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It is popular in some corners to say “space” for “$\infty$-groupoid” /”homotopy type”. Taking the risk of coming across as a hair-splitter trying to waste everybody’s time (and yes, I will, so better stop reading), I hereby voice the opinion that this is a step in the wrong direction. I kow it’s an uphill battle sociologically, but I think it’s for a just cause. While I won’t convince anyone who already sticks to the habit, maybe at least making young students aware of the problem is a just cause for wasting bandwidth with this.
The reason: all along the history of mathematics, the word “space” always referred to a notion of geometric space. All those people in all those centuries who studied vector spaces, Euclidean spaces, affine spaces, Minkowski spaces would have studied the point if you insisted that “space = homotopy type”.
(It is precisely this clash with tradition that for instance in J. Lurie’s Formal Moduli Problems – which sticks to the terminology “space = homotopy type” – requires, below def. 1.2.1, a paragraph of disambiguation.)
The problem begins earlier. The word “space” as used for $\infty$-groupoids/homotopy types is already a truncation of “topological space” and that truncation by itself is already a step in the wrong direction. Even outside of homotopy theory it is a bad idea to declare that by default “space” is to mean “topological space”. Even if we do, it is a bad idea to conflate homotopy types with the topological spaces that present them. Elsewhere one sees the opinion voiced that choosing a basis of a vector space is “non-gentleman like” behaviour. If so, then conflating topological spaces with their homotopy types is considerably more rude even.
One can see people run into the inconsistency of using “space” to mean “homotopy type” all over the place. For instance everyone accepts the time-honored notion of diffeological spaces. But when I go and say that these are but a special case of “smooth spaces” only to observe that these are naturally embedded into an ambient homotopy theory of smooth infinity-groupoids, I frequently draw flak: They say: “You shouldn’t say ’smooth space’ for what is just a sheaf, for we are likely to confuse it with what you call ’smooth $\infty$-groupoid’.”
Clearly there is a source of confusion here, but it is not the fault of saying “smooth space” for something that generalizes non-homotopy theoretically what has always been called “diffeological spaces”, and “Euclidean spaces” and has always been a notion of space in geometry.
I could go on and maybe point to Lurie’s Structured Spaces which crucially are not just structured homotopy types. I could point out how in homotopy type theory it has lead to confusion to speak of the homotopy types there as “spaces”, and will lead to more confusion later as the theory develops. And so forth.
In summary: The word “space” in mathematics has always referred to geometric notions of space. Hijacking the word “space” for “$\infty$-groupoid”/”homotopy type” just because topological spaces serve as a presentation for the latter is something that is certainly convenient in, say, the bounded context of a single-focus textbook, but applied more globally to mathematics it goes against the grain of tons of old and good conventions and tradition.
And most importantly: there is no need. The words “$\infty$-groupoid” and “homotopy type” are available and serve their purpose perfectly.
I agree 100% and I have another gripe on a related topic which I will air here.
This is the problem of ’space= simplicial set’ as used by certain homotopy theorists. A simplicial set is also just a presentation of a certain form of homotopy type, but then I find myself in difficulties when wearing a shape theorist hat as there the notions of homotopy type and weak homotopy type are distinct and whilst a singular complex presents the weak homotopy type, I am interested in the actual homotopy type and also the strong shape. Both of which are more geometric. Calling simplicial sets spaces confuses the issue.
I am also finding ‘profinite space’ is being used as a synonym for simplicial profinite space i.e. simplicial objects in the category of Stone spaces. If my ideas go off towards orbifolds etc, then I hit what Urs is complaining about as well. I fear that the same sloppiness may invade the land of stacks. There are enough terms around without multiple usage in this way.
I generally agree, that the word “space” is very misleading in a homotopy theoretic context. I’ve often wished for a new word, as you mention.
I’m not crazy about your suggestions, though. “Homotopy type” is also a bit misleading, since you can talk about the “homotopy type” of objects in a general setting. (E.g., “let X be a spectrum with the homotopy type of product of Eilenberg-Mac Lane spectra”: no spaces here).
An $\infty$-groupoid is just plain ugly. I will always prefer “space” to that.
I’m happy to join the chorus of agreement. But are there specific instances you have in mind which moved you to rant about this here and now (please take “rant” in the friendliest, most kidding, and finally most sympathetic way you can (-: )?
