In
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, cate ...
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.
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 is a refinement of
in the category
of locally convex spaces by means of the subcategory
of
normed space
In mathematics, a normed vector space or normed space is a vector space over the real or complex numbers, on which a norm is defined. A norm is the formalization and the generalization to real vector spaces of the intuitive notion of "length" i ...
s:
# The saturation
of a pseudocomplete
[A topological vector space is said to be ''pseudocomplete'' if each totally bounded Cauchy net in converges.] locally convex space is a refinement in the category
of locally convex spaces by means of the subcategory
of the
Smith spaces:
See also
*
Envelope
Notes
References
*
*
*
{{Category theory
Category theory
Duality theories
Functional analysis