Foundations Of Category Theory Research Articles

Mac Lane\u2019s Comparison Theorem for the Kleisli Construction Formalized in Coq

(co)Monads are used to encapsulate impure operations of a computation. A (co)monad is determined by an adjunction and further determines a specific type of adjunction called the (co)Kleisli adjunction. Mac Lane introduced the comparison theorem which allows comparing these adjunctions bridged by a (co)monad through a unique comparison functor. In this paper we specify the foundations of category theory in Coq and show that the chosen representations are useful by certifying Mac Lane’s comparison theorem and its basic consequences. We also show that the foundations we use are equivalent to the foundations by Timany. The formalization makes use of Coq classes to implement categorical objects and the axiom uniqueness of identity proofs to close the gap between the contextual equality of objects in a categorical setting and the judgmental Leibniz equality of Coq. The theorem is used by Duval and Jacobs in their categorical settings to interpret the state effect in impure programming languages.

Approximating Cartesian Closed Categories in NF-Style Set Theories

I criticize, but uphold the conclusion of, an argument by McLarty to the effect that New Foundations style set theories don’t form a suitable foundation for category theory. McLarty’s argument is from the fact that Set and Cat are not Cartesian closed in NF-style set theories. I point out that these categories do still have a property approximating Cartesian closure, making McLarty’s argument not conclusive. After considering and attempting to address other problems with developing category theory in NF-style set theories, I conclude that NF-style set theories are not a good foundation for category theory, because of numerous limitations introduced by their stratification restrictions.

NEITHER CATEGORICAL NOR SET-THEORETIC FOUNDATIONS

AbstractFirst we review highlights of the ongoing debate about foundations of category theory, beginning with Feferman’s important article of 1977, then moving to our own paper of 2003, contrasting replies by McLarty and Awodey, and our own rejoinders to them. Then we offer a modest proposal for reformulating a theory of category of categories that would actually meet the objections of Feferman and Hellman and provide a genuine alternative to set theory as a foundation for mathematics. This proposal is more modest than that of our (2003) in omitting modal logic and in permitting a more “top-down” approach, where particular categories and functors need not be explicitly defined. Possible reasons for resisting the proposal are offered and countered. The upshot is to sustain a pluralism of foundations along lines actually foreseen by Feferman (1977), something that should be welcomed as a way of resolving this long-standing debate.

A Systems Theoretical Formal Logic for Category Theory

A systems theoretical thinking on the categorial object and morphism is developed, leading to a reflection on the philosophical and mathematical foundations of category theory, which allows for the introduction of a formal language for category theory and of a categorial calculus as a morphic web-based logical calculus. A formal system, built from such calculus, is proposed and the logical semantics is addressed. Both syntax and semantics are independent from set theory.

Local theory of sets as a foundation for category theory and its connection with the Zermelo-Fraenkel set theory

Local theory of sets as a foundation for category theory and its connection with the Zermelo-Fraenkel set theory

The MacLane Problem on the Set-Theoretic Foundation for Category Theory. Ii

The crisis that arose in naive set theory at the beginning of 20th century led to the appearance of several precise axiomatic constructions of theories of mathematical totalities. Among them, set theory in Zermelo–Frankel’s axiomatics (ZF) (see [1]) and the class theory and set theory in the axiomatics of Neumann–Bernays–Goedel (NBG) (see [2] and [3]) are the most widely used. These axiomatic theories remove all known paradoxes of naive set theory at the expense of a sharp restriction of possible representational tools. At the same time, they give the possibility of involving almost all mathematical objects and constructions existing at that time in the framework of these theories. However, even in 1945, in the fundamental work [4], Eilenberg and MacLane introduced a new mathematical concept of a category. Since that time, category theory has become an independent field of mathematics. However, from the very beginning, category theory encoutered the unpleasant fact that it cannot be included in the framework of the axiomatic set theory ZF and that of the axiomatic theory of classes and sets NBG (see [4]). Therefore, MacLane in [5], written in 1959, has posed the general problem of constructing a new and more flexible axiomatic set theory which can serve as an adequate logical foundation of the whole naive category theory. Variants of new axiomatic theories of mathematical totalities adopted for one or another need of category theory were suggested by Ehresmann [6], Dedecker [7], Sonner [8], Grotendieck [9], da Costa [10] and [11], Isbell [12], MacLane [13] and [14], Fefferman [15], Herrlich and Strecker [16], and others. The axiomatics of Ehresmann–Dedecker and Sonner–Grotendieck contains a very strong axiom in addition to the axioms of ZF, which asserts that every set is contained in a certain universal set. In the axiomatics of MacLane, it is assumed that there exists at least one universal set. Versions close to them were suggested by Isbell and Feferman. The axiomatics of Herlich–Strecker deals with objects of the following three types: sets, classes, and conglomerates. Within the framework of each of these axiomatic theories, some definitions of a category and a functor were given. However, the concept of a category given in [12–16] is not closed under important operations of naive set theory such as the “category of categories” and the “category of functors” (see [17, 8.4]). Within the framework of the axiomatic theories in [6–9], the definitions of a U -category and a U functor which consist of elements of a universal set U were given. Such a concept of a category is closed under operations such as the V-category of U-categories and the V-category of U -functors, where V is a certain universal set containing the universal set U . However, U -categories can contain objects of a bounded cardinality, which is not natural from the point of view of naive category theory. Therefore, in [13], MacLane noted the following about this approach: “This stronger assumption evidently provides for each universe U a category of all those groups which are members of U . However, this does not provide any category of all groups.” The second axiomatics of da Costa in [10] and [11] has the same shortcoming. His first axiomatics in [10] and [11] is free from this shortcoming, and, at the same time, it is closed with respect to the operation of naive category theory mentioned above. Therefore, it is the most preferable among all the

Failure of Cartesian closedness in NF

On any reasonable definition of functions, neither the category of sets nor the category of small categories is cartesian closed in New Foundations (NF). The latter category is sometimes proposed as a foundation for category theory since it is among its own objects. Our result shows it is a poor one.In NF, as in other set theories, a "function" f from a set A to a set B is defined to be a set f of ordered pairs 〈x, y〉 with x in A and y in B, such that (a) if 〈x, y〉 ∈ f and 〈x, y′〉 ∈ f then y = y′, and (b) for every x in A there is some y in B with 〈x, y〉 ∈ f. But in NF different definitions of ordered pairs give significantly different functions. I say a reasonable definition must give:1. The formula z = 〈x, y〉 is stratifiable.2. For every set S there is a set {〈x, x〉 ∣ x ∈ S}.3. If f is a function from A to B, and g one from B to C, there is a set {〈x, z〉∣(∃y)〈x, y〉∈ f & 〈y, z〉∈ g}.Principles 2 and 3 are needed for identity functions and composites. By principle 1, any sets A and B have a set A × B of all ordered pairs 〈x, y〉 with x in A and y in B, but it does not follow that functions exist making A × B a categorical product of A and B.

Základy teorie kategorií [Výtah z přednášky A. G. Kuroše

Základy teorie kategorií [Výtah z přednášky A. G. Kuroše