(I don’t suppose you really mind if someone announces at the beginning of an article that whenever he says ’space’, he will mean, for the purpose of that article, ’topological space’. Or do you?)
I griped not ranted. :-)
Todd’s suggestion is fine with me except that that is not the end of it. You often find that space means ‘topological space having the homotopy type of a CW-complex’ and most of that is not said. Why then the author does not say CW-space or something similar i do not know.
I think that all these words of wisdom need to be applied throughout the n-Lab, i.e. to improve this sort of thing if and when we see it (and when we have time!)
And most importantly: there is no need. The words “∞-groupoid” and “homotopy type” are available and serve their purpose perfectly.
The problem with the term “homotopy type” is that some people mean an object in the homotopy category of the ∞-category S of ∞-groupoids when they say “homotopy type”, and the homotopy category of S is very different from S itself.
I fully agree with Urs (except that out of specialized context I would say “weak homotopy type” instead of homotopy type).
I agree that the common usage of the word space is misleading, but I would like something else than $\infty$-groupoid or homotopy type as an alternative.
I don’t like homotopy type precisely for the reason mentioned by Dmitri. In principle we should be able to reuse homotopy type as an “object of the $(\infty, 1)$-category of topological spaces” which is also in the spirit of HoTT, but it seems to me that too many people think of homotopy types as “objects in the homotopy category of topological spaces” so we run into a similar problem as with spaces.
I don’t like $\infty$-groupoids because not all models for the homotopy theory of topological spaces are higher categorical. Some of them are topological and I would prefer a name that doesn’t privilege any particular model (or family of models).
I don’t like $\infty$-groupoids because not all models for the homotopy theory of topological spaces are higher categorical.
I think this is a non-issue, because “$\infty$-groupoid” is not a name for any model but for the invariant objects presented by all of the models. Some of the models are topological, some are combinatorial, some are globular-algebraic, etc.
I do agree that the previous use of “homotopy type” for an object in the homotopy category of $\infty$-groupoids is a problem. I hope that eventually people will realize that this is not nearly so useful and important a notion, and so the nice word “homotopy type” deserves to be taken for a much more important object, but currently it’s an unfortunate barrier.
I generally try to use “$\infty$-groupoid” when talking about the invariant objects; the only problem I see with it is that it’s six syllables, as opposed to one for “space”. At least the implicit infinity-category theory convention cuts it down to the two syllable word “groupoid”.
It seems to me that it would be good to have some kind of adjective/prefix that means ‘$(\infty, 1)$’ without implicitly assuming the homotopy hypothesis. An obvious candidate is the word ’homotopy’ itself, but then we would be saying ‘homotopy category’ instead of ‘$(\infty, 1)$-category’, which is surely problematic. Perhaps some word along the lines of ‘soft’, ‘pliant’, ‘flexible’, ‘plastic’, ‘fluid’, …
I think this is a non-issue, because “$\infty$-groupoid” is not a name for any model but for the invariant objects presented by all of the models. Some of the models are topological, some are combinatorial, some are globular-algebraic, etc.
My point was that I don’t like to use the name $\infty$-groupoid in such an all encompassing sense. Some of the models feel like $\infty$-groupoids to me. Some, like topological spaces, don’t. And for example simplicial sets look like $\infty$-groupoids if I squint one eye and they don’t look like them at all if I squint the other one. That’s merely a matter of taste and I will not oppose this nomenclature (but I know many people who would give me very skeptical looks if I suddenly replaced all occurrences of “homotopy type” or “space” by “$\infty$-groupoid” in a talk.)
I agree that “$\infty$-groupoid” feels model dependent. Using it in a context where “space” is also appropriate does amount assuming the homotopy hypothesis … but maybe that is a good thing.
I’ve tried to come up with a good neologism. The best I can do is “genotope” (from Greek “genos” (race/kind) and “topos” (place)). Etymologically you can think of it as meaning “homotopy type”, but it’s meant to be used in a model independent way, and not tied to the notion of homotopy category. So any space/simplicial set/$\infty$-groupoid/whatever, gives a presentation of a “genotope”.
Also, it sounds kind of like “genotype”, and that kind of science is where the money is these days. :)
Genotope! That’s very cute!
Look at: Dennis Sullivan, Genetics of homotopy theory and the Adams conjecture, Annals of Math., 100, (1974), 1–79
I also say ‘$\infty$-groupoid’, meaning by it what Mike means. But ‘genotope’ is intriguing! And certainly ‘space’ shouldn't mean topological space by default, since there are many other kinds of spaces even within ordinary $0$-categorial mathematics.
