Related papers: Defining Functions on Equivalence Classes
Taking a quotient roughly means changing the notion of equality on a given object, set or type. In a quantitative setting, equality naturally generalises to a distance, measuring how much elements are similar instead of just stating their…
We apply some tools developed in categorical logic to give an abstract description of constructions used to formalize constructive mathematics in foundations based on intensional type theory. The key concept we employ is that of a Lawvere…
Quotients and comprehension are fundamental mathematical constructions that can be described via adjunctions in categorical logic. This paper reveals that quotients and comprehension are related to measurement, not only in quantum logic,…
The filter quotient construction is a particular instance of a filtered colimit of categories. It has primarily been considered in the context of categorical logic, where it has been used effectively to construct non-trivial models, for…
This work is meant to be a step towards the formal definition of the notion of algorithm, in the sense of an equivalence class of programs working "in a similar way". But instead of defining equivalence transformations directly on programs,…
We investigate two constructive approaches to defining quasi-compact and quasi-separated schemes (qcqs-schemes), namely qcqs-schemes as locally ringed lattices and as functors from rings to sets. We work in Homotopy Type Theory and…
We construct an A_infinity-category D(C|B) from a given A_infinity-category C and its full subcategory B. The construction is similar to a particular case of Drinfeld's quotient of differential graded categories. We use D(C|B) to construct…
The notion of a categorical quotient can be generalized since its standard categorical concept does not recover the expected quotients in certain categories. We present a more general formulation in the form of $\mathcal{F}$-quotients in a…
Presentations of categories are a well-known algebraic tool to provide descriptions of categories by means of generators, for objects and morphisms, and relations on morphisms. We generalize here this notion, in order to consider situations…
It is well known that the R, the set of real numbers, is an abstract set, where almost all its elements cannot be described in any finite language. We investigate possible approaches to what might be called an epi-constructionist approach…
We define notions of generically and coarsely computable relations and structures and functions between structures. We investigate the existence and uniqueness of equivalence structures in the context of these definitions
We extend the notion of exact completion on a weakly lex category to elementary doctrines. We show how any such doctrine admits an elementary quotient completion, which freely adds effective quotients and extensional equality. We note that…
The mathematical formulation of Quantum Mechanics is derived from purely operational axioms based on a general definition of "experiment" as a set of transformations. The main ingredient of the mathematical construction is the postulated…
Many algorithms use concrete data types with some additional invariants. The set of values satisfying the invariants is often a set of representatives for the equivalence classes of some equational theory. For instance, a sorted list is a…
We introduce some classes of genuine higher categories in homotopy type theory, defined as well-behaved subcategories of the category of types. We give several examples, and some techniques for showing other things are not examples. While…
We demonstrate that, in certain cases, quantization and the classical limit provide functors that are "almost inverse" to each other. These functors map between categories of algebraic structures for classical and quantum physics,…
A quantum set is defined to be simply a set of nonzero finite-dimensional Hilbert spaces. Together with binary relations, essentially the quantum relations of Weaver, quantum sets form a dagger compact category. Functions between quantum…
We propose a definition of quantum computable functions as mappings between superpositions of natural numbers to probability distributions of natural numbers. Each function is obtained as a limit of an infinite computation of a quantum…
In this paper we formalize some foundation concepts and theorems of group theory in a variant of type theory called the Calculus of Constructions with Definitions. In this theory we introduce definition of a group, which is both general and…
A description is an entity that can be interpreted as true or false of an object, and using feature structures as descriptions accrues several computational benefits. In this paper, I create an explicit interpretation of a typed feature…