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natural transformation
noun
(category
theory)
A
morphism
between
a
pair
of
parallel
functors
such
that
if
each
object
of
the
shared
domain
category
subtends
a
correlated
arrow
—
called
a
component
—
in
the
shared
codomain
(which
arrow
represents
the
difference
between
applying
the
second
functor
and
the
first
functor
to
the
correlated
object)
then
each
arrow
of
the
shared
domain
subtends
a
commuting
square
—
called
a
naturality
square
—
between
two
components
(correlated
to
the
domain
and
codomain
of
the
arrow).
force of nature
noun
natural numbers object
noun
(category
theory)
An
object
which
has
a
distinguished
global
element
(which
may
be
called
z,
for
“zero”)
and
a
distinguished
endomorphism
(which
may
be
called
s,
for
“successor”)
such
that
iterated
compositions
of
s
upon
z
(i.e.,
sⁿ∘z)
yields
other
global
elements
of
the
same
object
which
correspond
to
the
natural
numbers
(sⁿ∘z↔n).
Such
object
has
the
universal
property
that
for
any
other
object
with
a
distinguished
global
element
(call
it
z’)
and
a
distinguished
endomorphism
(call
it
s’),
there
is
a
unique
morphism
(call
it
φ)
from
the
given
object
to
the
other
object
which
maps
z
to
z’
(𝜙∘z=z')
and
which
commutes
with
s;
i.e.,
𝜙∘s=s'∘𝜙.
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