point-set-topology

[tiny ridge]

You can take the 2D example I have in mind and then product with an S^2 to produce examples

You want a simply connected example?

[feral copper]

Sure!

[umbral panther]

Right, the Whitney trick

The signs take values in the group ring of the fundamental group, or something. You have a path in S between intersection points and a path in S’. If you can find a Whitney disk you can form an isotopy to remove the intersections. In dimension > 4 you just use general position. In dimension 4 it’s false in the smooth category. It might be true in the topological category, but it’s very difficult for good fundamental groups and open for general ones

[feral copper]

Ah yes

Good catch

[tiny ridge]

This is not about Whitney

Theres no reason you can find a topological disk in the first place, in the complement of the pair of surfaces

Let me think of an example

Just imagine the following scenario.

Take S^4

Take two intersecting spheres S1, S2 which intersect at exactly two points, one with oriented # 1 and other with -1

In the complement, perform a surgery so that these are no longer compressible

(This is how you produce 2D examples)

[feral copper]

Ah yeah you'll have \pi_2 but you can make sure you don't have \pi_1

[tiny ridge]

Yes

I need to run but I can cook up an example for you later if yall dont figure it out by the time

[feral copper]

I mean you just gave me an example!

Well, I can believe that it's a working one for now

[tiny ridge]

I haven’t exactly told you explicitly what you need to do in the complement. But this should do the trick. One has to attach 2 handles in the complementary regions

Working in analogy in 2D where you attach 1 handles in the complement

[feral copper]

Thinking of what you described as a stabilized Heegaard splitting, it makes me wonder if S²xS² could be a (simpler) counter-example

But anyway, thank you!

[tiny ridge]

Most likely (S^2 x S^2) # (S^2 x S^2) will work

[unreal stratus]

Why #s ugh twitter has ruined gen z's minds

[feral copper]

[trail charm]

when $\operatorname{O}(n)$ acts on $V_k(\bR^n)$ transitively, why is its isotropy group $\operatorname{O}(n-k)$?

[gentle osprey]

anamono

[trail charm]

where V_k(R^n) denotes the stiefel manifold

[dry jolt]

The isotropy subgroup in this case is just the space of orthogonal matrices which fix the k-frame and act however they want on the orthogonal complement of the subspace spanned by the k-frame

In particular, the isotropy subgroup of the frame given by, say, the first k vectors of the standard basis would consist of the matrices whose upper left k x k minor is the identity and whose lower right (n-k) x (n-k) minor is whatever orthogonal matrix

[trail charm]

oh gotcha

so an element of O(n-k), an (n-k) x (n-k) matrix, acts on a k-frame of R^n, an n x 1 matrix, by "extending" the element of O(n - k) to an n x n matrix?

[tiny ridge]

@feral copper I have a nonoptimal example, if you still want one.

[trail charm]

oh wait sorry k-frame is set of vectors

[feral copper]

Sure do!

[tiny ridge]

Take the following Kirby diagram, all links with 0 framing

[feral copper]

What are colours?

Red is 1-handles?

[dry jolt]

well an element of O(n-k) is an (n-k) x (n-k) matrix

[trail charm]

right

[tiny ridge]

No, all are 2 handles. Colors are meant for exposition, for the following claim: boundary of this handlebody is S^3

[feral copper]

Oh okay!

[trail charm]

i guess im just wondering what the group action looks like, bc in my head it's matrix multiplication

[tiny ridge]

Proof: Boundary of a handlebody given by Kirby diagram doesnt change if you flip a 0 framed handle unknot to a circle-with-dot ie a make it a 1-handle.

[trail charm]

and im not sure how we can multiply an (n-k) x (n-k) matrix by an element of the k-frame

[dry jolt]

Yeah, the group action of O(n) on V_n, k is matrix multiplication, but you look at the image of each vector in your k-frame under multiplication by the matrix

[tiny ridge]

Flip all the red ones, then theyre 1 handles which cancel with the green 2-handles

[feral copper]

Yup

[dry jolt]

It's like, where does a matrix A send the standard basis of of R^n? To the frame consisting of the vectors Ae_1, ..., Ae_n

[tiny ridge]

So cap off the whole handlebody by D^4 to get your 4 manifold. Now the sphere given by the white 2-handle has intersection number 0 with the sphere given by any of the green ones

But they cannot be isotoped to be disjoint

[trail charm]

ohhhh gotcha

[feral copper]

Any of the green? Not just the right one? But yeah this does work!

Ah no, both indeed

[tiny ridge]

Yup

[feral copper]

Thank you for this great example!

[tiny ridge]

In general the intersection form matrix is just the linking matrix for a Kirby diagram

np

[feral copper]

How did you come up with that so quickly? xD

[tiny ridge]

I was thinking about it on my way to buy myself some clothes

Got bored in the mall

[feral copper]

So now I kinda want to ask the weaker question:

If Q(S,S')=n, are there surfaces F and F' respectively homologous to S and S' such that F intersects F' in exactly |n| points?
Which actually is terrible in my case, because I need that F and F' have the same genus as S and S'...

[tiny ridge]

Ought to be true.

[feral copper]

And actually, now that I think about it, being able to control the genus is equivalent to the Thom conjecture, and that's definitely not an easy thing x')

[tiny ridge]

Effectively

[feral copper]

Screw me x')

Alright, thanks for the discussion!

[tiny ridge]

Thanks for the question

Cut and paste topology, as always, rocks

[white oxide]

Who had that 60 page "All you will ever need" general topology pdf?

[hoary breach]

ibsen lol

you're taking about hatcher's notes right?

[white oxide]

If it fits the description, probably yeah

[coarse night]

@tiny ridge

[hoary breach]

just search ibsen bookworms in the search bar

[white oxide]

Thanks, now I can start to convert interested people form other fields into our cult mathematicians

[thorny agate]

Hatcher's notes are 40ish pages, Chapter 1 of Bredon is 60ish pages

Bredon covers alot alot more though, Hatcher's notes are relatively gentle

Very much appreciated

https://pi.math.cornell.edu/~hatcher/Top/Topdownloads.html

For example Urysohn's Lemma and the Tietze Extension Theorem are notable omissions.

[white oxide]

Oh, I already found it sorry if I didn't make it clear

[thorny agate]

Ah

[pseudo coral]

Glen Brendon’s Topology & geometry ?

[thorny agate]

indeed

[thorny agate]

7 weeks,
7 sections of topology to cover

self studying looking good

[unreal stratus]

Anywhere good to look for p-adic / p-completed topological complex K theory

I can't find much on it specifically beyond it being a special case of Morava E-Theory

[umbral panther]

What do you want to know?
What more could you want than an integral model? It tells you everything

[unreal stratus]

What is an integral model

I am new to this stuffs, I know about complex K-theory but not the p-adic variant basically

[umbral panther]

p-adic K-theory is the p-completion of K-theory. You destroy information when you pass from K-theory to p-adic K-theory. By integral model, I just mean something that’s leads to it by completion

[unreal stratus]

Sure yes

Are you basically just saying like all I need to know is that you p complete KU?

and then work w that lol using properties of p completion and so forth

Or am I misinterpreting

[umbral panther]

Pretty much. That’s why you won’t find people talking about it

[unreal stratus]

Sure

(It is useful tho right?)

Idk I am doing a summer project w some computations w this and cohomology lol gonna talk more to supervisor ig

Thank u

[umbral panther]

There are Morava type things you can say about it. Morava says things should not 2-periodic, but 2(p-1)-periodic. You can build an idempotent that splits off the part with the larger period. That’s a reason to isolate the p-part

Sometimes you run across a spectrum that isn’t K-theory, but whose p-completion is. The algebraic K-theory of the algebraic closure of a finite field of different characteristic has p-completion equal to the p-completion of connective K-theory. Quillen talks about this in his ICM address, cohomology of groups

[unreal stratus]

Thanks interesting

[tiny ridge]

snoozefest; aka i try to learn category theory

[unreal stratus]

Tfw category theory thread in topology

[tiny ridge]

going to educate myself on infinity topos crap because you all gave me a hard time about it

at least until i get bored

[limpid hedge]

yo, I need to see to which covering of S^1xS^1 the subgroupe 0xZ sub ZxZ corresponds. I know that this subgroup corresponds to the covering R^2/0xZ iso RxS^1. Now I have to see what is the map RxS^1 -> S^1 xS^1 and I have no clue how to get it. Intuitively it hsould be the map (s,e^it)->(e^is, e^it) but how to get it without "guessing"?

