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Stream: theory: category theory

Topic: universal constructions

Christian Williams (Mar 23 2020 at 18:27):

one of my favorite things I've learned is the way to express Kan extensions as co/ends:

given $F:A\to C$ and $p:A\to B$, we have

$Lan_pF \simeq \int^a B(p(a),-)\cdot F(a)$
$Ran_pF \simeq \int_a B(-,p(a))\uparrow F(a)$

where $\cdot$ is "copower" and $\uparrow$ is "power"

Christian Williams (Mar 23 2020 at 18:29):

when $C = \mathrm{Set}$, then copower is product and power is hom.

Christian Williams (Mar 23 2020 at 18:30):

Then if $F$ is a presheaf, the Yoneda lemma is right Kan extension along the identity!
And the coYoneda lemma is left Kan extension along the identity.

Christian Williams (Mar 23 2020 at 18:41):

the Yoneda lemma says if $F:A\to \mathrm{Set}$, then
$F(a')\simeq \mathrm{Nat}(C(-,a'),F)$
and natural transformations can be constructed as an end
$F(a') \simeq \int_a [C(a,a'),F(a)]$
and that's the formula for right Kan extension where $p = id_A$
$F \simeq Ran_{id_A} F \simeq \int_a [C(-,a),F]$

Christian Williams (Mar 23 2020 at 18:42):

hm, something is tricky with the variance. but that's the general idea.

sarahzrf (Mar 23 2020 at 20:59):

i really fuckin need to read the coend book one of these days :sob:

Christian Williams (Mar 23 2020 at 21:05):

it's great! I haven't finished yet though, it gets pretty advanced...
I don't suppose Fosco is on here yet.

Stelios Tsampas (Mar 23 2020 at 21:06):

Speaking of books, these two are great! @David Spivak @Brendan jacobs-act.jpg