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1. Local cohomology and support for triangulated categories

   Benson, Dave; Iyengar, Srikanth B.; Krause, Henning (January 2008, Annales Scientifiques de l'Ecole Normale Supérieure)

   We propose a new method for defining a notion of support for objects in any compactly generated triangulated category admitting small coproducts. This approach is based on a construction of local cohomology functors on triangulated categories, with respect to a central ring of operators. Special cases are, for example, the theory for commutative noetherian rings due to Foxby and Neeman, the theory of Avramov and Buchweitz for complete intersection local rings, and varieties for representations of finite groups according to Benson, Carlson, and Rickard. We give explicit examples of objects, the triangulated support and cohomological support of which differ. In the case of group representations, this allows us to correct and establish a conjecture of Benson. 

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2. Noncommutative Tensor Triangular Geometry and the Tensor Product Property for Support Maps

   https://doi.org/10.1093/imrn/rnab221  

   Nakano, Daniel K; Vashaw, Kent B; Yakimov, Milen T (August 2021, International Mathematics Research Notices)

   Abstract The problem of whether the cohomological support map of a finite dimensional Hopf algebra has the tensor product property has attracted a lot of attention following the earlier developments on representations of finite group schemes. Many authors have focused on concrete situations where positive and negative results have been obtained by direct arguments. In this paper we demonstrate that it is natural to study questions involving the tensor product property in the broader setting of a monoidal triangulated category. We give an intrinsic characterization by proving that the tensor product property for the universal support datum is equivalent to complete primeness of the categorical spectrum. From these results one obtains information for other support data, including the cohomological one. Two theorems are proved giving compete primeness and non-complete primeness in certain general settings. As an illustration of the methods, we give a proof of a recent conjecture of Negron and Pevtsova on the tensor product property for the cohomological support maps for the small quantum Borel algebras for all complex simple Lie algebras. 

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3. Homotopy Theoretic and Categorical Models of Neural Information Networks

   https://doi.org/10.46298/compositionality-6-4  

   Manin, Yuri; Marcolli, Matilde (September 2024, Compositionality)

   In this paper we develop a novel mathematical formalism for the modeling ofneural information networks endowed with additional structure in the form ofassignments of resources, either computational or metabolic or informational.The starting point for this construction is the notion of summing functors andof Segal's Gamma-spaces in homotopy theory. The main results in this paperinclude functorial assignments of concurrent/distributed computingarchitectures and associated binary codes to networks and their subsystems, acategorical form of the Hopfield network dynamics, which recovers the usualHopfield equations when applied to a suitable category of weighted codes, afunctorial assignment to networks of corresponding information structures andinformation cohomology, and a cohomological version of integrated information. 

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4. COHOMOLOGICAL MACKEY 2-FUNCTORS

   https://doi.org/10.1017/S1474748022000408  

   Balmer, Paul; Dell’Ambrogio, Ivo (January 2024, Journal of the Institute of Mathematics of Jussieu)

   Abstract We show that the bicategory of finite groupoids and right-free permutation bimodules is a quotient of the bicategory of Mackey 2-motives introduced in [2], obtained by modding out the so-called cohomological relations. This categorifies Yoshida’s theorem for ordinary cohomological Mackey functors and provides a direct connection between Mackey 2-motives and the usual blocks of representation theory. 

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5. Bi-incomplete Tambara functors

   Blumberg, Andrew J.; Hill, Michael A. (January 2021, Equivariant Topology and Derived Algebra)

   Balchin, S.; Barnes, D.; Kędziorek, M.; Szymik, M. (Ed.)

   For an equivariant commutative ring spectrum R, \pi_0 R has algebraic structure reflecting the presence of both additive transfers and multiplicative norms. The additive structure gives rise to a Mackey functor and the multiplicative structure yields the additional structure of a Tambara functor. If R is an N_\infty ring spectrum in the category of genuine G-spectra, then all possible additive transfers are present and \pi_0 R has the structure of an incomplete Tambara functor. However, if R is an N_\infty ring spectrum in a category of incomplete G-spectra, the situation is more subtle. In this chapter, we study the algebraic theory of Tambara structures on incomplete Mackey functors, which we call bi-incomplete Tambara functors. Just as incomplete Tambara functors have compatibility conditions that control which systems of norms are possible, bi-incomplete Tambara functors have algebraic constraints arising from the possible interactions of transfers and norms. We give a complete description of the possible interactions between the additive and multiplicative structures. 

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