Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userCategorical Syntax and Consequence Relations
75
0
0.0
(
0
)
Authors
Lingyuan Ye
Ask ChatGPT about the research
No Arabic abstract
In this paper, we use a categorical and functorial set up to model the syntax and inference of logics of algebraic signature, extending previous works on algebraisation of logics. The main feature of this work is that structurality, or invariance under substitution of variables, are modelled by functoriality in this paper, resulting in a much clearer framework for algebraisation. It also provides a very nice conceptual understanding of various existing results already established in the literatures, and derives several new results as well.
rate research
Read More
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 discuss the relationship between (co)homology groups and categorical diagonalization. We consider the category of chain complexes in the category of finitely generated free modules on a commutative ring. For a fixed chain complex with zero maps as an object, a chain map from the object to another chain complex is defined, and the chain map introduce a mapping cone. We found that the fixed object is isomorphic to the (co)homology groups of the codomain of the chain map if and only if the chain map is injective to the cokernel of differentials of the codomain chain complex and the mapping cone is homotopy equivalent to zero. On the other hand, the fixed object is regarded as a categorified eigenvalue of the chain complex in the context of the categorical diagonalization introduced by B.Elias and M. Hogancamp arXiv:1801.00191v1. It is found that (co)homology groups are constructed as the eigenvalue of a chain complex.
Ultrafilters are useful mathematical objects having applications in nonstandard analysis, Ramsey theory, Boolean algebra, topology, and other areas of mathematics. In this note, we provide a categorical construction of ultrafilters in terms of the inverse limit of an inverse family of finite partitions; this is an elementary and intuitive presentation of a consequence of the profiniteness of Stone spaces. We then apply this construction to answer a question of Rosinger posed in arXiv:0709.0084v2 in the negative.
Markov categories are a recent category-theoretic approach to the foundations of probability and statistics. Here we develop this approach further by treating infinite products and the Kolmogorov extension theorem. This is relevant for all aspects of probability theory in which infinitely many random variables appear at a time. These infinite tensor products $bigotimes_{i in J} X_i$ come in t
We consider Ribenboims construction of rings of generalized power series. Ribenboims construction makes use of a special class of partially ordered monoids and a special class of their subsets. While the restrictions he imposes might seem conceptually unclear, we demonstrate that they are precisely the appropriate conditions to represent such monoids as internal monoids in an appropriate category of Ehrhards finiteness spaces. Ehrhard introduced finiteness spaces as the objects of a categorical model of classical linear logic, where a set is equipped with a class of subsets to be thought of as finitary. Morphisms are relations preserving the finitary structure. The notion of finitary subset allows for a sharper analysis of computational structure than is available in the relational model. For example, fixed point operators fail to be finitary. In the present work, we take morphisms to be partial functions preserving the finitary structure rather than relations. The resulting category is symmetric monoidal closed, complete and cocomplete. Any pair of an internal monoid in this category and a ring induces a ring of generalized power series by an extension of the Ribenboim construction based on Ehrhards notion of linearization of a finiteness space. We thus further generalize Ribenboims constructions. We give several examples of rings which arise from this construction, including the ring of Puiseux series and the ring of formal power series generated by a free monoid.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