[coarse night]

you can try to prove your guess is correct

finite product of covering maps is a covering

[limpid hedge]

Ye I can prove my guess but I'd like to know how to find this map without guessing

[empty grove]

If you have a space X with universal cover U and fundamental group G, then X is homeomorphic to U/G. The cover corresponding to H < G is homeomorphic to U/H. The map from this cover to X is the same as the natural map from U/H to U/G.

This is either the definition of "cover corresponding to the subgroup H" or a theorem depending on how you have defined things

[limpid hedge]

alright thank you

[mint rose]

Hey, I'm struggling with this basic fact, why is $\pi_H$ closed if $H$ is compact?

[gentle osprey]

digiterate

[mint rose]

Second question, I see why the example 3.6 is true if X is Hausdorff, but not without that hypothesis... Is it true in the general setting or not?

[unreal stratus]

Good question - I've a feeling it's not true but trying to find an example lol

[tiny ridge]

How do you remember which ones are face maps and which ones are degeneracy maps in a simplicial set?

The mnemonic "inclusion of a face is a face map, collapse onto a face is a degeneracy map" works well but for the simplex category. In a simplicial set everything gets contravariant'ed

In fact, the standard notation is also backwards. d_i denotes face maps, s_i denotes degeneracy map.

Hm, not really. d_i is like "part of the boundary", from the contravariant POV

[cosmic socket]

It’s not backwards d_i:X[n] -> X[n-1] assigns to an n-simplex which n-1 simplex that lives on the boundary omitting it ‘i’th vertex

[tiny ridge]

I wouldn't call that a face map, I would call that the ith boundary map

But yeah I guess I should really imagine d_i as the ith boundary face of a simplex, and s_i as the (n+1) simplex given by fudgifying the ith vertex

[empty grove]

What would be the difference between ith boundary and ith face?

[cosmic socket]

I mean that’s just terminology what does ith boundary mean

[tiny ridge]

Face map feels like inclusion of a face into the simplex

Different domain/target

[tiny ridge]

Yeah it was a terminology question 😛

"How do you remember X"?

[cosmic socket]

I think of it as asking the question what is the ‘ith’ face of this simplex?

[empty grove]

So you interpret the ith face as the inclusion of the ith boundary?

As in one is an object and one is a map

[tiny ridge]

Yeah.

When I think of a stratum, it is always an inclusion of the stratum in the big space

Boundary goes the other way

[empty grove]

In that case why don't you think of boundary as inclusion of the boundary rather than the boundary space itself?

[tiny ridge]

No, because I write dX = Y

Boundary of X is Y

X -> Y

Language has directionality for topologists who cannot use diagrams

[empty grove]

Stop being difficult ibsen

[tiny ridge]

?

This is how I think

[empty grove]

[tiny ridge]

Sorry if it's difficult for you 😛

[empty grove]

No but I think in this case we do regard ith face as the object rather than the map

So it is assigning to a simplex in X its ith face

[tiny ridge]

Which makes perfect sense from the POV that it's called d_i

"ith component of the d map"

d_i : (n-simplex) -> (n-1 simplex)

So that does answer my question 🙂

[empty grove]

[empty grove]

When you write this, do you mean set of n-simplices and set of n-1-simplexes?

[tiny ridge]

Yes, throw away the simplex category and the presheaf POV

That's useless

[cosmic socket]

It’s probably helpful to just draw out a random simplicial complex and right down the simplicial set that has it as the geometric realization

[tiny ridge]

I have done this exercise before, bigraded. The question was one of terminology

[empty grove]

That's the best part to me

Because of the nerve realization adjunction

[cosmic socket]

presheaf perspective IMO is useful because it encapsulates all the coherences you need very simply

[tiny ridge]

Sure

It is currently not useful to me

Once it is I will switch POV

[cosmic socket]

For example what is the set of maps $\Delta^n \rightarrow X$

[gentle osprey]

bigradedSphere

[cosmic socket]

It’s not obvious to me without yoneda

[tiny ridge]

wtf lol

ok

It is obviously just X[n]

[empty grove]

You should have clarified that these are both simplicial sets

[tiny ridge]

Yeah that took me a minute

A simplex is a simplex

What is the big deal

It can be degenerate but those are all in X[n]

[empty grove]

How would you formally prove this though

[tiny ridge]

Yes, Yoneda. But it is obvious without it

[empty grove]

[tiny ridge]

Do you guys seriously formally machine verify everything everytime

I do math by having a network of intuitive understanding

[empty grove]

I think of Δ^n as the free sset generated by one n-simplex

[tiny ridge]

lol ok

[empty grove]

Which is equivalent to Yoneda ofc

[tiny ridge]

Its just a simplex to me (with all the buffing)

The hobuff

[cosmic socket]

I think of simplicial sets as basically their geometric realization

[tiny ridge]

I am going to call simplicial set a hobuffed simplicial complex from now on

[empty grove]

Both of you are basically monkeys damn

Clearly inferior perspectives

[cosmic socket]

How to think of sSets?

[empty grove]

presented without my perspective

[tiny ridge]

Well, yes, but I don't try to hide it behind poetic ornamentations

[cosmic socket]

I think I go back and forth between perspectives for what is useful

[tiny ridge]

That's what the homotopy theorists do

Not my fault if the notions at its core are actually all very simple

[empty grove]

They're not I need to feel proud for understanding them

[cosmic socket]

No one said they aren’t simple. That’s probably why they are useful

[tiny ridge]

Inner horn fillability is just saying that the composable morphisms (homotopies) can be hocomposed

If Thurston wrote this, he'd have just written "Composables can be hocomposed"

But instead we have linguists

[tiny ridge]

Let C be a quasicategory. For any two objects x, y in C, Hom(x, y) also admits the structure of a quasicategory. Namely, its a simplicial set whose vertices are 1-cells [x, y] of C, edges are degenerate 2-cells [x, y, y] of C, etc. Inner horn fillability of Hom(x, y) follows from (1 dim higher) inner horn fillability of C, yes?

I guess alternatively its the simplicial set Hom_x,y(Delta^1, C)

simplicial Hom (with boundary conditions)

If C is Kan, this is clearly Kan in analogy with fibrant objects in topology. I can't entirely check the inner horn condition in my head though

When are simplicial Hom's of quasicats a quasicat, in general?

Wait. Hom_x,y(Delta^1, C) feels like it is actually Kan for any quasicat C

[empty grove]

The usual one or the Hom_x,y?

[empty grove]

Yes it needs to be

[tiny ridge]

Morphisms are hoinvertible for example

In Hom_xy(Delta^1, C)

[empty grove]

Ye

[tiny ridge]

I guess this is what people mean by (infty, 1) categories

[empty grove]

[tiny ridge]

No?

[empty grove]

Ye lol

[tiny ridge]

Ok

[empty grove]

Another way to think about it is that a category enriched over (n,r)-Cat is an (n+1, r+1)-category

So (infty, 1) should be those enriched over Kan complexes

[tiny ridge]

Makes sense

Thats not a "way to think about it"

That's just an elaborate language rigamarole

[empty grove]

That is so way to think about it

You should focus on studying instead of focusing on starting another half hour argument with me

[tiny ridge]

? This is how topologists study

[empty grove]

[empty grove]

Not gonna start another argument smh

[tiny ridge]

So higher category theory is just a bunch of blind men trying to visualize high-dimensional horns of simplices

Makes sense

[empty grove]

Visualize?

[tiny ridge]

Oh, they can't.

Good point

They're blind

[empty grove]

[mint rose]

Any idea of how to show this?

I've been struggling for a while, I'm at loss

[arctic relic]

Take the real numbers since they form a topological group. Take a discrete subgroup, so just a set of numbers. Left translate the real numbers by this discrete subgroup. Can you find an element in the real numbers that isn’t in the translation? Or in other words, take an open set U in R and a the left translated U and look at the intersection. Is this action properly discontinuous?

[mint rose]

I can see why it's properly discontinuous: take $U$ a neighbourhood of $e$, such that $U\cap \Gamma = {e}$, then $m^{-1}(U) \subseteq G\times G$ contains $(e, e)$ and so there exists an open set $V$ such that $e$ is in $V$ and $V \times V \subseteq m^{-1}(U)$. Up to replacing $V$ by $V \cap V^{-1}$ we may suppose $V = V^{-1}$, so we have $VV^{-1} \subseteq U$. For any $x \in G$, $V_x = xV$ is a neighbourhood of $x$ and $V_x \cap hV_x = \emptyset$ if $h \neq e$, since otherwise we have $h \in V_x^{-1}V_x \cap \Gamma \subseteq U \cap \Gamma = {e}$ which contradicts $h \neq e$

[gentle osprey]

digiterate

[coarse night]

what's your definition of proper action then?

