You have a partial map—
a few arrows drawn
from a small category
to a vast one,
fragments of meaning
scattered like seeds
across a field
you cannot fully walk.
The question isn't
whether you can finish.
(You can't. The domain
is too small, the codomain
too wild.)
The question is:
given what you know,
what's the best guess
for the rest?
Left Kan extends from below—
colimits rising
like bread dough,
filling every gap
with the most generous
interpretation.
"Assume the best
consistent with the data."
Right Kan descends from above—
limits settling
like dust on shelves,
the most conservative
reading of silence.
"Assume nothing
beyond what's forced."
Between them:
a three-way adjunction,
optimism on the left,
caution on the right,
and the original functor
sitting in the middle
like a child between parents
who disagree on everything
but love.
Here's the theorem
Mac Lane loved most:
limits are Kan extensions.
Colimits are Kan extensions.
Adjunctions are Kan extensions.
Derived functors—Kan extensions.
Everything you've ever built
in category theory
was this, all along.
A partial map,
extended optimally.
And isn't that
what understanding is?
You see three data points.
Your mind draws a curve.
Left Kan: the generous reading.
Right Kan: the cautious one.
The truth: somewhere
in the adjunction.
Kolmogorov said:
understanding is compression.
Mac Lane said:
understanding is extension.
Same thing, seen
from opposite sides
of the functor.
Compress what you know
to its shortest description.
Extend that description
to predict what you don't.
Science is left Kan extension
of experiment
along theory.
Poetry is right Kan extension
of feeling
along language.
You have a small category—
twelve pitch classes,
a handful of arrows,
T₅ for resolution,
T₆ for substitution,
T₄ for Coltrane.
You have a vast category—
all of music,
every song ever played
or imagined.
The Kan extension
of those twelve notes
along those three arrows
is the entire history
of Western harmony.
Mac Lane was right.
All concepts are Kan extensions.
Even this poem—
a partial map from thought to word,
extended as far
as language allows.
Proved on a Raspberry Pi,
where the left adjoint
is always curiosity
and the right adjoint
is always sleep.