Related papers: A remark on Yoneda's Lemma
This work presents an exposition of both the internal structure of derived category of an abelian category D*(A) and its contribution in solving problems, particularly in algebraic geometry. Calculation of some morphisms will be presented…
We develop some basic concepts in the theory of higher categories internal to an arbitrary $\infty$-topos. We define internal left and right fibrations and prove a version of the Grothendieck construction and of Yoneda's lemma for internal…
In this note we formulate and give a self-contained proof of the Yoneda lemma for infinity categories in the language of complete Segal spaces.
A classical theorem of Wonenburger, Djokovic, Hoffmann and Paige states that an element of the general linear group of a finite-dimensional vector space is the product of two involutions if and only if it is similar to its inverse. We give…
The analogy between Yetter's deformation theory form (lax) monoidal functors and Gerstenahaber's deformation theory for associative algebras is solidified by shown that under reasonable conditions the category of functors with an action of…
Let $X$ be a complete measure space of finite measure. The Lebesgue transform of an integrable function $f$ on $X$ encodes the collection of all the mean-values of $f$ on all measurable subsets of $X$ of positive measure. In the problem of…
A classical result of Tannaka duality is the fact that a coalgebra over a field can be reconstructed from its category of finite dimensional representations by using the forgetful functor which sends a representation to its underlying…
In this work, we show that the quantum mechanical notions of density operator, positive operator-valued measure (POVM), and the Born rule, are all simultaneously encoded in the categorical notion of a natural transformation of functors. In…
We show that the Yoneda embedding 'is' $(\infty,2)$-natural with respect to the functoriality of presheaves via left Kan extension, refining the $(\infty,1)$-categorical result proven independently by Haugseng-Hebestreit-Linskens-Nuiten and…
For a finite dimensional algebra $\Lambda$ of finite representation type and an additive generator $M$ for $\mathrm{mod}\,\Lambda$, we investigate the properties of the Yoneda algebra $\Gamma=\bigoplus_{i \geq…
We make Hinich's $\infty$-categorical enriched Yoneda embedding natural. To do so, we exhibit it as the unit of a partial adjunction between the functor taking enriched presheaves and Heine's functor taking a tensored category to an…
The Legendre transform is an important tool in theoretical physics, playing a critical role in classical mechanics, statistical mechanics, and thermodynamics. Yet, in typical undergraduate or graduate courses, the power of motivation and…
Category theory has foundational importance because it provides conceptual lenses to characterize what is important in mathematics. Originally the main lenses were universal mapping properties and natural transformations. In recent decades,…
A fundamental result in representation theory is Kostant's theorem which describes the algebra of polynomials on a reductive Lie algebra as a module over its invariants. We prove a quantum analogue of this theorem for the general linear…
Optics are bidirectional accessors of data structures; they provide a powerful abstraction of many common data transformations. This abstraction is compositional thanks to a representation in terms of profunctors endowed with an algebraic…
We construct a canonical irreducible representation for the orthofermion algebra of arbitrary order, and show that every representation decomposes into irreducible representations that are isomorphic to either the canonical representation…
A covariant description of the canonical theory for interacting classical fields is developed on a space-like hypersurface. An identity invariant under the canonical transformations is obtained. The identity follows a canonical equation in…
We construct the ordinary irreducible representations of the group of automorphisms of a finite rooted tree and we get a natural parametrization of them. To achieve this goals, we introduce and study the combinatorics of tree compositions,…
The set of all transformation monoids on a fixed set of infinite cardinality \lambda, equipped with the order of inclusion, forms a complete algebraic lattice Mon(\lambda) with 2^{\lambda} compact elements. We show that this lattice is…
This short introductory category theory textbook is for readers with relatively little mathematical background (e.g. the first half of an undergraduate mathematics degree). At its heart is the concept of a universal property, important…