Category-theoretic Structures Research Articles

Category-theoretic Structure for Independence and Conditional Independence

Relations of independence and conditional independence arise in a variety of contexts. Stochastic independence and conditional independence are fundamental relations in probability theory and statistics. Analogous non-stochastic relations arise in database theory; in the setting of nominal sets (a semantic framework for modelling data with names); and in the modelling of concepts such as region disjointness for heap memory. In this paper, we identify unifying category-theoretic structure that encompasses these different forms of independence and conditional independence. The proposed structure supports the expected reasoning principles for notions of independence and conditional independence. We further identify associated notions of independent and local independent product, in which (conditional) independence is represented via a (fibred) monoidal structure, which is present in many examples.

Category theoretic structure of setoids

A setoid is a set together with a constructive representation of an equivalence relation on it. Here, we give category theoretic support to the notion. We first define a category Setoid and prove it is Cartesian closed with coproducts. We then enrich it in the Cartesian closed category Equiv of sets and classical equivalence relations, extend the above results, and prove that Setoid as an Equiv-enriched category has a relaxed form of equalisers. We then recall the definition of E-category, generalising that of Equiv-enriched category, and show that Setoid as an E-category has a relaxed form of coequalisers. In doing all this, we carefully compare our category theoretic constructs with Agda code for type-theoretic constructs on setoids.

Toward Foundations of Near Sets: (Pre-)Sheaf Theoretic Approach

The formal content of near set theory can be summarised in terms of three concepts: a perceptual system, a nearness relation and a near set. Perceptual systems and different forms of nearness relations have been already successfully related to important mathematical structures (e.g., approach spaces) and described in the frameworks of general topology and category theory. However, since near sets actually do not form any regular structure, there is lack of similar results about the concept of a near set. The main goal of this paper is to fill this gap and provide a mathematical basis for near sets on a similar abstract level as it has been already done for nearness relations. However, we take in the paper a down-to-earth approach; that is, instead of seeking the richest mathematical structures which express our intuitions (call it up-to-sky approach), we present in the paper the simplest category theoretic structures which, however, are rich enough to convey our ideas. Thus, although many times we actually deal with sheaves, we speak about them as pre-sheaves. The main reason is that a sheaf (in contrast to a pre-sheaf) embodies the idea of gluing (in the very similar way like manifolds, which are obtainable by gluing open subsets of Euclidean space), which is irrelevant to the study of near sets and rough sets. Furthermore, the concept a pre-sheaf is much simpler than a sheaf and should be easily “digested” by readers who are not very familiar with category theory.

Category-theoretic structure and radical ontic structural realism

Radical ontic structural realism (ROSR) claims that structure exists independently of objects that may instantiate it. Critics of ROSR contend that this claim is conceptually incoherent, insofar as, (i) it entails there can be relations without relata, and (ii) there is a conceptual dependence between relations and relata. In this essay I suggest that (ii) is motivated by a set-theoretic formulation of structure, and that adopting a category-theoretic formulation may provide ROSR with more support. In particular, I consider how a category-theoretic formulation of structure can be developed that denies (ii), and can be made to do work in the context of formulating theories in physics.

Pseudo-distributive laws and axiomatics for variable binding

We give a general category theoretic formulation of the substitution structure underlying the category theoretic study of variable binding proposed by Fiore, Plotkin, and Turi. This general formulation provides the foundation for their work on variable binding, as well as Tanaka's linear variable binding and variable binding for other binders and for mixtures of binders as for instance in the Logic of Bunched Implications. The key structure developed by Fiore et al. was a substitution monoidal structure, from which their formulation of binding was derived; so we give an abstract formulation of a substitution monoidal structure, then, at that level of generality, derive the various category theoretic structures they considered. The central construction we use is that of a pseudo-distributive law between 2-monads on Cat, which suffices to induce a pseudo-monad on Cat, and hence a substitution monoidal structure on the free object on 1. We routinely generalise that construction to account for types.

A Category Theoretic Formulation for Engeler-style Models of the Untyped λ-Calculus

We give a category-theoretic formulation of Engeler-style models for the untyped λ-calculus. In order to do so, we exhibit an equivalence between distributive laws and extensions of one monad to the Kleisli category of another and explore the example of an arbitrary commutative monad together with the monad for commutative monoids. On Set as base category, the latter is the finite multiset monad. We exploit the self-duality of the category Rel, i.e., the Kleisli category for the powerset monad, and the category theoretic structures on it that allow us to build models of the untyped λ-calculus, yielding a variant of the Engeler model. We replace the monad for commutative monoids by that for idempotent commutative monoids, which, on Set, is the finite powerset monad. This does not quite yield a distributive law, so requires a little more subtlety, but, subject to that subtlety, it yields exactly the original Engeler construction.

Semi-graphs of Anabelioids

In this paper, we discuss various “general nonsense” aspects of the geometry of semi-graphs of profinite groups [cf. [Mzk3], Appendix], by applying the language of anabelioids introduced in [Mzk4]. After proving certain basic properties concerning various commensurators associated to a semi-graph of anabelioids , we show that the geometry of a semi-graph of anabelioids may be recovered from the category-theoretic structure of certain naturally associated categories — e.g., “ temperoids ” [in essence, the analogue of a Galois category for the “tempered fundamental groups” of [André]] and “ categories of localizations ”. Finally, we apply these techniques to obtain certain results in the absolute anabelian geometry [cf. [Mzk3], [Mzk8]] of tempered fundamental groups associated to hyperbolic curves over p-adic local fields.

A representable approach to finite nondeterminism

We reformulate denotational semantics for nondeterminism, taking a nondeterministic operation V on programs, and sequential composition, as primitive. This gives rise to binary trees. We analyse semantics for both type and program constructors such as products and exponential types, conditionals and recursion, in this setting. In doing so, we define new category-theoretic structures, in particular premonoidal categories. We also account for equivalences of programs such as those induced by associativity, symmetry and idempotence of V, and we study finite approximation by enrichment over the category of ω-cpos with least element. We also show how to recover the classical powerdomains, especially the convex powerdomain, as three instances of a general, computationally natural, construction.

Lax naturality through enrichment

We develop the relationship between algebraic structure and monads enriched over the monoidal bicloscd category LocOrd of small locally ordered categories, with closed structure given by Lax( A, B). We state the theorem, give a series of examples, and incorporate an account of sketches and contravariance into the theory. This was motivated by C.A.R. Hoare's use of category theoretic structures to model data refinement.

A note on russell's paradox in locally cartesian closed categories

Working in the fragment of Martin-Lofs extensional type theory [12] which has products (but not sums) of dependent types, we consider two additional assumptions: firstly, that there are (strong) equality types; and secondly, that there is a type which is universal in the sense that terms of that type name all types, up to isomorphism. For such a type theory, we give a version of Russell's paradox showing that each type possesses a closed term and (hence) that all terms of each type are provably equal. We consider the kind of category theoretic structure which corresponds to this kind of type theory and obtain a categorical version of the paradox. A special case of this result is the degeneracy of a locally cartesian closed category with a morphism which is generic in the sense that every other morphism in the category can be obtained from it via pullback.