[mint rose]

That the preimage of a compact set via the map $G\times \Gamma \to G\times G, (g, h) \mapsto (g, hg)$ is compact

[gentle osprey]

digiterate

[coarse night]

O don't think that's what they mean by a proper action

also, G acts by homeomorphism so your properness is obvious

[mint rose]

this is the definition in the script of the course I'm following

[mint rose]

I'm not sure why?

[coarse night]

by a proper action, it means that for all x, there is a nbd N of x s.t. gN ∩ N is empty for all but finitely many g ∈G

[coarse night]

compactness is preserved by homeomorphisms

[mint rose]

is this equivalent?

[coarse night]

no it's not equivalent but it what means for an group action to be proper

[mint rose]

but this map is not a homeo?

it's not surjective for example

[coarse night]

completely different terminology, it's unfortunate

ok wait, what you sent contradicts the definition I know

where did you get that?

[mint rose]

from my course on adelic number theory

it also is the same definition in General Topology by Bourbaki

[umbral panther]

Only if the group is discrete

[coarse night]

yeah just looked up

[mint rose]

Is there a way one can deduce properness in the above sense from being properly discontinuous in this context?

[umbral panther]

If G is discrete and X is locally compact, properness follows from proper discontinuous

[mint rose]

local compactness includes Hausdorff in your def?

[umbral panther]

Yes

I think the argument goes through without Hausdorff

[mint rose]

Because for context, the goal is to show local compactness is preserved when taking the quotient under this action

Hausdorff is preserved because a discrete subgroup is closed

Then local compactness I guess because the projection map is open...

I guess properness is not really needed then... Out of curiosity, what's the argument you had in mind?

[umbral panther]

Properness is local on the target

To prove properness near x,y, if they are not in the same orbit, take a neighborhood whose pre image is empty. If they are in the same orbit, reduce to x,x. Take the neighborhood N given by the hypothesis of proper discontinuity. Shrink so that it is preconpact. The pre image of the closure of NxN is finitely many copies of the closure of N, which is compact

[urban zinc]

Btw where does the name "proper" come from? as in a map where inverse images of compact sets are compact

Is it just a random name

[umbral panther]

Probably they wanted it to be close to projective

[unreal stratus]

Really?

I don't see how those are close

[mint rose]

was the geometric or topological notion first 🙂

[coarse night]

https://math.stackexchange.com/questions/2434982/why-is-the-definition-of-a-proper-group-action-the-way-it-is#2437237
somewhat relevant

[umbral panther]

I just meant the same first letter.
But, no, I was not serious

[fading vale]

I feel like it's probably the same type of terminology as normal

[umbral panther]

I think properly discontinuous is older. I doubt it influenced proper map, though

[fading vale]

I.e could just as well be called "good"

[mint rose]

I'm a bit confused on the last part, won't the preimage of NxN be just Nx{e}?

since for any h != e, we have hN disjoint from N

I mean the end result is the same

[umbral panther]

The definition given of proper discontinuous of that there finitely many. You could shrink N to get rid of them. So why isn’t the definition just one? I don’t know. Maybe to prepare us for groups that aren’t discrete? Or maybe two versions of the definition got mixed up? One is about there exists N. The other is about all pre compact N

[mint rose]

Ahh I thought there was just one in proper discontinuous

[late iron]

If I have a continuous map between CW complexes f: X to Y, is there a well-defined notion of an induced map between some vector bundles over them?

[fading vale]

I don't imagine there would be anything that would particularly preserve a lot of information

Like given a map between the zero sections as subspaces I dont see a natural extension to the entire bundles

[late iron]

Yeah I was just wondering abt this due to a question on a math competition I saw

The original question was as follows

“Let X be a CW complex whose second integral Cohomology is 0, let V be a real vector bundle on S^2 and f: X to S^2 be a continuous map, show that f^*(V) is a trivial bundle.”

Any idea what f^* means here?

[fading vale]

Maybe I misunderstood what you meant

Give me a moment

[bright acorn]

https://math.stackexchange.com/questions/903678/embedded-lie-subgroups-are-closed

How can I show that xH open implies xH dense in the closure of H, as the op describes?

[fading vale]

Given bundles E -> X, E' -> Y and a continuous map f: X -> Y

there is no natural notion of an induced map f*: E -> E'

Given a bundle E -> Y and a continuous map f: X -> Y, there is a well defined bundle f^*(E) -> X over X with a natural map f^*(E) -> E making the natural diagram commute

(there is a sort of notion of pushforward, taking bundles over X and yielding spaces over Y, as well but it doesnt yield bundles in general so i would not recommend thinking about it rn)

[gentle osprey]

mothemotics

[fading vale]

Does that make sense

[umbral panther]

nlab seems to say that Bourbaki invented the term “proper map” and “proper action” in the same book. So maybe proper map is named after properly discontinuous. Except that nlab cites different editions. If proper action were added in 1960, proper map in 1951 was not named after it

[mint rose]

Bourbaki defines proper action as an action for which the map (g, x) -> (gx, x) is proper, not much about properly discontinuous

I mean if the book is called General Topology

[urban zinc]

Oh that's interesting

[tiny ridge]

I beg to differ

[umbral panther]

Proper actions are a generalization of properly discontinuous actions, which only makes sense for discrete groups. I’m pretty sure this use of proper is named after properly discontinuous

[mint rose]

but properly discontinuous actions are proper if X is locally compact, is it true in general though?

is it really a generalization?

Also if Bourbaki came up with this nomenclature, he did not mention properly discontinuous anywhere in the book

[umbral panther]

Yes, if the group is discrete, the action is proper iff properly discontinuous, regardless of hypotheses on X

Britannica 1902 defines properly discontinuous as a property of groups, not actions. It seems to mean discrete, and improperly discontinuous as, I dunno, maybe not connected, such as the group of rational numbers. In fact, it says that PGL2(R) is improperly discontinuous, so it really just means disconnected

Not that I trust Britannica as an authority on mathematics. But it’s surprising that it’s a very early example of the use in English. The earliest use I see in google books is 1899. Maybe Britannica is translating a usage popular in another language?

[eager vigil]

Would the closure in (iii) be any different in X+ than in X?

[unreal stratus]

It definitely can be

I mean consider Y = X

[marsh forge]

You yourself have already argued this

in (i)

If Y is a subset of X with noncompact closure then it will have a different closure in X^+

Part (iii) wants you to use parts (i) and (ii) to break the possibilities into cases

[eager vigil]

Ok, but if x is in the closure of Y in X but not in X+, then there must some compact set C of X containing Y but not x, right?

Or am I missing smth?

[marsh forge]

sure

oh sorry no

Some compact subset C of X^+

[eager vigil]

Oh? But doesn't C have to be compact of X?

[marsh forge]

The closed sets in the definition of a closure are taken from the ambient space

[eager vigil]

Any new open set in X+ is of the form X+ - C

[marsh forge]

if you're talking about closure in X, then C is a subset of X

if in X^+, then C is a subset of X^+

[marsh forge]

I'm not sure what you're trying to write here

[eager vigil]

When we topologize X+ from X, don't we define all the open sets to be all open sets in X and all sets X+ - C where C is compact in X?

[marsh forge]

I have no clue what X+ - C is supposed to mean

Oh do you mean $X^+ - C$

[gentle osprey]

themaxj

[eager vigil]

Yes

[marsh forge]

ah okay

That seems reasonable 2 me

[eager vigil]

Well, then any point which is in the closure of Y in X but not in X^+, must have some neighborhood in our new space which is disjoint from Y, right?

This new set must be of the form X^+ - C for a compact C in X

[marsh forge]

Sure

[eager vigil]

But then since X is Hausdorff, C is closed

And doesn't contain x

So we can take a neighborhood disjoint from C in X of x

and hence it is not a limit point of Y in X either

[marsh forge]

seems reasonable to me

[eager vigil]

But then the closure of Y in the new space cannot contain fewer points, right?

[marsh forge]

yep

[eager vigil]

But the reverse direction: if x is a limit point of Y in X^+, then all normal open sets in X of x intersect Y

Hence x is also a limit point of Y in X

[marsh forge]

That doesn't quite work

(although its close to working)

you're making an implicit assumption that isn't true

[eager vigil]

Hmm

[marsh forge]

Think about whether what you just said is compatible with parts (i) and (ii)

You'll get a contradiction

(Hint, part (iii) follows extremely formally from (i) and (ii))

(Hint 2, you can basically see everything that could happen here in the example of X=(0,1) with compactification the circle)

[eager vigil]

Just to be understand

Could you tell me what about the statement is wrong cause I'm having a hard time figuring it out

[marsh forge]

You are assuming that x is in X

[eager vigil]

Ohhhhhhh

Jesus

Ofc

[marsh forge]

your proof does work for any x in X though

[eager vigil]

So, we would then get that the closure is the same if the set is compact, and if not compact it is its closure in X adjoined with the point at infinity?

