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\begin{document}

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\section*{tensor triangulated category}

\hypertarget{context}{}\subsubsection*{{Context}}\label{context}

\hypertarget{homological_algebra}{}\paragraph*{{Homological algebra}}\label{homological_algebra}

[[!include homological algebra - contents]]

\hypertarget{stable_homotopy_theory}{}\paragraph*{{Stable homotopy theory}}\label{stable_homotopy_theory}

[[!include stable homotopy theory - contents]]

\hypertarget{contents}{}\section*{{Contents}}\label{contents}

\noindent\hyperlink{idea}{Idea}\dotfill \pageref*{idea} \linebreak
\noindent\hyperlink{definition}{Definition}\dotfill \pageref*{definition} \linebreak
\noindent\hyperlink{examples}{Examples}\dotfill \pageref*{examples} \linebreak
\noindent\hyperlink{related_concepts}{Related concepts}\dotfill \pageref*{related_concepts} \linebreak
\noindent\hyperlink{references}{References}\dotfill \pageref*{references} \linebreak


\hypertarget{idea}{}\subsection*{{Idea}}\label{idea}

A \emph{tensor triangulated category} is a [[category]] that carries the structure of a [[symmetric monoidal category]] and of a [[triangulated category]] in a compatible way.

\hypertarget{definition}{}\subsection*{{Definition}}\label{definition}

\begin{defn}
\label{}\hypertarget{}{}
A \textbf{tensor triangulated category} is a [[category]] $Ho$ equipped with

\begin{enumerate}%
\item the structure of a [[symmetric monoidal category]] $(Ho, \otimes, 1, \tau)$ (``[[tensor category]]'');


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


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

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


\end{enumerate}
such that

\begin{enumerate}%
\item (tensor product is additive) for each object $V$ the functor $V \otimes (-) \simeq (-) \otimes V$ is an [[additive functor]];


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

\begin{displaymath}
X 
    \overset{f}{\longrightarrow}
  Y
    \overset{g}{\longrightarrow}
  Y/X
    \overset{h}{\longrightarrow}
  \Sigma X
\end{displaymath}
in $CofSeq$, then also

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



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

\end{defn}
(\hyperlink{Balmer05}{Balmer 05, def. 1.1})

In addition one may ask that

\begin{enumerate}%
\item (coherence) for all $X, Y, Z \in Ho$ the following [[commuting diagram|diagram commutes]]

\begin{displaymath}
\itexarray{
    ( \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) )
  }
\end{displaymath}
is in $CofSeq$, where $\alpha$ is the [[associator]] of $(Ho, \otimes, 1)$.


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

\begin{displaymath}
\itexarray{
    (\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
  }
  \,,
\end{displaymath}
where the horizontal isomorphisms are composites of the $e_{\cdot,\cdot}$ and the braidings.



\end{enumerate}
This is (\hyperlink{HoveyPalmieriStrickland97}{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''.

\hypertarget{examples}{}\subsection*{{Examples}}\label{examples}

\begin{itemize}%
\item The archetypical example is the [[stable homotopy category]] equipped with the [[smash product of spectra]]. For details see at \emph{[[model structure on orthogonal spectra]]} the section \emph{\href{model+structure+on+orthogonal+spectra#TheMonoidalStableHomotopyCategory}{The monoidal stable homotopy category}}.

\end{itemize}
\hypertarget{related_concepts}{}\subsection*{{Related concepts}}\label{related_concepts}

\begin{itemize}%
\item [[spectrum of a tensor triangulated category]]

\end{itemize}
\hypertarget{references}{}\subsection*{{References}}\label{references}

\begin{itemize}%
\item [[Mark Hovey]], [[John Palmieri]], [[Neil Strickland]], \emph{Axiomatic stable homotopy theory}, Memoirs of the AMS 610 (1997) (\href{https://www.math.rochester.edu/people/faculty/doug/otherpapers/axiomatic.pdf}{pdf})


\item [[Paul Balmer]], \emph{The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math., 588:149--168, 2005 (\href{http://arxiv.org/abs/math/0409360}{arXiv:0409360})}



\end{itemize}
Review is for instance in

\begin{itemize}%
\item [[Greg Stevenson]], \emph{Tensor actions and locally complete intersection} PhD thesis 2011 (\href{http://www.math.uni-bielefeld.de/~gstevens/Stevenson_thesis.pdf}{pdf})

\end{itemize}
[[!redirects tensor triangulated categories]]



\end{document}