In 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.
Suppose
K
X
K
\Gamma
\Phi
K
X
\Gamma
\Phi
\sigma:X'\toX
K
X
\Gamma
\Phi
\sigma\in\Gamma
\varphi:B\toX
\Phi
\varphi':B\toX'
K
\varphi=\sigma\circ\varphi'
\rho:E\toX
X
\Gamma
\Phi
X
\Gamma
\Phi
\sigma:X'\toX
X
\Gamma
\Phi
\upsilon:E\toX'
K
\rho=\sigma\circ\upsilon
E
X
\Gamma
\Phi
| \Gamma | |
| \rho=\operatorname{ref} | |
| \Phi |
X,
| \Gamma | |
| E=\operatorname{Ref} | |
| \Phi |
X.
In a special case when
\Gamma
L
K
\Gamma
L
| L | |
| \rho=\operatorname{ref} | |
| \Phi |
X,
| L | |
| E=\operatorname{Ref} | |
| \Phi |
X.
Similarly, if
\Phi
M
K
\Phi
M
| \Gamma | |
| \rho=\operatorname{ref} | |
| M |
X,
| \Gamma | |
| E=\operatorname{Ref} | |
| M |
X.
For example, one can speak about a refinement of
X
L
M
| L | |
| \rho=\operatorname{ref} | |
| M |
X,
| L | |
| E=\operatorname{Ref} | |
| M |
X.
X\operatorname{born
X
X
\operatorname{LCS}
\operatorname{Norm}
X\operatorname{born
X\blacktriangle
X
\operatorname{LCS}
\operatorname{Smi}
| \blacktriangle=\operatorname{Ref} | |
| X | |
| \operatorname{Smi |
X
X