[marsh forge]

yep

[eager vigil]

I see

Thanks a lot

[marsh forge]

np

[tiny ridge]

What is a nice way of thinking about the simplicial set corresponding to a chain complex under Dold-Kan? You can of course define X[n] = Hom_Ch(S_*(Delta^n), C_*). This is a bit unappetising to me

[marsh forge]

I want to say that there should be some correspondence between simplicial homotopy groups and homology of the resulting complex

but i forget the precise statement

oh its literally just the same according to nlab

idk if thats satisfying but at least its pretty concrete?

[tiny ridge]

I like that

[umbral panther]

Nah, that’s a homotopy statement, not a concrete statement. The Dold Kan equivalence is of categories, not just up to equivalence

[marsh forge]

huh?

I'm not trying to say its the full strength of dold kan I am just saying its one way to interpret it. Notice that a priori its not obvious that an equivalence of categories would respect this (of course in general equivalences of categories can completely ignore model structures)

[umbral panther]

You might as well with work with topological abelian groups

[marsh forge]

?

[umbral panther]

Anyhow, compare the Dold-Kan theorem to the Dold-Thom theorem

[tiny ridge]

DT is a good one

[tiny ridge]

The Dold Kan simplicial realization of the hom complex Hom(C, D) is much easier for me. A map from the 1-simplex into this is the same information as a chain homotopy.

By definition

higher simplices are evidently higher homotopies etc

I suppose now understanding the derived category of an abcat as a \infty category is walking distance from here

the Hom sets are clearly simplicially enriched

Well, this simplicially enriches Ch(A). One has to do a simplicial localization next

“Of course” that should be possible but I don’t know how to actually do it

[umbral panther]

Yesterday there was the discussion of face maps and degeneracy maps. d stands for boundary. s stands for stupid. Because you don’t need degeneracy maps.

You don’t need degeneracies for semi simplicial sets to model spaces and you don’t need degeneracies for semi simplicial abelian groups to model chain complexes. Where by model I mean an equivalence of infinity categories

The reason to do full fledged simplicial sets is more precise statements, like that simplicial abelian groups are the same as chain complexes, before inverting equivalences

[marsh forge]

This is explained in HTT ch1 if you’re interested

Well how you get an infinity category anyway

[tiny ridge]

Thanks, Ill read that

[tiny ridge]

Good remark

Now you’re talking like a topologist

[umbral panther]

I remember the other technical application of degeneracies. They make the nerve a product persevering functor. This allows you to iterate the clarifying space functor and build K(pi,n). This sounds close to DK, so this is the more general principle of what degeneracies are for

[tiny ridge]

I remember degeneracies being essential for products being correct

But I don’t remember anything other than that. I think one runs into trouble with I x I

Oh this is what you said, nerve isnt product preserving otherwise

Its not hard to see, I dont need to remember. You’ll always have degenerate simplices like [0, 1, 1] in the product

[silver umbra]

is this claim even true?

even if u just take a single 2-simplex

[umbral panther]

A single triangle has three edges, which is odd, so you can’t pair them off for identification

A point inside a triangle is a manifold point. A point on the inside of paired edge is part of two triangles and so is a manifold point. The only problem is vertices, which may be part of many triangles

[silver umbra]

ahh ok

this might sound really dumb but is there an easy to see then that the closed disk is not locally homeomorphic to R^2?

for points on the boundary u can only draw like part of an open disk around them

[umbral panther]

If you delete a point from the boundary, it doesn’t change the fundamental group, even the fundamental group of a small neighborhood

[urban zinc]

Btw what are the prerequisites for that?

[marsh forge]

For ch1? Probably just category theory and knowing what simplicial sets are

The issue with HTT and HA isn’t really that there’s a ton of prereqs, more so it requires absurd mathematical maturity (there are no exercises for example) as well as being really hard to motivate if you aren’t familiar w homotopy theory already

Like in context HTT is clearly just reformulating lots of old ideas with better machinery

But out of context it’s like “what the fuck”

[urban zinc]

Ahh okay

Thank you! I'll take a little look

What does HA stand for?

[marsh forge]

Higher algebra

[urban zinc]

👍

[hidden crag]

potato why did you delete

😭

[marsh forge]

I gave the canonical higher algebra rely

Reply

[unreal stratus]

Well Max was already saying it and I wanted to give it to him to say lol idk

Idk

[hidden crag]

you were first even on my screen

should've taken the snipe

[unreal stratus]

I was first lol

[novel acorn]

(some people very much don't like HTT but personally to me it's one of the best texts on the subject out there)

[tiny ridge]

im looking at bit sof HTT and HA; its fairly natural, but boring

[marsh forge]

Yeah it’s also dry as hell

[tiny ridge]

ill force myself to learn tho

i think i understand what Max meant when he said you should only begin to read this stuff if you encounter it

[marsh forge]

It’s like the pilot fish meme w higher categories as the light and quasicstegories as the scary fish

[tiny ridge]

lmao

[marsh forge]

I will say tho, with every pass of HTT and HA content I pay more attention to details and start to find them beautiful

Maybe this is Stockholm syndrome

[tiny ridge]

thats probably also because youre much more naturally inclined

[marsh forge]

I think my brain is just irreparably broken

[tiny ridge]

lol now if you roast yourself i wont have a purpose for doing this

[marsh forge]

Lmao dw knowing I’m a lost cause doesn’t stop me from posting homotopy brained takes

Here’s a good one

From a paper of my advisors

(Sorry for shit quality it’s a phone pic of an e-ink tablet)

[tiny ridge]

i like that notation

[marsh forge]

You like Ab for the derived infinity category of Z?

[tiny ridge]

its by Lurie right

better than writing D^b(Ab) every time if youre using it

[marsh forge]

Burklund Hahn Senger

I actually don’t mind it but it’s very homotopy brain

[marsh forge]

(Altho this is the unbounded guy)

[tiny ridge]

ah

oh i meant the notation is originally by lurie

[marsh forge]

Oh maybe

[tiny ridge]

oh youre reading the BHS paper nice

[marsh forge]

Oh this paper and I have spent much time together

The appendices are super cool

[tiny ridge]

its a bit like how a perverse sheaf is just called a perverse sheaf instead of a perverse complex of sheaves

or constructible sheaves for the matter

[marsh forge]

Right yeah that’s fair

[tiny ridge]

do you know why its a good question to ask which homotopy spheres are stein fillable; i know some results in this direction but do not know the origin story of this particular q

if Yasha asked it there must be a reason to it

theres some results of Abouzaid which indicates that the symplectic topology of T*S^n detects smooth structure of S^n

[unreal stratus]

Ohy eah lol

The heart of the category or smth right lol

[tiny ridge]

heart of the t-structure rather

[unreal stratus]

I remember finding that funny seeing someone write it heh

[marsh forge]

The heart notation is Good

I love that notation

[tiny ridge]

A simplicial abelian group ought to automatically be a Kan complex, yes?

[marsh forge]

I believe so

[tiny ridge]

Ok, just rememered I needed Ch(A) to be not just simplicially enriched but Homs to be Kan

for \infty cat

its because we have inverses

[unreal stratus]

Yeah

I remember yes this was really nice cause you can like define homotopy groups relatively painlessly for Kan complexes and this shows we have a nice class of objects for which the def works ig lol

[tiny ridge]

i vaguely remember this was useful at some point for me but i am not sure how

its been a long time since i learnt simplicial sets

I actually think you don't need "abelian" for this. Any simplicial group is a Kan complex.

