesis 2011 (pdf)