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Triangle Functor

A triangle functor between triangulated categories C and D is an additive functor F:C->D equipped with a natural isomorphism whose components are phi_X:F(X[1])->F(X)[1] and which preserves distinguished triangles. For a distinguished triangle with maps u:X->Y, v:Y->Z, and w:Z->X[1], its image

F(X)->^(F(u))F(Y)->^(F(v))F(Z)->^(phi_X degreesF(w))F(X)[1]

is a distinguished triangle in D. The notation [1] denotes the shift functor in the respective category (The Stacks Project Authors 2026).

A natural transformation alpha:F->G between triangle functors (F,phi) and (G,psi) is compatible with their triangle structures when

psi_X degreesalpha_(X[1])=alpha_X[1] degreesphi_X.

Here alpha_X[1] is the shift of the morphism alpha_X. An isomorphism of triangle functors is a natural isomorphism satisfying this condition (The Stacks Project Authors 2026).

See also

Additive Functor, Functor, Natural Isomorphism, Natural Transformation, Triangulated Category

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References

The Stacks Project Authors. "The Definition of a Triangulated Category." §13.3 in The Stacks Project, Tag 014V, 2026. https://stacks.math.columbia.edu/tag/05QK.

Cite this as:

Weisstein, Eric W. "Triangle Functor." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/TriangleFunctor.html

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