All you need is inverses

Ya, this is OK

If you have a 2-horn, worst case an outer one, define the missing edge as e_0^-1 e_1. But that doesn't have the right boundaries, so scale it by an appropriate degenerate 1-simplex

The filling 2-simplex is the degenerate guy dropping down to this

I think this is useful for something obviously important, but I can't remember what lol

[marsh forge]

I mean Kan complexes are the fibrant objects in the usual model structure on sSet

So it comes in handy if you get them for free

Instead of having to do replacement games

[tiny ridge]

Yes, I think what I had in mind was some sort of an application eg, Diff(N) -> Emb(M, N) is a Serre fibration without having to check a bunch of things

Make simplicial models of everything, apply the above and get fibrant

(That's isotopy extension theorem, btw)

[unreal stratus]

Sorry yes groups is en7ff

[tiny ridge]

I think a small modification of the above proves that a morphism of simplicial groups G -> H which is levelwise surjective is a Kan fibration (of the underlying simplicial sets)

This shows, for instance, BH -> BG is a fibration (with fibers G/H) with relatively little effort, in the simplicial model of everything

@marsh forge @umbral panther I feel a little scammed. I looked at HTT and the way Lurie defines the derived \infty category of an abcat A is by looking at the nerve of the full subcategory of injective objects in Kom(A). In #1119853874749128764 I suggested dg enriching D(A) by simply taking Kom(Injectives of A) and using Hom-complexes for as the dg hom sets, but apparently that is "noncanonical". I thought there was a more intuitive way to localize \infty-categories that you guys had in mind 😉

Because, if you're going to take injective resolutions anyway, why bother with the infty nuttiness?

[marsh forge]

He uses a dg/simplicial nerve right?

[tiny ridge]

Should I read how to do simplicial localization or is there a better ref

[marsh forge]

That’s what I was intending to showcase

[tiny ridge]

Ah. Is that not essentially equivalent to simplicially enriching Ch(Injectives)?

I thought there'd be a way to avoid talking about injectives at all

[marsh forge]

It should be I think. The point is more so about how you change the naive nerve functor to get it to care about the simplicial enrichment

I thought you could too

[tiny ridge]

Oh I misspoke, this is not HTT but HA (Ch 1.3.2)

[marsh forge]

Hm

[tiny ridge]

Most likely I'm lost in the maze that's all

[marsh forge]

No I often over simply this construction

I might’ve done so again

Oh

Okay here’s a slightly slicker thing

You can construct D(R) as localization of the infinity category of complexes at the complexes with homology 0

One way to express this formally is to form the cofiber in Cat^ex which is known as the Verdier localization

This probably the most “natural” construction

[tiny ridge]

Aha.

OK, thanks, I will learn this POV

I need to know how to glue or coglue infty cats anyway

[marsh forge]

The theory of (Bousfield) localization at a system of objects you want to force to be 0 is somewhere in HTT as well

[tiny ridge]

Eventually I'll deal with sheaves of them

[marsh forge]

Lurie doesn’t use the name bousfield tho

[tiny ridge]

Gotcha

[plain raven]

It suffices for it to be a group. This is a theorem of moore.

Actually it suffices for it to be a monoid satisfying some weird technical conditions

[marsh forge]

Interesting, ibsens sketch relied pretty substantially on the existence of inverses

[tiny ridge]

yeah im surprised by that

[plain raven]

Max I learned this by following up on something you told me.
If (X,x) is a pointed topological space then there's a (strict) monoid whose objects are loops from x to x defined on some real interval [0, t]; if you have loops [0, t] -> (X,x) and [0, s] -> (X,x) then their composition is a map [0, t+s] -> X.
Moore figured out some way to construct a simplicial set out of this which had a true monoid structure and proved it was a Kan complex by appeal to some weird technical condition on the monoid.

[marsh forge]

Ag

Ah*

Interesting

I wonder if the condition is related to the grouplike condition on a loop space

[plain raven]

I might be misremembering this. I'm re-reading the paper one sec

Ah, ok, yes, I am misremembering.

Instead, what Moore proved is that if X is a simplicial monoid satisfying a weird technical condition, and is a Kan complex, you can compute its homotopy groups by taking its Moore normalization and computing the homology.

Moore proved that if (X,x) is a pointed space, and (X,x)^R is its Moore path space, then the singular simplicial set of this space is a simplicial monoid satisfying the weird technical condition, and thus its homotopy groups agree with the homology groups of the Moore normalization.

[tiny ridge]

😵‍💫

[plain raven]

Basically this just means it's pretty easy to compute the homotopy groups from a combinatorial point of view, as the Moore normalization is a relatively simple object.

[marsh forge]

What’s the technical condition

[plain raven]

There is an involution isomorphism M \cong M^op

which is an isomorphism of (simplicial) monoids

[marsh forge]

Oh this is actually kinda neat, this is a weird way of noting that the fundamental group of a loop space is abelian

[plain raven]

This makes sense because groups are obviously equipped with an isomorphism G\cong G^op, namely the inverse map

and so is Moore's path space, you just reverse the direction of the paths.

so this proves that simplicial groups have homotopy equal to their homology as a corollary.

[marsh forge]

I wonder if this is equivalent to grouplike somehow

If a simplicial monoid has such an involution will it’s pi_0 be a group?

[plain raven]

I don't see that theorem in the paper. He does note what you just said tho, which is that the fundamental group of any simplicial monoid satisfying this hypothesis will be Abelian

I think that the technical hypothesis is necessary for the homology groups to be well-defined

Moore was in the Cartan seminar for a couple years and talked to them about Kan complexes.

[marsh forge]

Wait that’s slightly different

Oh wait nvm

You were talking about the loop space thing

[marsh forge]

Well it was already known (not too hard to see directly)

[plain raven]

sorry that's confusing

yeah

[marsh forge]

But this says that a Moore space (which is a loop space) has Kan homotopy groups (iso to normal homotopy groups) given by homology of normalized chains, which in particular is abelian in all degrees

[plain raven]

ok, nice

there's so much good stuff in the cartan seminar papers

and they're all free online

I learned all about computing the homology of K(pi,n) spaces there

[marsh forge]

Do you know of Manuel Rivera

[plain raven]

It doesn't ring a bell.

[marsh forge]

I saw him give a talk about some of his work I feel like you’d find it interesting

[plain raven]

oh is this the guy you were mentioning the other day

yeah i think you sent me his paper.

[marsh forge]

Oh did I

[plain raven]

I saved it

[marsh forge]

Wow I have no memory of this

[plain raven]

It was about projective lifting of model structures along adjunctions under general hypotjeses

right?

[marsh forge]

No no no

[plain raven]

oh ok.

[marsh forge]

That was Maru

Manuel does like algebraic models for topological categories

Like generalizations of that Sullivan model for rational stuff

[plain raven]

i see

[plain raven]

I think Lie groups are pretty important, right? And so it stands to reason that if you think of like, simplicial complexes as a basic combinatorial prototype for spaces, then simplicial groups are potentially very important, maybe almost as important as Lie groups!
But you cannot have a sensible notion of a "simplicial group" unless you have a sensible notion of Cartesian product, and the geometric realization functor doesn't preserve products for semisimplicial sets. Yes it has a product in the presheaf category but imo it is not geometrically meaningful, so semisimplicial group theory is a really inadequate proxy for topological group theory.

[umbral panther]

I’m confused what you’re saying. Semsimplicial groups are sensible. Yeah, the geometric realization preserves products only up to homotopy equivalence, but that’s a very fine grained technical condition. Ultimately you want it, but for most purposes it doesn’t matter

[umbral panther]

I dunno. Most abelian categories don’t have enough injectives. Finitely generated abelian groups don’t have enough. I thought one of Lurie’s big things was working with small categories without enough fibrant objects, just like the original version of Quillen model categories. Except he wanted even smaller than Quillen.
Worse, the opposite of the category of sheaves doesn’t have enough injectives

[grim knot]

hey guys, I am reviewing the topology notes and by going through the proof of this lemma, I don't get why they argue in such a complicated way to show transitivity (the continuity part of the proof))

[empty grove]

You can show continuity using the pasting lemma. The screenshot is proving the pasting lemma itself instead of just citing it.

The pasting lemma is equivalent to saying that if a space Z is a union of A and B, both open (or both closed), then Z is the pushout of A and B along their intersection.

Here,
Z = X × [0, 2],
A = X × [0, 1],
B = X × [1, 2]

https://en.wikipedia.org/wiki/Pasting_lemma

I assume that you were asking why we can't just use this

[grim knot]

yes exacctly, I thought it was elaborated much more complicated than it really is

Thanks a lot

Does somebody have a nice proof on why the fundamental group of Sn is trivial? I do not fully get the concept

[empty grove]

Which proof are you looking at? There are multiple

[grim knot]

this one

I also have one in german

I can't really visualize it

[empty grove]

Another way to phrase this is that S^n minus a point is contractible

All the things being said about pushing the loop towards the North pole are about how you can contract S^n - South pole to the North pole

So if your loop misses the South pole, it is homotopic to the trivial loop by horizontal composition of homotopies (compose the identity homotopy on the loop with the contracting homotopy of S^n - South pole)

The rest is arguing that any loop is homotopic to one that misses the South pole

[grim knot]

Ah okay, that explains it a lot better, thank you for elaborating it

The english kinda confused me, but it makes sense now!

