Charles Rezk pointed out that the 0-sphere doesn't have this property (consider a weak homotopy equivalence from the 0-sphere to the topologists' sine curve which sends one point to (0,0)). So I would guess that very few spaces have this property.

wait the 0-sphere isn't weak homotopy equivalent to the topologist's sine curve, is it?

isn't the topologist's sine curve weakly contractible?

Isn't it not path connected?

There's no path from (0,0) to any other point.

Yeah but you can get close enough, so it's fine.

Oh wait -- I'm thinking of the version where you connect up the "limit segment" to the other side of the curve.

via some non-wild path

I agree -- the closure of the graph of $sin(1/x)$ is not path-connected.

and now I believe that it's weak homotopy equivalent to $S^0$. Cool!

And the same goes for the graph of $sin(1/x)$ union'ed with the origin.

where when I say "graph of $sin(1/x)$" I always mean for $x>0$.

Well, $X = D^0$ and $X = \emptyset$ have this property :)

This is giving me flashbacks to a paper I'm just finishing. We could have done everything simplicially, but we decided to work with arbitrary topological spaces (not even a convenient category!) because we thought it might be psychologically reassuring for our intended audience of model theorists.

big mistake 0/10 would not recommend.

Ha, for me it's a talk I'm preparing that has got me weighing up the expository benefits of topological spaces vs simplicial sets.

I'll just go for infinity-groupoids.

Just call them "anima" to avoid confusion.

Just as long as I don't have to figure out what the plural form is.

The other day it occurred to me that there's a perfectly good word for the objects of the free cocomplete $\infty$-category on a point --

homotopy type

I think the only reason we haven't settled on using is that at some point some decades ago, we decided that "homotopy type" meant "object of the homotopy category"

And we want to be clear when we're avoiding doing something so crude as passing to the homotopy category.

But the homotopy category has the same objects as the the free cocomplete $\infty$-category on a point.

So there's really no good objection to using "homotopy type" as far as I can see.

Unlike "space", it doesn't carry the implication that spaces not living in the homotopy category are "not spaces"

Unlike "infinity groupoid", it doesn't sound technical or feel like it ties you down to "algebraic" models somehow

And it's suggestive of the idea that you started with some more complicated sort of object -- whether it be a topological space or a simplicial set or whatever -- and now you consider it only up to homotopy.

Anyway, Charles' example shows that spaces $X$ with Alex's wacky property must be path-connected... Does the unit interval have this property?

(I'm thinking that by taking suspensions of Charles' example you can probably show that any such $X$ is empty or weakly contractible...)

I used to like anima but when I realized Scholze plans to use the same word for singular and plural I decided I didn't like it.

not animae?

animas?

Yeah he's been public about not liking animæ.

But I guess if we all say it enough he might cave.

I feel like the ae or æ is pronounced "ē," am I wrong? As in, it rhymes with "me" or "pea."

it depends on your accent

I've always liked "homotopy type" too. Of the two of us, I was the more conservative in terms of wanting to stick to terms that already exist. But I'm not so sure about one of your justifications. The notion of two categories "having the same objects" is way too broad because it could also justify using "topological spaces" to mean the same thing as anima. In some sense the whole point was to have a word which acknowledges that the important thing is the category up to equivalence

and the set of objects is not invariant under equivalence!

Also, "homotopy type" gets clunky when you put it in larger semantic constructions, e.g "E_1-homotopy type" feels a bit strange, whereas a little word like "anima" slides in nicely everywhere, just like "set"

Moreover there's the connection with the general notion of "animation" which is a good and useful terminology for non-abelian derived functors, e.g. animated abelian groups, animated rings, etc etc

The pluralization issue is a bit annoying. We just both mildly preferred "anima" over the other options so we went with that.

animoi

I mean, when I say they have the same objects, I mean that the forgetful functor Spaces -> ho(Spaces) is bijective on isomorphism classes of objects. I don't know what it would mean to literally say that the object set are the same -- it would depend how you constructed either category. But I guess the need for a "verbable noun" does seem like a good argument

But if I use "anima" in a talk, I reserve the right to pluralize it differently :)

somehow my brain just can't wrap itself around a feminine singular and neuter plural

Oh, thanks for clarifying, Tim, that makes a lot of sense.

Speaking personally I don't care how people pluralize it. I bet Peter doesn't either.