And it's also OK to say things like ‘contractible space’, noting that the property ultimately refers only to the homotopy type.
Why should “$\infty$-groupoid” feel model dependent? I’ve met plenty of people who say “$\infty$-category” and emphasize that when they say that, they mean to not depend on a particular model (as opposed to saying “quasicategory” or “complete Segal space” or whatever when they want to refer to a particular model).
Mike, it’s probably no more model dependent than “space”. But remember that the discussion began with an objection to that.
I don’t care so much that it’s model dependent. I do care than it is an ugly, awkward, and unevocative term.
While ranting: I also dislike (for much the same reasons) the term “$(\infty,1)$-category”. There is at least a serviceable replacement here: “homotopy theory”.
(I first learned about what we now call $(\infty,1)$-categories from a remarkable talk Bill Dwyer gave at a conference in Boston in 1993, while I was still a graduate student. The punchline of the talk was the $(\infty,1)$-categorical Yoneda lemma. He called them “homotopy theories”. I wish that had stuck.)
I agree that it’s ugly and awkward, but I don’t find it unevocative. For me it evokes exactly the right meaning. And I think it is less model dependent than “space”. An $\infty$-groupoid has points, paths, higher paths, and ways to compose all of them, etc. Topological spaces have those, but they also have lots of other things (like open sets). Simplicial sets have those too, but they also have lots of other things (like orientations on their paths). Ditto algebras for a contractible cofibrant globular operad. An $\infty$-groupoid abstracts away from all the superfluous structure in all the models to describe the underlying invariant object.
I’ve met plenty of people who say “$\infty$-category” and emphasize that when they say that, they mean to not depend on a particular model (as opposed to saying “quasicategory” or “complete Segal space” or whatever when they want to refer to a particular model).
Actually, everything I’ve said so far applies to $(\infty, 1)$-categories as well. That’s because not all models of “$(\infty, 1)$-categories” look like higher categories to me, for example relative categories don’t. After all they don’t have any higher morphisms and I need to apply some non-trivial procedure to see them. (Similarly, topological spaces don’t have morphisms on the face of it, I still need to do something to recover them.) On the other hand I definitely see relative categories as homotopy theories.
Here my preference is to use homotopy theory as an invariant notion and $(\infty, 1)$-category as an umbrella term for some family of models. From that perspective it seems attractive to reuse homotopy type as an invariant notion. Then we would say that a homotopy type is an object of the homotopy theory of topological spaces (or simplicial sets or something equivalent.) We would just have to insist on making a clear distinction between “homotopy theory” and “homotopy category.”
I don’t think you can argue against a name for something because it has some model that doesn’t look so obviously like that name. For instance, there’s a model structure on the category of topological spaces which presents the same homotopy theory as the trivial model structure on the category of sets. But would you argue against calling sets “sets” because topological spaces don’t look like sets?
Well, you are right that if I followed this rule strictly I could arrive at absurd conclusions. But I think it makes sense if it’s taken informally. I would say that X is not a good name if there is some natural and important model that doesn’t obviously look like X. By a “natural and important” model I mean for example one which is a standard source of examples or one which is used to formalize standard techniques for working with the notion in question. So topological spaces satisfy both criteria as models of homotopy types and relative categories satisfy at least the first one as models of homotopy theories. But topological spaces as model of sets satisfy neither and I wouldn’t call this model natural or important.
The meaning of “natural and important” is of course debatable but when we choose names for mathematical concepts we need to rely on some informal conventions anyway. And I think the convention I described here is not so bad.
How about ’archetype’? That gives the sense of something original, basic or foundational, characterising some mould or form.
Group presentations are a standard source of examples of groups, and I bet there is a model structure on a category of group presentations which presents the category of groups. But I don’t think a group presentation looks any more like a group than a relative category looks like an $(\infty,1)$-category.
The way I see it, the whole point of higher category theory is to free ourselves from model-dependent intuitions and preferences.