[empty grove]

[steel glen]

is there an easy way to see that S^n - a pt is contractible for n > 2?

[empty grove]

It is homeomorphic to ℝ^n

[grim knot]

should be homeomorph to Rn, which is contractible because it is convex

[steel glen]

ah right

thank you

[unreal stratus]

Works for any n btw lol

[empty grove]

S^(-1):

[unreal stratus]

Lol

[tiny ridge]

You know how when localizing a category at a multiplicative set, you define the morphisms as (equivalence classes of) roofs X <- Y -> Z where X -> Y is in the multiplicative set? Compositions of zig-zags can be replaced by a big zig-zag upto equivalence, by "completing the house of cards".

[empty grove]

Yeah a big-zag

[novel acorn]

I love left roofs

[tiny ridge]

The Dwyer-Kan idea is to not forget the chain of zig-zags. Let M be a model category, and W be the weak equivalences. For any pair of objects X, Y in Ob(M), define a simplicial set whose n-simplices are length n zig-zags from X to Y

The left arrows always lying in W

The face and degeneracy maps are given by factorizing the identity and inserting it in the diagram

This is the Dwyer-Kan localization M[W^-1]

Which is a simplicially enriched category

[empty grove]

This isn't exactly it though is it?

Aren't the n-simplices the hammocks of length n

Or width n I think

[tiny ridge]

Oh you're right

[empty grove]

Though I guess its like hozigzags

Because the connections in the hammocks are wequivalences

[rough cedar]

hozigizigizagzag

[tiny ridge]

Yeah OK

I guess my description is fundamentally wrong because these aren't n-simplices.

[empty grove]

Is a neat way to think of it though

I can now see much more clearly why hodwyerkan is the usual thing

[tiny ridge]

I like the hozigzag comment

I am unhappy though because I cannot see what the hom SSets are anymore

It's a weird coproduct of various chunks

[empty grove]

Ye I find it not so intuitive

Hammocks are not how I would guess higher homotopies would be defined

[tiny ridge]

n-simplices are sort of just n-chains (n-cliques, actually) of hammocks

bizarre

[empty grove]

Have you seen Hirschhorn's version of the simplicial localization for a model category?

[tiny ridge]

To be fair, hammocks in and of of themselves is a natural idea because even in the derived category you mod out roofs by equivalence of roofs

[hidden crag]

These words are insane

[tiny ridge]

Which are hammocks

Nope

Where can I find it

[empty grove]

Hirschhorn's book

It's even less intuitive and I don't yet understand it

He constructs cofibrant and fibrant resolutions of objects

Which are (co)simplicial objects

So that the Hom from the cofibrant resolution to the fibrant resolution is a bisimplicial set

And the diagonal on that is somehow the correct Hom

[tiny ridge]

Insanity

I'll have a look

[empty grove]

Tell me if you figure out some nice intuitive explanation

[tiny ridge]

This is the Bousfield localization?

[empty grove]

Nope

Let me check

He calls them homotopy function complexes

Chapter 17 in his book

[tiny ridge]

Thanks

Fuck this looks too beef brained for me

[empty grove]

It is very systematic with citing past results

But It's hard to jump into a random later chapter because it's a whole tree of citations you have to backtrack through

Part 2 is the appendix btw

The main content is in part 1, which is up to chapter 6

[tiny ridge]

fucking hell

BTW what is "Reedy"? I asked someone a long time back the following question: suppose you have two diagrams of spaces P -> Spaces and a natural transformation between them which is a pointwise weak equivalence, when can I invert it? They said idk man look up Reedy fibrant diagrams

I still dont know the answer to this question

I ended up avoiding it altogether

[empty grove]

Suppose you want to put a projective model structure on a category of diagrams

[tiny ridge]

In my situation there was a concrete inverse lol

[empty grove]

Projective meaning that fibrations and weaquivalences are pointwise

[tiny ridge]

Ok

[empty grove]

If you target category is not cofibrantly generated this is difficult in general

But works out when you domain category is Reedy, which is the condition you need to give a nice characterization of the cofibrations inductively

Like if you take your domain category to be . → . → . → ...

You can work out the projective model structure inductively

[tiny ridge]

Yeah, makes sense

[empty grove]

Reedy is the most general kind of category in which this inductive argument will work

And the simplex cat is Reedy which is why it is often used from computing homotopy (co)limits

[tiny ridge]

So in paricular if P is Reedy, this does answer my question?

Say if all the arrows are fibrations

Can inductively construct an inverse, maybe

By inverse I mean homotopy-inverse

[empty grove]

I didn't bother reading the part of the message after "what is Reedy?"

[tiny ridge]

You should tell me the answer to that one, sounds like something homotopy theorists should know

[empty grove]

Homotopy inverse as in inverse with respect to the left and right homotopy relations?

[tiny ridge]

Yes

I want a natural transformation in the inverse direction which is a homotopy-inverse, pointwise, for all original diagram

Commutation should be on the nose

[empty grove]

Since the weaquivs and fibrations in the projective model structure are levelwise, I reckon right homotopies are levelwise right homotopies

So between bifibrant diagrams, the levelwise weak equivalences should be homotopy equivalences

Since fibrations are levelwise, fibrant diagrams are levelwise fibrant

Cofibrant diagrams are also in particular levelwise cofibrant

[tiny ridge]

Ok, interesting

[empty grove]

So this is from the usual model cat whitehead

[tiny ridge]

I should Reedy up on this stuff then

Maybe someday

[empty grove]

Same whatever I said now is almost all I know about them

[tiny ridge]

One should encounter this all the time, no? Suppose you have two simplicial sheaves F, G on a space X, and a pointwise weak heq F -> G. When can I say it's an actual heq? I guess the question is what is the model structure on simplicial sheaves

This sounds like something Goerss would know

[empty grove]

Yeah no clue lmao

[tiny ridge]

I did a brief literature survey when I wanted to understand this and I concluded "fuck this" after a day or two

[empty grove]

[tiny ridge]

Looking it up again because of low attention span

Ah yes, "Simplicial presheaves - j.f. jardine" by S. Presheaves

[empty grove]

[tiny ridge]

One theory is where the weak equivalences are "combinatorial weak equivalences", which is just a stalkwise weak equivalence

This is severely underwhelming for me so I'm looking at a better one

[empty grove]

It sounds good though

Given that exactness can be checked at the level of stalks for sheaves valued in abcats

Seems like the right analogue for simplicial stuff

[tiny ridge]

I'm underwhelmed because of the following

Dunno if I want to say it

[empty grove]

[tiny ridge]

Suppose F, G are two simplicial sheaves on X whose stalks are Kan fibrant. Say F -> G is a local weak equivalence, i.e., on stalks it is a weak equivalence. If X is sufficiently nice and the restriction maps of F, G are Kan fibrations, then F -> G should be a global weak equivalence ie on every open set F(U) -> G(U) is a weak equivalence

This is essentially a theorem of Gromov

But this doesn't give me an insight into when I can invert F -> G

You also only need the restriction maps from a compact to a compact to be a Kan fibration IIRC

Gromov doesn't use this language so some hypothesis might get lost in translation

I cant be bothered

[umbral panther]

What do you mean “can I invert“?

You can invert whatever you want.

It might be hard to work with. You could follow up with if this is what I want to invert, is there a model category?
Usually with diagram categories that’s exactly what you want to invert. There are several model categories with those equivalences. Projective, injective, and sometimes Reedy

[tiny ridge]

I dont mean formal beef brained invert

[empty grove]

Is there no way to stitch a locally defined map into a global map of sheaves?

[tiny ridge]

Thats why I asked the question for a diagram

That's like a constructible simplicial sheaf version of the same q

[umbral panther]

Maps of sheaves form a sheaf. If you have actual maps of sheaves, they glue
But if you only have homotopy coherent maps of sheaves, you have to have a lot of coherence

[tiny ridge]

Suppose you have two diagrams P -> SSet, and a natural transform which is pointwise weak eq. Can I invert every arrow to make a natural transform in the reverse direction?

No hocobolobotomy sheaf

Just a sheaf

Res res = res on the nose

But maybe this is why later on in manifold calculus people started using hosheaves

I dunno

[empty grove]

Where should I learn sheaf theory from if I don't want to be pipelined into AG?

[tiny ridge]

depends on what you want to do

with them

[empty grove]

Is there sheaf theory in homtopi

[tiny ridge]

Yes

[empty grove]

Yes

[tiny ridge]

Read Weiss' stuff on immersion theory

Let me see if I can find an article

[empty grove]

Nice

[tiny ridge]

https://gradmath.org/wp-content/uploads/2020/10/GJM2020_Weiss.pdf

[empty grove]

Thank

[white oxide]

Can't hurt to read relatively self-contained sections on sheaves in AG books, unless you already know sheaves well and need some more specialized results

[empty grove]

I am familiar with the basics, a bit more than is needed for schemes.