I consider one advantage of the term "homotopy type" to be that you can parse it as the type (in some vague type theoretic sense) central to homotopy theory (especially if you see it for the first time). The term space seems misleading as it suggests some intrinsic geometry, which however in my opinion is only there because many homotopy types are the underlying homotopy type (shape) of some familiar geometric object such as a circle.

Fun fact : "Anima sana in corpore sano" was the way "Asics" avoided being called "Msics"

I guess I disagree with the idea that "space" refers to topological spaces or geometric objects - in my mind, "topological spaces" are just a model for what we "intuitively understand" as being "spaces", and that oo-groupoids/anima/e /homotopy types are actually a good model for what we understand to be spaces, which is why I don't feel so bad about the name. I guess you could make an argument for overloading, but then you would also have to find a new name for vector spaces :D

But then again people around me have started using "anima" so I'm slowly caving in too - I like "space" but it's not a hill I will die on and if going "anima" is the price to pay to be understood...

@MaximeRamzi I must admit to thinking a notion of "space" which does not distinguish between a point and 3-dimensional Euclidean space undeserving of the name "space".

@AlexanderCampbell I can see where you're coming from (obviously), but I feel like the same objection could be said about a sphere vs a cube, in which case one would have to say that a space should come equipped with form of knowledge about differentiability; which topological spaces certainly don't have

I think my point of view on this is sort of skewed by my fascination with the Königsberg problem :D this shapes my understanding of the word "space" to some extent

What is a nice proof showing that core Tw(C) is equivalent to core Ar(C)? Here C is an infinity category and Tw is the twisted arrow category and Ar(C) is the category of arrows.

One needs to construct a zig-zag probably

I guess core means the functor "throw away noninvertible 1-morphisms"? If so, then both cores are equivalent to the core of C itself.

@ClarkBarwick Is that true ? That doesn't sound correct, $Ar(C)^\simeq = Fun(Delta^1,C)^\simeq$ which is in general different from $Fun(Delta^1,C^\simeq) = C^\simeq$

those should be $\Delta^1$'s

Yes, by core C I just mean the maximal infinity-groupoid inside of C

Oh sorry. I completely misunderstood. You're right.

Can you compare both sides to a thing whose n-simplices are maps $\Delta^1\times(\Delta{n,op} \ast \Delta^n) \to C$ in which each of the two $\Delta^n$'s go to equivalences?

That doesn't sound "nice."

Actually, we can be smaller. Take the simplicial set whose n-simplices are maps Δⁿ ∗ (Δⁿ)ᵒᵖ ∗ Δⁿ → C such that 0 → ... → n, n' → ... 0', 0'' → ... → n'', and i' → i'' all go to equivalences.

I may be off by an op (depending on which version of Tw you want, I guess), but I think that should compare to each side.

@ClarkBarwick I think you’re right. Why is it the comparison maps equivalences? Sorry if this is obvious

@DustinClausen yeah sorry I didn't mean to imply that you or Peter were somehow drawing a hard line about how others could pluralize it, haha

And I am personally a big fan of the term "animate" since I think it makes the relationship between classical algebra/topos theory and derived or animated algebra/topos theory much clearer.

Although I think I sort of like the Latin pronunciation of animae as something like anim-eye

@JonathanBeardsley I remember hearing it would be something closer to "anime" ;)

Yeah, I think it depends. I've tried to find information about it online at least, and it seems like the pronunciation varies quite a bit by country and discipline. From what I can tell it would be "anime" or maybe more like "animeh" if one used "classical Church Latin" or something, and if you're German that's also how you'd say it? But in actual Latin, like if you learned Latin in school, it'd be "anim-eye"? And if you're American you're more likely to say "anim-ee"?

Maybe we should try to get someone from the languages Stack Exchange in here...

In my latin class a few years back (in France, but I don't know enough to say it's the same everywhere) I was taught the "anime" version

(I'm obviously not saying it means it's right, just to add to the examples you gave )

Does someone know a reference for a model structure on the category of A-infinity algebras over a non-field base ring?

