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Written to be contributed as the "mathematical modeling" chapter of a book, edited by Elaine Landry, to be titled "Categories for the Working Philosopher". In this chapter, category theory is presented as a mathematical modeling framework…
A concept of "evolving categories" is suggested to build a simple, scalable, mathematically consistent framework for representing in uniform way both data and algorithms. A state machine for executing algorithms becomes clear, rich and…
The purpose of this paper is to show that the dual notions of elements & distinctions are the basic analytical concepts needed to unpack and analyze morphisms, duality, and universal constructions in the Sets, the category of sets and…
We introduce generalized Frobenius-Schur indicators for pivotal categories. In a spherical fusion category C, an equivariant indicator of an object in C is defined as a functional on the Grothendieck algebra of the quantum double Z(C) via…
G\"odel's Dialectica has been introduced and developed in the tradition of the so-called functional interpretations. Only recently has it been related with the a priori unrelated notion of differentiation, by taking a program-theoretic…
The Euclidean algorithm in algebra is applied to a class of gravitational lenses for which the lens equation consists of any set of coupled polynomial equations in the image position. In general, this algorithm allows us to reduce an…
Frames play significant role in various areas of science and engineering. In this paper, we introduce the concepts of frames for $End_{\mathcal{A}}^{\ast}(\mathcal{H, K})$ and their generalizations. Moreover, we obtain some new results for…
We revisit the gravitational lensing phenomenon using a new visualization technique. It consists in projecting the observers sky into the source plane, what gives rise to a folded and stretched surface. This provides a clear graphical tool…
Service robots are expected to operate effectively in human-centric environments for long periods of time. In such realistic scenarios, fine-grained object categorization is as important as basic-level object categorization. We tackle this…
We discuss strong gravitational lensing by multiple objects along any line of sight. The probability for strong gravitational lensing by more than one lens is small, but a number of strong lens systems in which more than one separate lens…
We compare various different definitions of "the category of smooth objects". The definitions compared are due to Chen, Fr\"olicher, Sikorski, Smith, and Souriau. The method of comparison is to construct functors between the categories that…
In this paper, we present a generalization of Grothendieck pretopologies -- suited for semicartesian categories with equalizers $C$ -- leading to a closed monoidal category of sheaves, instead of closed cartesian category. This is proved…
Gravitational lenses are a unique new constraint on the structure of galaxies. We review the evidence that most lenses are early-type galaxies, the optical properties of the lens galaxies, the evidence against constant M/L models, recent…
Categorical universal algebra can be developed either using Lawvere theories (single-sorted finite product theories) or using monads, and the category of Lawvere theories is equivalent to the category of finitary monads on Set. We show how…
Gravitational lensing refers to the deflection of light by the gravity of celestial bodies, often predominantly composed of dark matter. Seen through a gravitational lens, the images of distant galaxies appear distorted. In this paper we…
We define a monoidal semantics for algebraic theories. The basis for the definition is provided by the analysis of the structural rules in the term calculus of algebraic languages. Models are described both explicitly, in a form that…
Classification is an important goal in many branches of mathematics. The idea is to describe the members of some class of mathematical objects, up to isomorphism or other important equivalence in terms of relatively simple invariants. Where…
We show that differential calculus (in its usual form, or in the general form of topological differential calculus) can be fully imdedded into a functor category (functors from a small category of anchord tangent algebras to anchored sets).…
We develop category-theoretic framework for universal homogeneous objects, with some applications in the theory of Banach spaces, linear orderings, and in topology of compact spaces.
Recall that the definition of the $K$-theory of an object C (e.g., a ring or a space) has the following pattern. One first associates to the object C a category A_C that has a suitable structure (exact, Waldhausen, symmetric monoidal, ...).…