# nLab tensor triangulated category

Contents

### 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

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

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

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

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”.

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

Review: