# Reflective subcategories

@article{Rada2000ReflectiveS, title={Reflective subcategories}, author={Juan Rada and Manuel Saori{\^a} and Alberto del Valle}, journal={Glasgow Mathematical Journal}, year={2000}, volume={42}, pages={97 - 113} }

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…

## 44 Citations

The envelope of a subcategory in topology and group theory

- MathematicsInt. J. Math. Math. Sci.
- 2005

It is shown that reflective subcategories are orthogonality classes, that the Morphisms orthogonal to a reflective subcategory are precisely the morphisms inverted under the reflector, and that each subcategory has a largest “envelope” in the ambient category in which it is reflective.

Reflective and coreflective subcategories

- Mathematics
- 2021

Given any additive category C with split idempotents, pseudokernels and pseudocokernels, we show that a subcategory B is coreflective if, and only if, it is precovering, closed under direct summands…

Categorical algebra Semi-abelian categories, semi-localizations and factorization systems

- Mathematics
- 2016

Semi-abelian categories [9] provide a suitable context to study the (co)homology of non-abelian algebraic structures (such as groups, Lie algebras, compact groups, crossed modules, cocommutative Hopf…

A criterion for reflectiveness of normal extensions

- Mathematics
- 2016

We give a new sufficient condition for the normal extensions in an admissible Galois structure to be reflective. We then show that this condition is indeed fulfilled when X is the (protomodular)…

How many Adjunctions give Rise to the same Monad?

- MathematicsAppl. Categorical Struct.
- 2017

This paper single out a class of adjunctions with especially good properties, and develops methods for computing all such adjunctions, up to natural equivalence, which give rise to a given monad.

Global dimension and left derived functors of Hom

- Mathematics
- 2007

It is well known that the right global dimension of a ring R is usually computed by the right derived functors of Hom and the left projective resolutions of right R-modules. In this paper, for a left…

On the existence of group localization under large-cardinal axioms

- Mathematics
- 2001

A long-standing open question in categorical group theory asks if every orthogonal pair (consisting of a class of groups and a class of group homomorphisms determining each other by orthogonality in…

## References

SHOWING 1-10 OF 31 REFERENCES

Partial cotilting modules and the lattices induced by them

- Mathematics
- 1997

We study a duality between (infinitely generated) cotilting and tilting modules over an arbitrary ring. Dualizing a result of Bongartz, we show that a module P is partial cotilting iff P is a direct…

On envelopes with the unique mapping property

- Mathematics
- 1996

We prove that (a) if R is a left coherent ring, then the weak global dimension w D(R) = 2) if and only if every (n – 2)th F–cosyzygy of a finitely presented right R–module has a flat envelope with…

Coherent rings of finite weak global dimension

- Mathematics
- 1998

The category of left modules over right coherent rings of finite weak global dimension has several nice features. For example, every left module over such a ring has a flat cover (Belshoff, Enochs,…

Injective and flat covers, envelopes and resolvents

- Mathematics
- 1981

Using the dual of a categorical definition of an injective envelope, injective covers can be defined. For a ringR, every leftR-module is shown to have an injective cover if and only ifR is left…

The Spectrum of a Module Category

- Mathematics
- 2001

Introduction The functor category Definable subcategories Left approximations duality Ideals in the category of finitely presented modules Endofinite modules Krull-Gabriel dimension The infinite…

The Ziegler spectrum of a tame hereditary algebra

- Mathematics
- 1998

Let A be a nite dimensional hereditary algebra of tame representation type. Let Com A be a complete set of indecomposable algebraically compact A-modules (one from each isomorphism class). We are…

Locally Noetherian categories and generalized strictly linearly compact rings. Applications.

- Mathematics
- 1969

Locally Noetherian categories and generalized strictly linearly compact rings. Applications.

Coherence of polynomial rings

- Mathematics
- 1976

The main result is that A (X), the polynomial ring in any number of indeterminates over a coherent ring A of global dimension two, is coherent.

Representation Theory of Artin Algebras II

- Mathematics
- 1974

(1974). Representation Theory of Artin Algebras II. Communications in Algebra: Vol. 1, No. 4, pp. 269-310.