nLab tensor triangulated category

Idea

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

(For a compatible non-symmetric monoidal category structure on a triangulated categories, some authors speak of monoidal triangulated categories, cf. Rowe 2024.)

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 $V$.

(Balmer 05, def. 1.1)

In addition one may ask that

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 1997, 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 2001.

Examples

• The archetypical example is the stable homotopy category equipped with the smash product of spectra. For details see at model structure on orthogonal spectra the section The monoidal stable homotopy category.

Related concepts

• spectrum of a tensor triangulated category

• tensor category, triangulated category

• tensor (∞,1)-category

References

Tensor triangulated categories

• Mark Hovey, John Palmieri, Neil Strickland, Axiomatic stable homotopy theory, Memoirs of the AMS 610 (1997) (pdf, doi:10.1090/memo/0610)

• Paul Balmer, The spectrum of prime ideals in tensor triangulated categories, J. Reine Angew. Math., 588:149–168, 2005 (arXiv:0409360, doi:10.1515/crll.2005.2005.588.149)

• Peter May, The Additivity of Traces in Triangulated Categories, Advances in Mathematics, Volume 163, Issue 1, 2001, Pages 34-73 (doi:10.1006/aima.2001.1995)

Review:

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

• Paul Balmer, A guide to tensor-triangular classification, in: Haynes Miller (ed.) Handbook of Homotopy Theory (arXiv:1912.08963)

Monoidal triangulated categories

On non-symmetric monoidal category structure on triangulated categories (“monoidal triangulated (mt) categories” in the terminology of Rowe 2024), just asing the tensor product to be an exact functor in each variable:

• Daniel K Nakano, Kent B Vashaw, Milen T Yakimov; §2.1 in: Noncommutative Tensor Triangular Geometry and the Tensor Product Property for Support Maps Free, International Mathematics Research Notices, Volume 2022, Issue 22, (2022) 17766–17796 [doi:10.1093/imrn/rnab221]

• Daniel K. Nakano: Monoidal Triangular Geometry with Applications to Representation Theory, talk slides [pdf]

• Hongdi Huang, Kent B. Vashaw: Group actions on monoidal triangulated categories and Balmer spectra, Documenta Mathematica, 30 5 (2025) 1055–1083 [doi:10.4171/DM/1016, arXiv:2311.18638]

• James Rowe; §2 in: Noncommutative tensor triangular geometry: classification via Noetherian spectra, Pacific Journal of Mathematics, 330 (2024) 355–371 [doi:10.2140/pjm.2024.330.355]

• Daniel K. Nakano, Kent B. Vashaw, Milen T. Yakimov: The homological spectrum for monoidal triangulated categories [arXiv:2506.19946]

In these articles, asking the tensor product just to be biexact appears to be sufficient for the intended application to nonsymmetric spectra of tensor triangulated categories. But beyond that it would seem natural to demand more coherence, for instance asking the tensor product to be a triangulated bifunctor in the sense of:

• Masaki Kashiwara, Pierre Schapira; Def. 10.3.6, 10.1.1(v) in: Categories and Sheaves, Grundlehren der Mathematischen Wissenschaften 332, Springer (2006) [doi:10.1007/3-540-27950-4, pdf]