tensor triangulated category


Homological algebra

homological algebra

(also nonabelian homological algebra)



Basic definitions

Stable homotopy theory notions



diagram chasing

Homology theories


Stable homotopy theory



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.



A tensor triangulated category is a category HoHo equipped with

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

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

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

    e X,Y:(ΣX)YΣ(XY) 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 VV the functor V()()VV \otimes (-) \simeq (-) \otimes V is an additive functor;

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

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

    in CofSeqCofSeq, then also

    VXid VfVYid VgVY/Xid VhV(ΣX)Σ(VX) 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 CofSeqCofSeq, where the equivalence at the end is e X,Vτ V,ΣXe_{X,V}\circ \tau_{V, \Sigma X}.

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

(Balmer 05, def. 1.1)

In addition one may ask that

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

    (Σ(X)Y)Z e X,Yid (Σ(XY))Z e XY,Z Σ((XY)Z) α ΣX,Y,Z Σα X,Y,Z Σ(X)(YZ) e X,YZ Σ(X(YZ)) \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 CofSeqCofSeq, where α\alpha is the associator of (Ho,,1)(Ho, \otimes, 1).

  2. (graded commutativity) for all n 1,n 2n_1, n_2 \in \mathbb{Z} the following diagram commutes

    (Σ n 11)(Σ n 21) Σ n 1+n 21 τ Σ n 11,Σ n 21 (1) n 1n 2 (Σ n 21)(Σ n 11) Σ n 1+n 21, \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 ,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”.



Review is for instance in

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

Last revised on May 17, 2017 at 23:03:43. See the history of this page for a list of all contributions to it.