No need to apologize for stating your opinion, @JonathanBeardsley! Besides, I didn't read anything like that into your statement anyway :)

@S.carmeli @MaximeRamzi that's great, thank you! i hadn't realized that the eilenberg--moore spectral sequence only converges when the cochains satisfy this (derived) tensor product condition -- that is very conceptually clarifying.

is anyone aware of any sufficient conditions for "the eilenberg--moore property over the sphere spectrum" of a pullback of spaces? i.e., when for a cospan $X \to Z \leftarrow Y$ the map $S^X \otimes_{S^Z} S^Y \to S^{X \times_Z Y}$ is an equivalence, where $S^X := map(X,S)$ denotes the mapping spectrum.

@MaximeRamzi i can tell you about my own experience with the parametrized stuff. i had always appreciated the "$C_2$-pushout" example, but the generalities beyond that had felt pretty opaque. then, suddenly i found that i needed the general theory, and all of a sudden it was all crystal-clear! (well, much closer to crystal-clear, anyways; i'm sure there's much i still don't fully appreciate.)

i find this to be a fairly general phenomenon: it's way, way easier to actually learn a new math once you actually "need" it -- said differently, when your heart is really in it. (non-math people probably learn this phenomenon in middle school, if not earlier. "i can't imagine ever needing to know how to solve a system of linear equations; therefore, it makes no sense to me.") i spoke to this previously regarding my attempts to force myself to learn representation theory in grad school.

regarding the "spaces" vs. "anima" vs. "homotopy type", i have come to the conclusion that it is extremely valuable to allow oneself to take "local" definitions and notation. for instance, i think i tried to fix the convention that the notation "$X$" denotes an arbitrary space (or something like that) throughout my entire thesis, which level of consistency was utterly needless and unhelpful, not to mention a huge waste of time to implement.

likewise, when i say the word "category", it means different things in different contexts, and i think that is just fine. i love math deeply for its utterly perfect consistency, but as a human being i need the flexibility not to enforce such consistency in my interactions with it.

all of which is to say: i really like to just say "space", and then clarify with the completely unambiguous term "$\infty$-groupoid" when appropriate.

at the same time, when i am talking a lot about topological spaces, i also just use the word "space", and likewise clarify with the completely unambiguous term "topological space" when appropriate.

@RuneHaugseng please tell me this is a real thing??!?

@AaronMazel-Gee Maybe if enough people contribute to my kickstarter campaign :-P

@AaronMazel-Gee Well I'm not even sure I know how to define the sequence without this condition :-D but I didn't learn about it the standard way (I learned it from an appendix in one of Toen's papers). I don't know about the case of the sphere, the proof I know for coefficients in a regular ring really uses the fact that we're over a ring and not the sphere - but that would be super nice to hear about !

And thanks for sharing your experience ! At the moment I think I need the parametrized stuff, but I'm not sure why I need it

(so I know what I need it for, but not why)

@RuneHaugseng please take my money

@Dedalus I'm sorry to have to be brief. Consider the restriction from the simplicial set I named above to core Ar(C). This is a surjection on 0-simplices. So it's enough to prove "full faithfulness." In this case that should to the fact that Tw(X) is equivalent to X when X is a Kan complex (which is roughly what I originally thought you were getting at).

Thanks!

@JonathanBeardsley Like this?

FWIW I took Latin in an English-speaking country, and according to English academic language pronunciation, "animae" would sound like "anime"

I think it would sound different in Church Latin, like if you were singing it in a song.

wait no

in English academic Latin pronunciation, "animae" would be pronounced like "anim-I"

long "I" sound

But conventions for actually importing Latin words into English differ from both English academic language pronunciation and Church Latin pronunciation, I think.

I think if I were to say "animae" in English, I'd make it sound like "anime".

fun stuff!

Oh right -- maybe I'll share the conclusion from the link I just shared -- the folks at Linguistics SE had to kind of reach to find another example of a word which "lost its plural form" when being imported into English, but the example they gave was "caribou". In the French from which it was borrowed, the plural of "caribou" is "caribous" as usual, but in English we leave it as "caribou" in the plural.

(I think the exception proves the rule -- it seems clear that "caribous" probably only lost its "s" when coming into English because the "s" is silent in French anyway, and because of the influence of words like "cattle"...)

@AaronMazel-Gee I would live to see an example of eilenberg moore convergence that doesn't essentiqlly come down to a nilpotence condition on the (generalized) homology of the pushforward as a parametrized spectrum.

Meaning an example where eilenberg moore property holds but not for nilpotent reasons (which essentially means bootstrapping the trivial case).