[umbral panther]

No, you can’t. If you invert locally, the inverses aren’t coherent, but only homotopy coherent, so they don’t glue

[tiny ridge]

The question is more "when can you?"

Suppose all the maps in the two respective diagrams are fibrations

Even then I don't know.

[umbral panther]

Sounds like a model category question

[tiny ridge]

If you have cycles in the diagram you run into issues

Yeah I have a feeling this is what homotopy theorists say when they dont know a concrete answer

[umbral panther]

Reedy is a concrete answer, it’s just a long answer

[tiny ridge]

I'll read about them

Soon

If the two diagrams are a tree of SSets, there should be no issue for sure. You use the fact that if A -> A', B -> B', C -> C' are weak equivalences, then A x_C B -> A' x_C' B' is a weak equivalence if all the maps A -> C <- B etc are fibrations

Inductively

Cycles I dunno

[umbral panther]

For general diagrams, even with cycles, there is a way of making the maps sufficiently fibrant. And a dual way of making them sufficiently cofibrant. These are two different model structures. The Reedy structure, for only special diagrams, does both at once

[plain raven]

EG is a contractible space and so is a singleton. But we don't identify them, because one has a well behaved group action and the other doesn't. So to me it's a meaningful distinction between preserving the terminal object and preserving it up to homotopy. It's subjective, but i don't think of it as a fine grained distinction.
At the end of the day I basically believe the constant map from the unit interval to a point is geometrically meaningful and should be regarded as a simplicial map, and it's a weakness of the formalism if you can't express non-injective maps.

[tiny ridge]

I think we understand that the concept of remembering collapse of homotopically irrelevant info is useful

Mapping cylinders!

[tiny ridge]

Going to try to learn \infty-localization

Sigh

[lunar yoke]

yeah, in HTT

or read something on six functors

[white oxide]

Math will completely run out of names for mathematical objects by 2031. "Six functors"?

[lunar yoke]

i mean that one goes back to grothendieck though afaik

its just gained more traction recently

[tiny ridge]

Happily there are people who do not care about terminology and linguistics

[lunar yoke]

since now there is a formal definition and what not

scholze gave a lecture on it last term in bonn, he has notes on his page

[abstract saffron]

Yeah, but we need Scholze-English dictionary first

[lunar yoke]

its genuinely quite tame, at least the start

he starts with the topology version

oh i guess it makes more sense if one knows about sym mon infty cats

[tiny ridge]

Lol

[lunar yoke]

but those are just like normal sym mon cats

[abstract saffron]

Yeah, sure, completely basic

[empty grove]

Should I read HTT

[tiny ridge]

You can just read something obviously more concrete and useful if you want to learn about the six functor formalism

[empty grove]

I also want to learn infty cat theory

[tiny ridge]

Like Kashiwara-Schapira

[lunar yoke]

not form start to finish

but chapter 5 is really good

like for practical stuff

adjoint functor theorem etc

[empty grove]

Requires ordinary topoi as a prereq though right?

[lunar yoke]

no

[empty grove]

oh

Nice

[lunar yoke]

topoi only starts in ch6

[abstract saffron]

topoi?

[empty grove]

Ah

[tiny ridge]

Ch 5 is what I'm reading now lol

[abstract saffron]

I know we have topos, but wtf is topoi?

[tiny ridge]

Plural

[lunar yoke]

same thing different name

[empty grove]

Toposeses

[abstract saffron]

And I thought I spoke English

[tiny ridge]

Well technically the convention is one from Latin

Torus -> tori

[lunar yoke]

so what I did is read lands book and do all the exercises, and then I started with some really nice notes on algebraic k theory from fabian hebestreit, and read a bit of HTT and HA on the side

in hindsight, I think lands book wasnt really that necessary

its mostly pointset stuff with simplicial sets that you don't care too much about when you actually wanna use the theory, like in stable homotopy theory

[abstract saffron]

lmao, but topos comes from Greek, not Latin

[lunar yoke]

the notes in algebraic k theory start with a 60 page recap of the most important stuff you need from infty cats, and there its done like you would in practice, can recommend a lot

[abstract saffron]

τόπος, means place

[lunar yoke]

here they are https://www.uni-muenster.de/IVV5WS/WebHop/user/fhebe_01/HigherCats2/KTheory.pdf

[lunar yoke]

like for example in lands book you kind of see straightening unstraightening and its used for a few things, but I didnt really get a feel for it until I saw it used "in practice" in these notes

maybe you can just start reading this and then look up stuff in kerodon or something if you need to

[empty grove]

This is exactly the kind of thing I have been looking for

Thank you

[lunar yoke]

you will have to get used to the scholze-school of calling spaces anima though xd

I'm already past the point of no return

[empty grove]

[tiny ridge]

Conventions of the clinically insane

[lunar yoke]

I think its important to distinguish though, you should definitely not call them spaces, since then ppl might confuse them with topological spaces etc

its like objects in the derived category D(R) technically arent chain complexes since its not well-defined to look at the n-th level

only n-th (co)homology

[tiny ridge]

I will continue to call a space "a space" and not know what an anima is

Thanks very much

[lunar yoke]

its an object in the infty cat of spaces

[white oxide]

Are they actually a cult? Trying to summon the great old ones for knowledge

[empty grove]

But people also call simplicial sets spaces sometimes

This would be in the same spirit I suppose

Since the hocats are the same

[lunar yoke]

I guess you also have the name infinity groupoid but this you usually only use when you realy wanna view it as an infty cat

[tiny ridge]

Believe it or not there are people who still care about spaces upto homeomorphism.

[lunar yoke]

yeah I know there are even quite a few here in bonn

[empty grove]

They should speak of topological spaces then

[lunar yoke]

they usually just speak of manifolds anyway

[umbral panther]

Simplicial sets aren’t spaces. They’re Spaces

[lunar yoke]

and then you know

[tiny ridge]

Right

[empty grove]

Simplicial paces

[tiny ridge]

Spaces means manifolds upto homeomorphism to us

[lunar yoke]

on that note, moldi, I think I remember you asking stuff about the HHR paper

did you read that

[abstract saffron]

I'm still super annoyed that ppl call polytopes the space bounded by simplices. It makes sense, but super weird

[empty grove]

Ye I wrote my masters thesis on the slice spectral sequence

But didn't read their full solution

[lunar yoke]

ooh cool

so can you explain norms to me

[empty grove]

No I cannot

[lunar yoke]

man

I've been learning about the spectral mackey functor picture of G-spectra and wondering how norms fit into that

[empty grove]

That is for constructing their C_8 spectrum

[lunar yoke]

and ive only been getting more and more confused

[empty grove]

I only applied the slice spectral sequence to KR

Didn't need norms to do that

[lunar yoke]

oki

[steel glen]

yeah the other n<=2 are easier to visualize though haha

[tiny ridge]

Take an \infty-category C, treat is as a quasicategory. A fibrant replacement is like C[C^-1], yes?

You have inverted all the arrows

So I suppose that given a set of edges W of C, the pushout Ex^infty(D) <- D -> C is the localization C[W^-1], where D is the faithful subcategory generated by W?

[tiny ridge]

OK, this is exactly what Lurie does in HTT 5.2.7. The point is fibrant replacement has a fully faithful right adjoint, given by assigning to a quasicategory its core

So Maps(C[W^-1], X) = Maps(Ex^infty(D), X) x_Maps(C, X) Maps(C, X) = Maps(D, Core(X)) x_Maps(C, X) Maps(C, X). A map C[W^-1] -> X is a map C -> X which sends all the arrows in W to the invertible morphisms

I think this is the construction of localization I like the best

And I think the reason this is exactly the same as Dwyer-Kan is as follows. Ex^infty is adjoint to the infinite barycentric subdivision

So a 1-cell in Ex^n(X) is a n-zigzag in X between vertices of X, factoring through various barycenters

2-cells are clearly hammocks between zigzags

@empty grove

[empty grove]

What's Ex^n and Ex^infty?

[tiny ridge]

Ex^infty(X) is a model of a fibrant replacement of X. Define sd^k(Delta^n) to be the k-fold barycentric subdivision of Delta^n

Then define Ex^k(X)[n] = Hom(sd^k(Delta^n), X)

Ex^infty is limit of Ex^k's

[empty grove]

Fibrant replacement in the Kan model structure?

