nLab
tensor triangulated category

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Homological algebra

homological algebra

(also nonabelian homological algebra)

Introduction

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diagram chasing

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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 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”.

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).

Examples

References

Review:

Last revised on January 25, 2021 at 00:50:54. See the history of this page for a list of all contributions to it.