In
category theory
Category theory is a general theory of mathematical structures and their relations. It was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Category theory ...
and related fields of mathematics, a refinement is a construction that generalizes the operations of "interior enrichment", like bornologification or saturation of a locally convex space. A dual construction is called
envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter (message), letter or Greeting card, card.
Traditional envelopes are made from sheets of paper cut to one o ...
.
Definition
Suppose
is a category,
an object in
, and
and
two classes of morphisms in
. The definition of a refinement of
in the class
by means of the class
consists of two steps.

* A morphism
in
is called an ''enrichment of the object
in the class of morphisms
by means of the class of morphisms
'', if
, and for any morphism
from the class
there exists a unique morphism
in
such that
.

* An enrichment
of the object
in the class of morphisms
by means of the class of morphisms
is called a ''refinement of
in
by means of
'', if for any other enrichment
(of
in
by means of
) there is a unique morphism
in
such that
. The object
is also called a ''refinement of
in
by means of
''.
Notations:
:
In a special case when
is a class of all morphisms whose ranges belong to a given class of objects
in
it is convenient to replace
with
in the notations (and in the terms):
:
Similarly, if
is a class of all morphisms whose ranges belong to a given class of objects
in
it is convenient to replace
with
in the notations (and in the terms):
:
For example, one can speak about a ''refinement of
in the class of objects
by means of the class of objects
'':
:
Examples
# The bornologification
of a
locally convex space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
is a refinement of
in the category
of locally convex spaces by means of the subcategory
of
normed space
The Ateliers et Chantiers de France (ACF, Workshops and Shipyards of France) was a major shipyard that was established in Dunkirk, France, in 1898.
The shipyard boomed in the period before World War I (1914–18), but struggled in the inter-war p ...
s:
# The saturation
of a pseudocomplete
[A ]topological vector space
In mathematics, a topological vector space (also called a linear topological space and commonly abbreviated TVS or t.v.s.) is one of the basic structures investigated in functional analysis.
A topological vector space is a vector space that is als ...
is said to be ''pseudocomplete'' if each totally bounded
In topology and related branches of mathematics, total-boundedness is a generalization of compactness for circumstances in which a set is not necessarily closed. A totally bounded set can be covered by finitely many subsets of every fixed “si ...
Cauchy net
In mathematics, more specifically in general topology and related branches, a net or Moore–Smith sequence is a function whose domain is a directed set. The codomain of this function is usually some topological space. Nets directly generalize ...
in converges. locally convex space
In functional analysis and related areas of mathematics, locally convex topological vector spaces (LCTVS) or locally convex spaces are examples of topological vector spaces (TVS) that generalize normed spaces. They can be defined as topological vec ...
is a refinement in the category
of locally convex spaces by means of the subcategory
of the
Smith spaces:
See also
*
Envelope
An envelope is a common packaging item, usually made of thin, flat material. It is designed to contain a flat object, such as a letter (message), letter or Greeting card, card.
Traditional envelopes are made from sheets of paper cut to one o ...
Notes
References
*
*
*
{{Category theory
Category theory
Duality theories
Functional analysis