# countedblessings

Karma: 360
Page 1
• My next se­ries of posts will be di­rectly about the Yoneda lemma, which ba­si­cally tells us that ev­ery­thing you could want to know about an ob­ject is con­tained in the mor­phisms go­ing into/​out of the ob­ject. More­over, we get this knowl­edge in a “nat­u­ral” way that makes life re­ally easy. It’s a pretty cool the­o­rem.

• In the end, we don’t re­ally care about sets at all. They’re just bags with stuff in them. Who cares about bags? But we do care about func­tions—we want those to be rule-based. We need func­tions to go “from” some­where and “to” some­where. Let’s call those things sets. Then we need these “sets” to be rule-based.

I’m grate­ful for your com­ments. They’re very use­ful, and you raise good points. I’ve got most of a post already about how func­tions give mean­ing to the el­e­ments of sets. As for how func­tions is a verb, think of prop­er­ties as ex­ist­ing in verbs. So to know some­thing, you need to ob­serve it in some way, which means it has to af­fect your sen­sory de­vices, such as your ears, eyes, ther­mome­ters, what­ever. You know dogs, for ex­am­ple, by the way they bark, by the way they lick, they way they look, etc. So prop­er­ties ex­ist in the verbs. “Legs” are a noun, but all of your knowl­edge about them has to come from verbs. Does that make sense?

• You raise a good point. Think of cat­e­gory the­ory as a lan­guage for ex­press­ing, in this case, the logic of sets and func­tions. You still need to know what that logic is. Then you can use cat­e­gory the­ory to work effi­ciently with that logic ow­ing to its gen­eral-ab­stract na­ture.

• That was my in­ten­tion. Thanks for point­ing it out. One of the mis­takes of this se­ries was the naive be­lief that sim­plic­ity comes from vague­ness, when it ac­tu­ally comes from pre­ci­sion. Dumb of me.

• Steam is run out of. This was poorly con­ceived to be­gin with, ar­ro­gant in its in­her­ent de­sign, and even I don’t have the pa­tience for it any­way. I’ll do a se­ries about ad­junc­tion di­rectly and Yoneda as well.

# Sets and Functions

11 Oct 2019 5:06 UTC
28 points
• Hon­estly my real jus­tifi­ca­tion would be “ad­joint func­tors awe­some, and you need cat­e­gories to do ad­joint func­tors, so use cat­e­gories.” More broadly...as long as it’s free to cre­ate a cat­e­gory out of what­ever you’re study­ing, there’s clearly no harm. The ques­tion is whether any­thing’s lost by treat­ing the sub­ject as a cat­e­gory, and while I fully ex­pect that there are en­tire uni­verses of math­e­mat­ics and re­al­ity out there where cat­e­gories are harm­ful, I don’t think we live in one like that. Cat­e­gories may not cap­ture ev­ery­thing you can think of, but they can cap­ture so much that I’d be stunned if they didn’t yield amaz­ing fruit even­tu­ally. I’d ac­knowl­edge that novel, ground­break­ing the­o­rems are still forth­com­ing.

• Thank you for the pos­i­tive feed­back. (A very un­der­rated thing in terms of en­courag­ing free con­tent pro­duc­tion.) I can go back to each post and add a link to the next one. I am con­cerned that I may want to add, re­ar­range, or even delete in­di­vi­d­ual posts at some point, but I sup­pose that’s no rea­son not to add in the links right now for con­ve­nience’s sake.

# Ex­am­ples of Categories

10 Oct 2019 1:25 UTC
24 points

# Cat­e­gories: mod­els of models

9 Oct 2019 2:45 UTC
42 points