From Matrices to Morphisms II: Examples of Computational Categories in MATLAB and Octave
Nelson Martins-Ferreira
arxiv.org/abs/2609.04982 [𝚖𝚊𝚝𝚑.𝙲𝚃]
Category Theory Papers
@CategoryPapers
New Category Theory papers from arxiv.org: enriched categories, topoi. Thank you to arXiv for use of its open access interoperability.
![This paper establishes the algebraic and categorical foundations of spectral duality structures (SDS). We show that the class of all SDS admits the structure of a graded monoidal category, in which the degree K is the fundamental invariant indexing the strata. We define the Cartesian product and the disjoint union of structures, together with morphisms preserving the involution and the weights, and we prove that C* = 1/(1+sqrt(K)) is a functor constant on each stratum. Objects are classified up to isomorphism by the combinatorial type (k,f), the degree K, and a multiset of inversion classes [r] = r, 1/r. We further prove that the response rank equals exactly the number of non-trivial pairs k, and we connect the categorical structure with the Fisher-Rao geometry developed in a companion paper. The SDS, originally identified in problems of stochastic resetting, thus emerges as an autonomous mathematical object with a rich algebraic structure and a natural geometric realisation.](https://pbs.twimg.com/media/HRl83_NaMAABOGI?format=webp&name=medium)
![In [B2] Balmer introduced the homological spectrum of a rigid tensor-triangulated category, and had the “nerves of steel” not to conjecture that it was in bijection with the prime spectrum, despite an “avalanche of examples”. Naturally, the statement came to be known as Balmer's Nerves of Steel Conjecture. More recently, in [NVY] the conjecture was stated explicitly and extended to a more general non-symmetric context. In this paper we provide a counterexample to the original suggestion in [B2], so also to the conjecture from [NVY]. A first counterexample was recently constructed in [BHR].](https://pbs.twimg.com/media/HRXDVQBbkAA_KP0?format=webp&name=medium)