Can I return towards Urs’ original point here? In using ’space’ as meaning ’homotopy type’, one gets in the way of the very rich use of ’space’ in related areas of mathematics. It loses the ’geometric’ nature of the term. Using $\infty$-groupoid as meaning ’homotopy type’ may lead to similar problems, (in particular with ‘backward compatibility’ but that cannot be avoided sometimes). There are combinatorial structures that yield $\infty$-groupoids directly, with initially little in the way of homotopy around. They lead to homotopy types of course, but they also have other structure around. I am thinking of extended systems of syzygies which give presentations of various objects (including groups) but have a deep combinatorial structure. That is well encoded in an $\infty$-groupoid and leads to a sort of algebraic or combinatorial homotopy theory (along the lines that Whitehead initiated and Hans Baues, and Ronnie espoused) but to say that it is just the homotopy type that it determines seems to be throwing away valuable insights into the interaction of that structure with other areas of mathematics.
Mike, you make valid objections and I could try to answer them but perhaps we don’t want to go down this road. Let me just summarize by saying that there are models of homotopy types and homotopy theories that I care about and that I don’t see as higher categories. So this higher categorical terminology just doesn’t seem like the best fit to me.
to say that it is just the homotopy type that it determines seems to be throwing away valuable insights into the interaction of that structure with other areas of mathematics
Indeed, and this is really the same issue (but run backwards) as with saying ‘space’ for a homotopy type, since again you are throwing away other aspects of a (say topological) space when you view it only for its homotopy type. I'm not familiar with these combinatorial structures involving syzygies, but if one cares about them for more than their homotopy type, then one should not call them simply ‘homotopy types’ or ‘$\infty$-groupoids’. (They might conceivably be strict $\infty$-groupoids.)
@Karol: One might argue (inversely to my previous arguments) that if something is a model of homotopy theories, then you should see it as a higher category, because that’s what homotopy theories are. (-:
@Tim, Toby: This seems to be just a question of what “$\infty$-groupoid” means. To me an $\infty$-groupoid is nothing more than a homotopy type (in the sense of an object of the $(\infty,1)$-category of such, rather than just the homotopy 1-category), so if something (like syzygies) has more structure than this that you want to remember, then as Toby says, you shouldn’t call it an $\infty$-groupoid.
This seems to be just a question of what “$\infty$-groupoid” means. To me an $\infty$-groupoid is nothing more than a homotopy type (in the sense of an object of the $(\infty,1)$-category of such…
I suppose this is just an $\infty$-version of the discussion about $Set$.
@Toby: my point with ’syzygies’ is just as an extension of group presentations. For instance, the Steinberg group (of a ring) has a presentation that encodes generators and relations encoding the elementary row operations on matrices. This gives a group! (of course). If you use that presentation and try to build a resolution of the group from it you get to add in new cells at each level and these can be chosen to be glued on so as to correspond to the use of decompositions of the n-sphere… and there is neat combinatorics here since Kapranov and Saito have conjectured that the decompositions are always labelled versions of Stasheff polytopes. (I do not know if the conjecture has been proved correct.) My question would be “Why?”
This corresponds to detailed structure of algebraic K-groups, or rather the homotopy type corresponding to them, and I feel may relate to higher categorical structure on the infinity groupoids. Seeing why a specific homotopy type has a particular structure often involves a combinatorial model for it.
There clearly are groups presentations that tell one more about the group being presented than others as they reflect, perhaps, the symmetry of some geometric figure.
@Tim: your explanation actually makes it less convincing to me as an argument against using “$\infty$-groupoid” for homotopy types. Certainly there are group presentations that tell more about the group being presented than others, just as there are topological spaces that tell more about the homotopy type being presented than others. In a sense that’s the whole point of having separate notions of “presentation” and “presented object”, and why they should have different names (e.g. “space” versus “$\infty$-groupoid” or similarly “syzygy” vs whatever).
OK, so a particular presentation of a group has more information than the group that it presents; and similarly a particular combinatorial syzygy structure (if this is the right way to put it) or a particular labelled Stasheff polytope (if this is the same thing) has more information than its homotopy type. And so I would not use the term ‘$\infty$-groupoid’ for such a combinatorial syzygy structure or for a labelled Stasheff polytope. (Just as I wouldn't use that term for a topological space and, conversely, wouldn't use ‘space’ for a weak homotopy type.)
@Mike: Perhaps I am old fashioned, but there does not seem to me much wrong with saying that a topological space determines a homotopy type, or that a particular infinity groupoid (given a particular choice of model for such things) determines a homotopy type. But it is not the homotopy type itself as if one adopts that terminology one disallows the statement that two spaces have the same homotopy type. I do not see what is wrong with saying along traditional lines that a homotopy type is an equivalence class of objects (of whatever type you like, spaces, etc.)
