Introduction to Introduction to Category Theory

Cat­e­gory the­ory is so gen­eral in its ap­pli­ca­tion that it re­ally feels like ev­ery­one, even non-math­e­mat­i­ci­ans, ought to at least con­cep­tu­ally grok that it ex­ists, like how ev­ery­one ought to un­der­stand the idea of the laws of physics even if they don’t know what those laws are.

We ex­pect ed­u­cated peo­ple to know that the Earth is round and the Sun is big, even though those facts don’t have any di­rect rele­vance to the lives of most peo­ple. I think peo­ple should know about Yoneda and ad­junc­tion in at least the same broad way peo­ple are aware of the ex­is­tence and use of calcu­lus.

But no one out­side of math­e­mat­ics and maybe pro­gram­ming/​data sci­ence has heard of cat­e­gory the­ory, and I think a big part of that is be­cause all of the ex­am­ples in text­books as­sume you already know what Sier­pin­ski spaces and abelian groups are.

That is to say: all ex­po­si­tions of cat­e­gory the­ory as­sume you know math.

Which makes sense. Cat­e­gory the­ory is the math­e­mat­ics of math. Try­ing to learn cat­e­gory the­ory with­out hav­ing most of an un­der­grad­u­ate ed­u­ca­tion in math already un­der your belt is like try­ing to study New­ton’s laws with­out hav­ing ever seen an ap­ple fall from a tree. You’re just go­ing to have ab­solutely no in­tu­ition to rely on.

Cat­e­gory the­ory gen­er­al­izes the things you do in the var­i­ous fields of math­e­mat­ics, just like how New­ton’s laws gen­er­al­ize the things you do when you toss a rock or push your­self off the ground. Ex­cept re­ally, cat­e­gory the­ory gen­er­al­izes what you do when you gen­er­al­ize with New­ton’s laws. Cat­e­gory the­ory gen­er­al­izes gen­er­al­iz­ing.

There­fore, with­out know­ing about any spe­cific gen­er­al­iza­tions, like alge­bra or topol­ogy, it’s hard to un­der­stand gen­eral gen­er­al­ities—which are cat­e­gories.

As a re­sult, there are no cat­e­gory the­ory texts (that I know of) that teach cat­e­gory the­ory to the ed­u­cated and in­tel­li­gent but math­e­mat­i­cally ig­no­rant per­son.

Which is a shame, be­cause you to­tally can.

Sure, if you’ve never learned topol­ogy, plenty of stan­dard ex­am­ples will fly over your head. But ev­ery ed­u­cated per­son has en­coun­tered the idea of gen­er­al­iza­tion, and they’ve seen gen­er­al­iza­tions of gen­er­al­iza­tions. In fact, cat­e­gory the­ory is very in­tu­itive, and I don’t think it nec­es­sar­ily benefits from re­lat­ing it all as quickly as pos­si­ble to more fa­mil­iar fields of math­e­mat­ics. In­stead, you should grasp the flow of cat­e­gory the­ory it­self, as its own field.

So this is (ten­ta­tively, hope­fully, un­less I get busy, bored, or it just doesn’t work out) a se­ries on the ba­sics of cat­e­gory the­ory with­out as­sum­ing you know any math. I’m think­ing speci­fi­cally of high school se­niors.

There is no sched­ule for the posts. They’ll just be up when­ever I make them.

Why cat­e­gory the­ory? And why less­wrong?

Well, cat­e­gory the­ory is a su­per-gen­eral the­ory of ev­ery­thing. Ra­tion­al­ity is also a su­per-gen­eral the­ory of ev­ery­thing. In fact, we’ll see how cat­e­gory the­ory tells us a lot about what ra­tio­nal­ity re­ally is, in a cer­tain rigor­ous sense.

Ba­si­cally...ra­tio­nal­ity comes from notic­ing cer­tain gen­eral laws that seem to emerge ev­ery time you try to do some­thing the “right” way. After a while, in­stead of fo­cus­ing so much on the speci­fics, it starts be­ing worth it to take a step back and study the gen­eral rules that seem to be emerg­ing. And you start to no­tice that do­ing things the “right” way gets a lot eas­ier when you start with the gen­eral rules and sim­ply fill in the speci­fics, like how the quadratic for­mula makes quadratic equa­tions a cinch to solve.

Cat­e­gory the­ory gives us all the gen­eral rules for do­ing things the “right” way.

(Don’t ac­tu­ally hold me to demon­strat­ing this claim.)

Why should you be in­ter­ested in cat­e­gory the­ory?

One is be­cause cat­e­gory the­ory is go­ing to rise in im­por­tance in the fu­ture. It offers pow­er­ful new ways of do­ing math and sci­ence. So get started!

Two is that cat­e­gory the­ory makes it much eas­ier to learn the rest of math. Well, maybe—this is an ex­per­i­ment, and a big mo­ti­va­tion for do­ing this. How fast and well do peo­ple learn reg­u­lar math if they can just say, “Oh, it’s an ad­junc­tion” ev­ery time they learn a new con­cept?

Three is that refer­enc­ing ho­mo­topy type the­ory in con­ver­sa­tion will make you sound cool and mys­te­ri­ous.

Please let me know if there’s any in­ter­est in this.