Related papers: On Adjoint and Brain Functors
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,…
The question "What is category theory" is approached by focusing on universal mapping properties and adjoint functors. Category theory organizes mathematics using morphisms that transmit structure and determination. Structures of…
These are the notes for a minicourse held in Odessa (2016) and Belo Horizonte (2017). My aim was to provide a short introduction to basic notions of category theory and representation theory of finite-dimensional algebras. We learnt the…
Voevodsky's derived category of motives is the main arena today for the study of algebraic cycles and motivic cohomology. In this paper we study whether the inclusions of three important subcategories of motives have a left or right…
1. This paper shows how the universals of category theory in mathematics provide a model (in the Platonic Heaven of mathematics) for the self-predicative strand of Plato's Theory of Forms as well as for the idea of a "concrete universal" in…
There is a lot of redundancy in the usual definition of adjoint functors. We define and prove the core of what is required. First we do this in the hom-enriched context. Then we do it in the cocompletion of a bicategory with respect to…
In this paper we investigate the categories of braided objects, algebras and bialgebras in a given monoidal category, some pairs of adjoint functors between them and their relations. In particular we construct a braided primitive functor…
We are checking the closed categories beginning with the category of sets and ending with the category of categories. The novelty is a generalizing the notion of adjoint functors to the joint pair of functors in the category of directed…
Given a right adjoint functor between triangulated categories and an object in the target category, we show that the unit map of adjunction on that object is a split monomorphism if and only if the object belongs to the additive closure of…
We show that contrary to appearances, Multimodal Type Theory (MTT) over a 2-category M can be interpreted in any M-shaped diagram of categories having, and functors preserving, M-sized limits, without the need for extra left adjoints. This…
The interaction between syntax (formal language) and its semantics (meanings of language) is one which has been well studied in categorical logic. The results of this particular study are employed to understand how the brain is able to…
Category theory has foundational importance because it provides conceptual lenses to characterize what is important and universal in mathematics---with adjunctions being the primary lense. If adjunctions are so important in mathematics,…
Adjoint functor theorems give necessary and sufficient conditions for a functor to admit an adjoint. In this paper we prove general adjoint functor theorems for functors between $\infty$-categories. One of our main results is an…
The functor that takes a ring to its category of modules has an adjoint if one remembers the forgetful functor to abelian groups: the endomorphism ring of linear natural transformations. This uses the self-enrichment of the category of…
In this work, we investigate an effective method for showing that functors between categories are left adjoints. The method applies to a large class of categories, namely locally finitely presentable categories, which are ubiquitous in…
We provide a criterion for the existence of right approximations in cocomplete additive categories; it is a straightforward generalisation of a result due to El Bashir. This criterion is used to construct adjoint functors in homotopy…
The postulates of comprehension and extensionality in set theory are based on an inversion principle connecting set-theoretic abstraction and the property of having a member. An exactly analogous inversion principle connects functional…
We apply the notion of relative adjoint functor to generalise closed monoidal categories. We define representations in such categories and give their relation with left actions of monoids. The translation of these representations under lax…
The main result concerns a bicategorical factorization system on the bicategory $\mathrm{Cat}$ of categories and functors. Each functor $A\xra{f} B$ factors up to isomorphism as $A\xra{j}E\xra{p}B$ where $j$ is what we call an ultimate…
The notion of a duality between two derived functors as well as an extension theorem for derived functors to larger categories in which they need not be defined is introduced. These ideas are then applied to extend and study the coext…