The question of presented object is similar but slightly different in its detail.
Getting back to Urs’s point the over use of ’space’ is a best confusing at worst ‘incorrect’. It is an overused word so does need additional adjectival specifications.
I do not see what is wrong with saying along traditional lines that a homotopy type is an equivalence class of objects (of whatever type you like, spaces, etc.)
I don’t think there is anything wrong with it but a more useful notion would be one where we do not mod out the homotopy equivalence relation (but which is still model independent). Just like the notion of a group is more useful than that of an isomorphism class of groups. From that perspective the name $\infty$-groupoid has an advantage because there is no risk of confusing it with equivalence class of $\infty$-groupoids while homotopy type is used in both senses.
When someone says that a homotopy type is an equivalence class of objects under the relation of homotopy that is precise. It does not mod out by the relation. It does not say that two objects with the same homotopy type are ‘the same’, merely that they are isomorphic in the ’homotopy category’. It is perhaps foolish to say that an infinity groupoid (modelled perhaps by a Kan complex) has the same homotopy type as a particular space. There is a pair of adjoint functors between Kan and CW that preserves homotopy and induce an equivalence on the homotopy categories. That does not say that there are not other functors that would do the same, but which do we deem to be the RIGHT one? Probably I am misunderstanding something about the use of homotopy type and infinity groupoid, but I prefer to be careful rather than rush into what perhaps I do not see.
My objection to over using infinity groupoid is partly due to the level of sophistication needed to understand, say, that a Kan complex can be thought of as an infinity groupoid. A groupoid is easy to understand, and going through the various steps to break and remend the intuition of structure until the penny drops is difficult. An infinity groupoid is not a homotopy type, since if it were we would have that a groupoid, being an inifinity groupoid, would be a homotopy type, yet a homotopy type is a class of (weakly) homotopy equivalent objects. (That leads to confusion as a terminology.)
I’m confused, I’ve always thought that talking about “equivalence classes” is the same thing as “modding out an equivalence relation”. If you give me a homotopy type in the form of an equivalence class of topological spaces under homotopy equivalence, then I can pick some representatives from this homotopy class and I can pick some equivalences between them, but I can’t pick those equivalences canonically. I think that we shouldn’t forget equivalences and how they relate to each other, but in my vocabulary talking about objects “isomorphic in the homotopy category” is exactly this: we remember if two objects are isomorphic, we don’t remember how they are isomorphic. (I suspect that we are really saying the same thing, we’re probably just having some communication problem.)
There is a pair of adjoint functors between Kan and CW that preserves homotopy and induce an equivalence on the homotopy categories. That does not say that there are not other functors that would do the same, but which do we deem to be the RIGHT one?
Well, this equivalence is essentially unique. The space of self-equivalences of the $(\infty, 1)$-category of Kan complexes is contractible.
I am now officially bored of this discussion, which I think is going around in circles.
Anyone have anything new to add?
I agree Mike that we are not getting very far with this. In any case we seem to have wandered from the original point that Urs made, which we all agree with (I think).
I do not quite think your third summary point is quite correct, but that is because I am coming from a more historical perspective than you are, and I think that is all.
I do think that we have to be careful in the choice of names, for pedagogic reasons amongst others.
One point for Karol: even in the simple case of forming the quotient of a group, the cosets are both considered as equivalence classes of elements of the group, AND elements of the quotient group. There is no difficulty there are there should be none here either. Both are valid views of the same things.
I put in my vote for genotope for what it’s worth.
While I’m sort of attracted to the idea of a totally new word without all the baggage of the old not-quite-right ones, I’m also hesitant to propose a totally new word for something that oodles of mathematicians have been studying for decades under half a dozen different names already.
What do you suggest?
Well, once homotopy type theory takes over the world as a foundational system, then we can just say “type”. (-:O
Until then, I’ll probably continue saying $\infty$-groupoid, or just “groupoid” in the presence of the IICC.
After thinking more about this, I’d like to argue in favor of “spaces” in homotopy theory.
Outside of homotopy theory the word “space” is always used with an adjective: topological space, affine space, vector space, Banach space, compact space, tangent space, etc. In particular, “space” doesn’t mean “topological space” unless presented in the appropriate context. Also, when one talks about vector spaces, it is not uncommon to refer to them as “spaces”, and similarly for other adjectives, so there is nothing special about topological spaces.
