Category Theory For Beginners: Topos Theory Essentials

[@JosiahWarren]

Thank you for your contributionnon popularizing category theory.

[@gucker]

6 hours! Intimidating :-) Thank you for your work!

[@xyzct]

Awesome. I'll have time this weekend to dive in.

[@RichardSouthwell]

enjoy

[@asalamkamal6365]

Allah bless your efforts I hope you will continue to create educational content ❤️

[@ryancress5974]

New video I love you!

[@RichardSouthwell]

Thanks man

[@him21016]

What is the difference between in and contained in? You've defined the subset relation and the in relation in exactly the same ways

[@user-gu5bo8ls1r]

In the case of the subset relation both arrows must be monomorphic (as I understand it)

[@alexanderlind9068]

Does this video overlap with your "Topos theory and subobjects"-video or is there a preferred order of viewing?

[@RichardSouthwell]

Neither depend on the other. This one is more advanced, if you get stuck "Topos theory and subobjects" gives a lighter introduction

[@alexanderlind9068]

@@RichardSouthwell Apologies for the late reply. Thank you very much for the help!

[@ronritekinamatigai]

I kind of understand the idea, but how do you formally define a morphism h:H→B being an element of a monomorphism m:A→B, that you write as h∈m? Edit: an, I think I found the answer myself, h∈m defined as: there is a morphism g:H→A, such that h = mg, right?

[@RichardSouthwell]

h in m if and only if there exists an arrow k such that h= m after k

[@ronritekinamatigai]

@@RichardSouthwell Thanks

[@jadpole]

Another thing I'd like to note, because this confused me originally: This is not the place to use the epsilon operator. "h in m" does _not_ mean "h is an element of m". Rather, in Set, it corresponds to "h(H) is a subset of m(A)". When you add logic, given a, b : X -> Ω, you get statements like "a AND b IN a", that is, A ∩ B ⊆ A, or {x∈X | x∈A and x∈B} ⊆ {x∈X | x∈A}

[@jadpole]

Looks like I am incorrect. I just noticed that Richard uses the epsilon notation at 1:13:00, in the forall. So I assume that "A ⊆ B" applies to objects, whereas "alpha ∈ beta" applies to arrows. I guess I'll have an "_of course, it had to be that way!_" moment at some point, while watching his 'Mitchell-Bénabou Language' video. EDIT: Also, just above, he wrote "a ⊆ b", where a, b are arrows. Hypothesis 1: ∈-notation is used on arbitrary arrows, wheras ⊆-notation is used when comparing monomorphisms. Hypothesis 2: He used ∈-notation when h is on the left, because in this context, h can be viewed as containing a single element.

[@him21016]

Is topos theory relevant to an undergraduate student? How does it relate to other areas of mathematics _other than foundations, set theory and logic_? (Which aren’t deeply relevant to most mathematicians)

[@ster2600]

Topos theory is also relevant to algebraic geometry and topology

[@anshanshtiwari8898]

Is it possible for you to trim this into six ~one hour long videos? It is a very intimidating. I will watch it regardless, just a small suggestion. :)

[@ohault]

Beyond set, does topos could help to support concept like indeterminism in non-classical logics ?

[@unplandivino]

Topos really gives a very direct, graphical intuition of how classical logic is a particular case. Lawvere/Schanuel's book, Conceptual Mathematics, is really good as an introduction. But it is amazing how many videos have made Richard to introduce all this (the links he put in the description!!) Also Alain Badiou's book, Logiques, that is an even more philosophical book, despite its ideological turn is a great, surprising work.

[@Achrononmaster]

Olivier, your question is too vague to answer. What is meant by "support" for a concept? What type of indeterminism are you talking about? As the video stated, topos theory and Cat theory are more like languages, and so any concepts expressible in that language are therefore "supported". Short answer then is for sure non-boolean logic is supported, fuzzy logic, any consistent logic that can be finitely schematized. But so were they supported in Set theory, just not explicitly at the axiom level. Set theory just makes constructing non-classical logics more cumbersome.

[@Achrononmaster]

@17:58 whoa man! Squaring an infinitesimal to "get zero" seems highly dubious. Zero relative to what? In the hyperreals we already have that sort of thing. Surely topos theory can plumb some of the Surreal Number concepts too, which basically are generalizing reciprocality. When you understand the transfinite cardinals just "go on forever" in ever higher cardinality, then the same must be true for infinitesimals... there can be no smallest type of infinitesimal. If you can square one of them and get literal zero it just means you've coarse grained too much and have filtered out "smaller" infinitesimals (thought of as reciprocals of higher transfinites). This, to my mind, is the real challenge for our generation, how to get some symmetry between the infinite hierarchy of transfinites and an equally infinite hierarchy of infinitesimals. If Conway did not truly complete that program with the Surreals, someone else will, or "should" (if there is such a thing as moral "aught" in sociology of mathematics!). Only then will we have any claim to have "understood" _the continuum_ (as a coherent meaningful concept). If you cannot identify (in principle) a unique generalized reciprocal of some arbitrary transfinite cardinal, then you've (I claim) missed a vast infinite class of infinitesimals (or you've glombed them together barbarically, so-to-speak).

[@Achrononmaster]

Mathematically you can maybe get a "perfect language" (whatever that means) but not humanely universal. Mathematics is a lot more like engineering than some pristine idealized heaven of reasoning. Topos and Cat theory likened to rivets, nuts, bolts, wiring, while most mathematicians will appreciate knowing that sort of stuff glues things together, they'll be working at a higher level on things like (metaphorically) chassis, engine blocks, whole vehicles, whole microchips, modular robots, etc. If not then they'll be making no progress in their field.

[@xicad1533]

Toquoid theorisy 😄