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Register a new userStrong shift equivalence as a category notion
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Added by
Emmanuel Jeandel
Publication date
2021
fields
Informatics Engineering
and research's language is
English
Authors
Emmanuel Jeandel
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No Arabic abstract
In this paper, we present a completely radical way to investigate the main problem of symbolic dynamics, the conjugacy problem, by proving that this problem actually relates to a natural question in category theory regarding the theory of traced bialgebras. As a consequence of this theory, we obtain a systematic way of obtaining new invariants for the conjugacy problem by looking at existing bialgebras in the literature.
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In the present paper we propose a new approach to quantum fields in terms of category algebras and states on categories. We define quantum fields and their states as category algebras and states on causal categories with partial involution structures. By utilizing category algebras and states on categories instead of simply considering categories, we can directly integrate relativity as a category theoretic structure and quantumness as a noncommutative probabilistic structure. Conceptual relationships with conventional approaches to quantum fields, including Algebraic Quantum Field Theory (AQFT) and Topological Quantum Field Theory (TQFT), are also be discussed.
We give a new characterization of silting subcategories in the stable category of a Frobenius extriangulated category, generalizing the result of Di et al. (J. Algebra 525 (2019) 42-63) about the Auslander-Reiten type correspondence for silting subcategories over triangulated categories. More specifically, for any Frobenius extriangulated category $mathcal{C}$, we establish a bijective correspondence between silting subcategories of the stable category $underline{mathcal{C}}$ and certain covariantly finite subcategories of $mathcal{C}$. As a consequence, a characterization of silting subcategories in the stable category of a Frobenius exact category is given. This result is applied to homotopy categories over abelian categories with enough projectives, derived categories over Grothendieck categories with enough projectives as well as to the stable category of Gorenstein projective modules over a ring $R$.
We make some beginning observations about the category $mathbb{E}mathrm{q}$ of equivalence relations on the set of natural numbers, where a morphism between two equivalence relations $R,S$ is a mapping from the set of $R$-equivalence classes to that of $S$-equivalence classes, which is induced by a computable function. We also consider some full subcategories of $mathbb{E}mathrm{q}$, such as the category $mathbb{E}mathrm{q}(Sigma^0_1)$ of computably enumerable equivalence relations (called ceers), the category $mathbb{E}mathrm{q}(Pi^0_1)$ of co-computably enumerable equivalence relations, and the category $mathbb{E}mathrm{q}(mathrm{Dark}^*)$ whose objects are the so-called dark ceers plus the ceers with finitely many equivalence classes. Although in all these categories the monomorphisms coincide with the injective morphisms, we show that in $mathbb{E}mathrm{q}(Sigma^0_1)$ the epimorphisms coincide with the onto morphisms, but in $mathbb{E}mathrm{q}(Pi^0_1)$ there are epimorphisms that are not onto. Moreover, $mathbb{E}mathrm{q}$, $mathbb{E}mathrm{q}(Sigma^0_1)$, and $mathbb{E}mathrm{q}(mathrm{Dark}^*)$ are closed under finite products, binary coproducts, and coequalizers, but we give an example of two morphisms in $mathbb{E}mathrm{q}(Pi^0_1)$ whose coequalizer in $mathbb{E}mathrm{q}$ is not an object of $mathbb{E}mathrm{q}(Pi^0_1)$.
We provide an intrinsic definition of the fundamental group of a linear category over a ring as the automorphism group of the fibre functor on Galois coverings. If the universal covering exists, we prove that this group is isomorphic to the Galois group of the universal covering. The grading deduced from a Galois covering enables us to describe the canonical monomorphism from its automorphism group to the first Hochschild-Mitchell cohomology vector space.
We introduce the category of finite strings and study its basic properties. The category is closely related to the augmented simplex category, and it models categories of linear representations. Each lattice of non-crossing partitions arises naturally as a lattice of subobjects.
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