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Idempotents in triangulated monoidal categories

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arxiv 1703.01001 v1 pith:QXMGWBWQ submitted 2017-03-03 math.CT

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keywords mathbfidempotentsmonoidalalgebracategoriescohomologyidempotentoperatorname
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abstract

In these notes we develop some basic theory of idempotents in monoidal categories. We introduce and study the notion of a pair of complementary idempotents in a triangulated monoidal category, as well as more general idempotent decompositions of identity. If $\mathbf{E}$ is a categorical idempotent then $\operatorname{End}(\mathbf{E})$ is a graded commutative algebra. The same is true of $\operatorname{Hom}(\mathbf{E},\mathbf{E}^c[1])$ under certain circumstances, where $\mathbf{E}^c$ is the complement. These generalize the notions of cohomology and Tate cohomology of a finite dimensional Hopf algebra, respectively.

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Cited by 1 Pith paper

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  1. Stable deformed $\mathfrak{gl}_N$ homology of torus knots

    math.GT 2025-06 accept novelty 7.0 of 10

    The authors compute the E2 page of the Rasmussen spectral sequence for stable gl_N Khovanov-Rozansky homology of torus knots, verifying the predicted algebraic description for all N.

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