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NikolajK

23:00

y on a function returns the function applied an infinite number of time. If you code the function with a break condition, you get loops

for logic, etc. see also

Church encoding

In mathematics, Church encoding is a means of representing data and operators in the lambda calculus. The data and operators form a mathematical structure which is embedded in the lambda calculus. The Church numerals are a representation of the natural numbers using lambda notation. The method is named for Alonzo Church, who first encoded data in the lambda calculus this way. Terms that are usually considered primitive in other notations (such as integers, booleans, pairs, lists, and tagged unions) are mapped to higher-order functions under Church encoding. The Church-Turing thesis asserts that...

if people speak of functional programming (as opposed to procedural or object oriented programming), they mean a language with features akin to the good old first programming language from the 30's.

"functions as arguments to other functions"

0celo7

@ACuriousMind "Chern classes were originally defined to be elements in $H^*(M,\mathbb{Z})$ via algebraic topology." What does $H^*(M,\mathbb{Z})$ stand for? The author doesn't explain it.

Something homology I bet.

Aha, cohomology ring.

ACuriousMind

@NikolajK Yeah, I have very vaguely heard of that

@0celo7 Either homology or cohomology, I think te latter, but I always forget which position of the asterisk is which

NikolajK

me too

I mean I also forget where the stars are

0celo7

@ACuriousMind Hmm, these notes mention a "singular cycle" and a "universal bundle"

NikolajK

I don't like homology, it's lengthy and in the end you just classify some topologies

cohomology (deRham) has differential equations, at least

0celo7

23:09

I'm considering asking a question on Math.SE.

@ACuriousMind You say that all characteristic classes have integer integrals?

ACuriousMind

@NikolajK I think it's nice on CW-complexes and such

NikolajK

universal bundles are like maps into universes in typed lambda calculus :)

ACuriousMind

@0celo7 Just the ones I've seen so far, which is really just $\int F\wedge F$ (think that's the second chern class or something).

NikolajK

for physics:

Classifying space for U(n)

In mathematics, the classifying space for the unitary group U(n) is a space BU(n) together with a universal bundle EU(n) such that any hermitian bundle on a paracompact space X is the pull-back of EU(n) by a map X → BU(n) unique up to homotopy. This space with its universal fibration may be constructed as either the Grassmannian of n-planes in an infinite-dimensional complex Hilbert space; or, the direct limit, with the induced topology, of Grassmannians of n planes. Both constructions are detailed here. == Construction as an infinite Grassmannian == The total space EU(n) of the universal bundle...

0celo7

What does $\text{Hom}$ mean?

NikolajK

23:11

the functor mapping objects X to Hom(X,Z)

the latter is the set/class of all homomorphisms from X to Z

0celo7

Jesus this is getting complicated

NikolajK

did you just edit this?

0celo7

Yes, I didn't care about the functor, just Hom

What does "$\mathbb{Z}$ is PID" mean?

NikolajK

principal ideal domain

ACuriousMind

@0celo7 Principal Ideal Domain

0celo7

23:13

:( I think this question is too advanced for me

user54412

@0celo7 \text? tsk tsk \mathrm won't screw up italics if placed into an italicized environment, and \operatorname will do even better and get the kerning right :P

ACuriousMind

@0celo7 You are really getting to the point where you don't want to go through the "dig until you find something you know" process, I think^^

0celo7

@ChrisWhite I know, lazy

NikolajK

contravariant hom functors such as Hom(-,Z) are also called pre-sheaves and the core notion of the categorical approach to geometry

0celo7

I'm hungry and have to do actual homework

But I can't let this go

NikolajK

23:15

and by "categorical approach to geometry" I could just as well as "algebraic geometry"

Grothendieck topoi, for example, are categories of such objects

and topoi have internal logics, such as lambda calculus :)

Categorical logic

Categorical logic is a branch of category theory within mathematics, adjacent to mathematical logic but more notable for its connections to theoretical computer science. In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor. The categorical framework provides a rich conceptual background for logical and type-theoretic constructions. The subject has been recognisable in these terms since around 1970. == Overview == There are three important themes in the categorical approach to logic: Categorical semantics Categorical logic...

but that's not relevant for you of course

0celo7

$$H^{2j}(M,\mathbb{Z})=\operatorname{Hom}(H_{2j}(M,\mathbb{Z}),\mathbb{Z})$$

I feel like I should know what this means...

NikolajK

it's hard before you haven't visualized each of the objects in the right frameworks/categories

0celo7

What the heck does that mean!?

NikolajK

you think where each of the arguments live, and then you see what it says

$\operatorname{Hom}(H_{2j}(M,\mathbb{Z}),\mathbb{Z})$

it's about homs between rings, I assume

so because the image is Z

0celo7

You forgot the dollar signs

NikolajK

23:20

I don't see them anyway

but okay

so $H_{2j}(M,\mathbb{Z})$ is taken as an object in the category of rings

0celo7

I don't know any category theory

NikolajK

"space of all rings"

0celo7

Ohh, I think that the naturality of characteristic classes has something to do with this.

NikolajK

hence the theorem says that the structural maps (homomorphisms) from $H_{2j}(M,\mathbb{Z}$ to $Z$ is nothing new - they coincide with $H^{2j}(M,\mathbb{Z})$ (left hand side of the equation)

0celo7

We can pull back to a bundle and cycle pair that gives integers.

NikolajK

23:23

you know I posted this pic at the beginning

the fact that you don't know about Hom makes me think you didn't really bother with it anyway :D