[tiny ridge]

Yeah

X -> Ex^infty(X) is a weak equivalence, and Ex^infty(X) is Kan

Nearly by definition

[empty grove]

I see

Seems cool, but will take me a while to fully process

[tiny ridge]

A map from sd^k(Delta^1) to X is a zig-zag being the point

Np

[empty grove]

Right yes

[tiny ridge]

Once again, it seems like a blind man trying to understand the geometry of barycentric subdiv

The way people write this stuff I mean

[empty grove]

No I am just playing minecraft on the side

[tiny ridge]

I wasn't directing it at you lol

[empty grove]

And can't be bothered rn

[tiny ridge]

Now I learn how to glue \infty-categories

Oh but that should be dumb, just colimits in SSet

[empty grove]

Doubt it, I think you will have to take colimit in SSet and then quasicategorify

Limits should be just limits in SSet

[tiny ridge]

Ah

Good point

[river granite]

Is Sato's AT book a good introduction to the subject? I don't mind having to fill in details or maybe complementing with other books

[marsh forge]

never heard of it

[gaunt laurel]

Hey, from what I can see one can't do the equivalent for normal spaces, right?

[thorny agate]

What the actual fuck is paracompactness

😭

most of this proof is gibberish to me

Hold on lemme get more specific

[gentle osprey]

spamakin

[thorny agate]

I understand the statement of the theorem (I think) but is there a better proof out there?

or can someone explain at least a couple of the points above (pick your favorite)

[unreal stratus]

This is a theorem I have always blackboxed

[cosmic socket]

Paracompactness is only important for this theorem. I would say to not worry about it too much until you actually need to know this proof.

[thorny agate]

okee

[cosmic socket]

Just remember paracompactness = have partition of unity

[river granite]

paracompactness is pretty much defined in such a way as to make partitions of unity work, for instance in your point 5) note that the function g is defined by a possibly infinite sum, however since the subcovering is locally finite, when you evaluate g(x) at any x you are actually summing the finitely many g_\alpha's that are supported in a nhood containing x.

the reason why one cares about paracompactness is because e.g. topological manifolds are paracompact

actually there might be a better explained proof of that result you posted on Lee's ITM

or ISM for the smooth mfold. case

[unreal stratus]

and cw complexes

[river granite]

and about this sure -- one often defines "regular" as T_0+T_3, and "normal" as T_1+T_4. You can find spaces that are T_0 and T_4 but not T_1

[thorny agate]

I'll look at this

[cedar jungle]

On a nonhausdorff topology, what do I need to have something like relative position of two points in a plane? e.g., x is above, below, to the side, to the left of y, etc.

[quick bough]

how is this a corollary from just this:

and what does it mean for the fundamental group to be a homotopy invariant of pointed spaces (in the first photo)

[hidden crag]

it means that homotopy equivalent spaces have the same fundamental group

[quick bough]

oh okay

how come it follows directly from the proposition

[tiny ridge]

with gusto

[quick bough]

gusto?

[marsh forge]

this is a good exercise

[unreal stratus]

Oh lol

I misread it

[quick bough]

oh wow, i just figured out its on the exercise sheet, i thought it was an immediate corollary

[marsh forge]

i mean

it kind of is

[quick bough]

well, thank you

oh

lmao

[unreal stratus]

I find 1 a bit weirdly worded

[quick bough]

like in corollary 9.19?

[unreal stratus]

"Is a homotopy invariant of pointed spaces" when usually homotopy equivalences of pointed spaces would be... pointed

like this is stronger

[quick bough]

yeh, i find the wording a bit misleading too

[unreal stratus]

But eh it says "hence" so nvm but still subtly different

Doesn't this need path connectedness anyway?

[marsh forge]

truly one should only ever work with pointed spaces

[unreal stratus]

Wait I need to check lol

[marsh forge]

pi_1(X,x) only depends on the path component of x

[unreal stratus]

Yeah this is true and any htpy equivalence would restrict to one of the corresponding path components of the poitns ig

Why did we always assume path connectedness in what I saw

Oh okay I guess it's so that they can justify omitting basepoints from notation (though the change of basepoint isomorphisms aren't canonical so eh)

[marsh forge]

taps sign

[plain raven]

what if i have a space and i don't know whether it's empty or not

[marsh forge]

bad space

[unreal stratus]

Freely adjoin one

[tiny ridge]

theres a reason a pointed space is called a based space

[unreal stratus]

I don't see the point

(jk i do just i wanted to make the joke)

[empty grove]

Every space has trivial homotopy groups.

[silver umbra]

so im currently wrestling with the difference between simplicial and singular homology rn (following hatcher's book)

my understanding currently is that in simplicial homology, you form the chain groups by first defining a triangulation on ur space that follows a bunch of rules

on the other hand in singular homology, any sort of simplex in your space is fair game

[empty grove]

You got it ✓

[silver umbra]

thanks

i'm also trying to understand the precise definition of a delta-complex structure that hatcher defines

i've done a bunch of exercises already so i have an intuitive understanding of what rules they need to follow; for example you can't just identify all edges to a point or smth like that

what would be some examples of delta-complex-like-but-not-quite structures on a topological space

that violate each of these rules?

[empty grove]

It might make it easier if I give another equivalent definition of a delta complex

It is like a CW complex, except that instead of disks, you attach simplices, and instead of arbitrary continuous maps as attaching maps, the attaching maps must map every face of the simplex that you are attaching "homeomorphically" onto one of the lower dimensional simplex

I put homeomorphically in quotes because it is not exactly it, let me come back to that

But basically the additional thing that you are allowed to do with CW complexes is that you don't need the attaching maps to respect the cells at all, only the skeleton. So your attaching maps can wind around a cell any number of times, can go halfway across the cell and come back etc (think of RP^2, where the attaching map for the 2-cell winds around the unique 1-cell twice).

[empty grove]

So the precise condition on the attaching map from the boundary of Delta^n to the (n-1)-skeleton is as follows. Of course, it should be continuous. Then if you restrict this attaching map to a face of Delta^n, which is homeomorphic to Delta^(n-1), this should give you the "inclusion" of one of the (n-1)-simplices into the (n-1)-skeleton.

Again, inclusion is in quotes because an (n-1)-simplex need not inject into the simplicial complex. For example, a circle can be constructed from a single 0-cell (a point) and then attaching a 1-cell to it by mapping both of its end points to this 0-cell. Then there is a natural map from the 1-cell into the circle, which I am calling the inclusion, but it is not injective.

[empty grove]

So basically any CW complex lol

[silver umbra]

hmmmmm ok

still trying to wrap my head around what exactly this inclusion is

ill draw an example

[empty grove]

Sure

[silver umbra]

so here's a 1-skeleton

whats wrong with attaching a 2-simplex like this

[empty grove]

Its boundary is not attaching to the 1-skeleton

That is a requirement for both Delta complexes and CW complexes

There should be an attaching map from the boundary of the simplex that you are attaching to the 1-skeleton to begin with

[silver umbra]

i see

[empty grove]

By attaching a simplex along an attaching map, I mean that you take the disjoint union of the complex you have till that point with this new simplex, and identify the points on the boundary of this new simplex with their images under the attaching map.

So there is always a natural map from a simplex in a Delta complex to the Delta complex: include the simplex into the above disjoint union and then apply the quotient map.

Note that this natural map will always be a homeomorphism on the interior - the only quotienting happened at the boundary of the simplex, and the interior of the simplex just includes into the Delta complex

Any point in the interior of this simplex will not appear in the interior of any other simplex - the only gluing that could happen for a point in the interior is a higher dimensional simplex gluing to it, but in that case, that point would be on the boundary of the higher dimensional simplex rather than in the interior.

This tells you how this definition and the given one are related. What I just said in the last message is (i) in the given definition. (ii) is part of the definition I gave anyway, and (iii) is saying that the Delta complex has the quotient topology after you glue each new simplex.

[silver umbra]

hmm ok, so (i) is going to be violated if u attach two k-simplices to a (k-1)-skeleton in a way where their interiors intersect each other

[empty grove]

Yep

[silver umbra]

for (ii), what im thinking about in particular

is the CW complex on S2

where u take a point (the 0-skeleton)

and then u immediately attach a 2-cell to it

by identifying the boundary of that 2-cell to that point

[empty grove]

Yes

[silver umbra]

the boundary of that 2-cell doesn't include into a 1-skeleton

bc there is no 1-skeleton?

[empty grove]

Yep

Ok well there is a 1-skeleton

Just that it is equal to the 0-skeleton

But the natural maps from the faces of the 2-cell to the Delta complex are all constant