Related papers: A logical and algebraic characterization of adjunc…
We study right quasi-representable differential graded bimodules as quasi-functors between dg-categories. We prove that a quasi-functor has a left adjoint if and only if it is left quasi-representable.
We introduce a new categorical framework for studying derived functors, and in particular for comparing composites of left and right derived functors. Our central observation is that model categories are the objects of a double category…
We study the relationship between cartesian bicategories and a specialisation of Lawvere's hyperdoctrines, namely elementary existential doctrines. Both provide different ways of abstracting the structural properties of logical systems: the…
We consider an intermediate category between the category of finite quivers and a certain category of pseudocompact associative algebras whose objects include all pointed finite dimensional algebras. We define the completed path algebra and…
We observe that an enriched right adjoint functor between model categories which preserves acyclic fibrations and fibrant objects is quite generically a right Quillen functor.
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…
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,…
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…
This article tackles categorical coherence within a two-dimensional generalization of Lawvere's functorial semantics. 2-theories, a syntactical way of describing categories with structure, are presented. From the perspective here afforded,…
We construct an explicit combinatorial model of the functor which adds right adjoints to the morphisms of an $\infty$-category, and we speculate on possible extensions to higher dimensions.
The aim of this paper is to establish a contravariant adjunction between the category of quasi-bialgebras and a suitable full subcategory of dual quasi-bialgebras, adapting the notion of finite dual to this framework. Various functorial…
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,…
There are many contexts in algebraic geometry, algebraic topology, and homological algebra where one encounters a functor that has both a left and right adjoint, with the right adjoint being isomorphic to a shift of the left adjoint…
This article is an introduction to the basic generalized category theory used in recent work on an extension of the theory of categories and categorical logic, including parts of topos theory. We discuss functors, equivalences, natural…
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,…
Two adjoint functors can be seen as generalisations of the two functions within a Galois connection. If instead the adjoints are not generalised from functions, but from relations, then analogously the object of study becomes a more general…
A survey of properties of the adjunction involving a semisymmetrization functor, which was suggested by J.D.H. Smith, and which maps the category of quasigroups with homotopies to the category of semisymmetric quasigroups with…
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…
The aim of this paper is to present a very simple original, purely formal, proof of Quillen's adjunction theorem for derived functors, and of some more recent variations and generalizations of this theorem. This is obtained by proving an…
We prove that a Quillen adjunction of model categories (of which we do not require functorial factorizations and of which we only require finite bicompleteness) induces a canonical adjunction of underlying quasicategories.