Abstract families and the adjoint functor theorems

1978 ◽  
pp. 1-125 ◽  
Author(s):  
Robert Paré ◽  
Dietmar Schumacher
Keyword(s):  
Adjoint Functor

scholarly journals Reflective subcategories

2000 ◽  
Vol 42 (1) ◽  
pp. 97-113 ◽  
Author(s):  
Juan Rada ◽  
Manuel Saorín ◽  
Alberto del Valle
Keyword(s):  
Category Theory ◽  
Full Subcategory ◽  
Subject Classification ◽  
Adjoint Functor ◽  
Direct Limits ◽  
The Relationship ◽  
Good Behaviour

Given a full subcategory [Fscr ] of a category [Ascr ], the existence of left [Fscr ]-approximations (or [Fscr ]-preenvelopes) completing diagrams in a unique way is equivalent to the fact that [Fscr ] is reflective in [Ascr ], in the classical terminology of category theory.In the first part of the paper we establish, for a rather general [Ascr ], the relationship between reflectivity and covariant finiteness of [Fscr ] in [Ascr ], and generalize Freyd's adjoint functor theorem (for inclusion functors) to not necessarily complete categories. Also, we study the good behaviour of reflections with respect to direct limits. Most results in this part are dualizable, thus providing corresponding versions for coreflective subcategories.In the second half of the paper we give several examples of reflective subcategories of abelian and module categories, mainly of subcategories of the form Copres (M) and Add (M). The second case covers the study of all covariantly finite, generalized Krull-Schmidt subcategories of {\rm Mod}_{R}, and has some connections with the “pure-semisimple conjecture”.1991 Mathematics Subject Classification 18A40, 16D90, 16E70.

scholarly journals Adjoint functor theorems for ∞‐categories

2019 ◽  
Vol 101 (2) ◽  
pp. 659-681
Author(s):  
Hoang Kim Nguyen ◽  
George Raptis ◽  
Christoph Schrade
Keyword(s):  
Adjoint Functor

Topological systems and Artin glueing

2012 ◽  
Vol 62 (4) ◽  
Author(s):  
Sergey Solovyov
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Fuzzy Topology ◽  
Adjoint Functor ◽  
And Topology ◽  
Necessary And Sufficient

AbstractUsing methods of categorical fuzzy topology, the paper shows a relation between topological systems of S. Vickers and Artin glueing of M. Artin. Inspired by the problem of interrelations between algebra and topology, we show the necessary and sufficient conditions for the category, obtained by Artin glueing along an adjoint functor, to be (co)algebraic and (co)monadic, incorporating the respective result of G. Wraith. As a result, we confirm the algebraic nature of the category of topological systems, showing that it is monadic.

scholarly journals An adjoint-functor theorem over topoi

1976 ◽  
Vol 15 (3) ◽  
pp. 381-394 ◽  
Author(s):  
B.J. Day
Keyword(s):  
Usual Sense ◽  
Adjoint Functor

The usual statements of the classical adjoint-functor theorems contain the hypothesis that the codomain category should admit arbitrary intersections of families of monomorphisms with a common codomain. The aim of this article is to formulate an adjoint-functor theorem which refers, in a similar manner, to arbitrary internal intersections of “families of monomorphisms” in the case where the categories under consideration are suitably defined relative to a fixed elementary base topos (in the usual sense of Lawvere and Tierney).

scholarly journals Categorical definitions and properties via generators

2019 ◽  
Vol 53 (2) ◽  
pp. 165-184
Author(s):  
Gustavo Arengas
Keyword(s):  
Adjoint Functor ◽  
Alternative Version ◽  
Universal Properties ◽  
Categorical Constructions

In the present work, we show how the study of categorical constructions does not have to be done with all the objects of the category, but we can restrict ourselves to work with families of generators. Thus, universal properties can be characterized through iterated families of generators, which leads us in particular to an alternative version of the adjoint functor theorem. Similarly, the properties of relations or subobjects algebra can be investigated by this method. We end with a result that relates various forms of compactness through representable functors of generators.

Regular and Strongly Finitary Structures Over Strongly Algebroidal Categories

1978 ◽  
Vol 30 (02) ◽  
pp. 250-261 ◽  
Author(s):  
Günter Matthiessen
Keyword(s):  
Left Adjoint ◽  
Adjoint Functor ◽  
Reduced Products

Most properties an algebraist needs in categories are reflected by regular functors, introduced in [6]. If is a regular and strongly finitary functor and has some nice properties, it can be shown that the left adjoint functor of G helps to characterize finitary and strongly finitary objects of . The property of being algebroidal can be lifted from to if a certain condition holds in . As an application, the implicational hull of subcategories can be constructed with the help of reduced products.

Adjunctions and Adjoint Functor Theorems

2021 ◽  
pp. 249-272
Author(s):  
Markus Land
Keyword(s):  
Adjoint Functor

scholarly journals On Dynamical Adjoint Functor

2011 ◽  
Vol 21 (5) ◽  
pp. 449-468 ◽  
Author(s):  
Andrey Mudrov
Keyword(s):  
Adjoint Functor

The theory of semi-functors

1993 ◽  
Vol 3 (1) ◽  
pp. 93-128 ◽  
Author(s):  
Raymond Hoofman
Keyword(s):  
General Theory ◽  
Natural Transformation ◽  
Lambda Calculus ◽  
Adjoint Functor ◽  
Typed Lambda Calculus ◽  
The World ◽  
Natural Transformations ◽  
Theoretical Characterization

The notion ofsemi-functorwas introduced in Hayashi (1985) in order to make possible a category-theoretical characterization of models of the non-extensional typed lambda calculus. Motivated by the further use of semi-functors in Martini (1987), Jacobs (1991) and Hoofman (1992a), (1992b) and (1992c), we consider the general theory of semi-functors in this paper. It turns out that the notion ofsemi natural transformationplays an important part in this theory, and that various categorical notions involving semi-functors can be viewed as 2-categorical notions in the 2-category of categories, semi-functors and semi natural transformations. In particular, we find that the notion ofnormal semi-adjunctionas defined in Hayashi (1985) is the canonical generalization of the notion of adjunction to the world of semi-functors. Further topics covered in this paper are the relation between semi-functors and splittings, the Karoubi envelope construction, semi-comonads, and a semi-adjoint functor theorem.

Subequalizers

1970 ◽  
Vol 13 (3) ◽  
pp. 337-349 ◽  
Author(s):  
Joachim Lambek
Keyword(s):  
Critical Reading ◽  
Ordered Sets ◽  
Adjoint Functor ◽  
Carry Over

The purpose of this exposition is threefold. Firstly, we wish to show that many concepts and arguments carry over from pre-ordered sets to categories. Secondly, we wish to make some propaganda for the notion of "subequalizer" of two functors, which appears to be more fundamental than Lawvere's so-called "commacategory", in the same sense in which equalizers are more fundamental than pullbacks. Thirdly, we wish to give simple proofs of the adjoint functor theorem and related theorems, which appear to be more economic than those in the literature. The author wishes to take this opportunity to refine some arguments that he has published earlier. He is indebted to Michael Barr, whose presentation of similar proofs in his course has provided the stimulation for preparing this note for publication, to John Isbell for his critical reading of the manuscript and to William Schelter for suggesting a shortcut in one of the proofs.

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