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Algebra of Programming summary
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Philip Kaluđerčić
Algebra of Programming summary
Commits
455e92aa
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455e92aa
authored
Apr 10, 2024
by
Philip Kaluđerčić
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Mention the "respect system dynamics" phrase
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coalgebra.tex
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455e92aa
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@@ -10,11 +10,12 @@
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@@ -10,11 +10,12 @@
\begin{definition}
\label
{
def:fcoalgebra
}
\begin{definition}
\label
{
def:fcoalgebra
}
In a
\de
{
category
}
\(
\C
\)
, given an
\de
{
object
}
\(
A
\in
\Ob
{
\C
}\)
In a
\de
{
category
}
\(
\C
\)
, given an
\de
{
object
}
\(
A
\in
\Ob
{
\C
}\)
and an
\de
{
endofunctor
}
\(
F :
\map
{
\C
}{
\C
}\)
the pair
and an
\de
{
endofunctor
}
\(
F :
\map
{
\C
}{
\C
}\)
the pair
\(
A, a :
\map
{
A
}{
F
(
A
)
}\)
is called a
\emph
{
\fca
}
. A
\fca
-homomorphism
\(
A, a :
\map
{
A
}{
F
(
A
)
}\)
is called a
\emph
{
\fca
}
. A
\(
f :
\map
{
(
A, a
)
}{
(
B, b
)
}\)
ensures
\(
f
\circ
a
=
b
\circ
F
(
f
)
\)
.
\fca
-homomorphism
\(
f :
\map
{
(
A, a
)
}{
(
B, b
)
}\)
ensures
\fca
{}
s and
\fca
-homomorphisms constitute a separate
\de
{
category
}
\(
f
\circ
a
=
b
\circ
F
(
f
)
\)
.
\fca
{}
s and
\fca
-homomorphisms
\(
\Coalg
{
F
}\)
\refsk
{
fca-category
}
, which is
\textbf
{
not
}
dual to
(which respect the system dynamics) constitute a separate
\(
\Alg
{
F
}\)
, but to
\(
\Alg
{
\op
{
F
}}\)
.
\de
{
category
}
\(
\Coalg
{
F
}\)
\refsk
{
fca-category
}
, which is
\textbf
{
not
}
dual to
\(
\Alg
{
F
}\)
, but to
\(
\Alg
{
\op
{
F
}}\)
.
Despite that qualification, results like
\cref
{
lem:lambek
}
or
Despite that qualification, results like
\cref
{
lem:lambek
}
or
\cref
{
def:initial-fa-construction
}
can mostly be derived analogously.
\cref
{
def:initial-fa-construction
}
can mostly be derived analogously.
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