There are many other names for geometric spaces: locales, schemes, manifolds, varieties, etc., whereas for homotopical spaces we only have some awkward substitutes like ∞-groupoid and homotopy type, which have their own problems, as explained above. But it does seem like there is no single word to refer to objects of some arbitrary fixed topos or ∞-topos, and I would certainly agree that “space” would be an appropriate way to refer to such objects. However, it doesn’t seem like there is a high demand for such a name, and I’m not sure that I have ever seen the word “space” used in this way.
Finally (and most importantly), names like A_∞-space, E_∞-space, loop space, classifying space, Segal space, Γ-space, path space use the word “space” in the sense of ∞-groupoids, and all of these objects have an underlying “space”, so it’s only natural to use this name for ∞-groupoids.
Note how this type of usage compares to the usage of “space” in names like affine space, topological space, tangent space; in the latter case there is no underlying “space” (without any adjectives), only an underlying set.
So I’d argue that using the word “space” for ∞-groupoids/homotopy types/genotopes does not create any conflicts or ambiguities: geometric “spaces” always have an adjective before them (possibly implied by the context), whereas “space” alone (and without any additional context) can only be interpreted as “∞-groupoid/homotopy type/genotope”.
… but a space is not an infinity groupoid. A (topological) space yields a singular complex which is one model for the infinity groupoid given by the space. Putting my old shape theorists hat on, that hat might be made from a cone on a Warsaw circle, what would be the infinity groupoid that gave its homotopy type. A (top) space has an infinity groupoid but the infinity groupoid is not the spaces as that will have spatial properties not represented by the infinity groupoid.
Note that Lurie’s chapter on applications to topology in HTT explicitly restricts which spaces he uses so that ’space’ can become synonymous with ’oo-groupoid/Kan complex’. Without that restriction one would really have to distinguish between the two (at least, that’s what I get out of it).
I think that was precisely Dmitri’s point: “space” by itself has no meaning without additional context, in particular it does not mean “topological space” until explicitly declared to, so that the term is free to be “occupied” by the homotopy-theoretic meaning.
Another thing to consider is that between any two objects in an $(\infty,1)$-category, there is a “mapping space” of morphisms. The terms “mapping homotopy type” or “mapping $\infty$-groupoid” are not as quite as easy on the tongue…
Hom-groupoid or hom-oo-groupoid are fine terms, and the former is well established for (2,1)-categories :-)
names like A_∞-space, E_∞-space, loop space, classifying space, Segal space, Γ-space, path space use the word “space” in the sense of ∞-groupoids, and all of these objects have an underlying “space”, so it’s only natural to use this name for ∞-groupoids.
I think you’ve got the causality backwards. Such things involve the word “space” because “space” is used to mean $\infty$-groupoid. If we called $\infty$-groupoids “phyles” (to take my most recent suggestion on G+, just for concreteness), then we would talk of $A_\infty$-phyles rather than $A_\infty$-spaces.
geometric “spaces” always have an adjective before them (possibly implied by the context), whereas “space” alone (and without any additional context) can only be interpreted as “∞-groupoid”.
Particular sorts of geometric space may always have an adjective, but that adjective is often omitted when it is clear from context. Moreover, it’s also very useful to be able to talk about “spaces” in a generic nonspecific sense to refer to “things with some kind of geometric structure”. I think it’s poor and inconsistent terminology if phrases like “red herring”, “blue herring”, and “green herring” are all instances of one general notion, but that general notion is not called a “herring”, while unadorned “herring” in fact refers to some totally different object.
A more practical problem is that one object can have both geometric space-structure and $\infty$-groupoid structure. It’s common practice when adding (say) a topology to something that you add the word “topological” in front, e.g. “topological group” or “topological vector space”. But if you want to add a topology to an $\infty$-groupoid and your word for $\infty$-groupoid is “space”, then when you do this you get a… “topological space”?
How does the caution what to call space work together with space and quantity ? There, all presheaves are descibed as spaces of sort, making a whole lot of topoi into categories of spaces, and they will in general be distant to infinity-groupoids.
My caution is really that ’space’ in any of the more usual mathematical senses is seen as a generalisation of the ’everyday’ sense of the word ’space’, so ’topological space’ means we are looking at something that abstracts and generalises something that considers certain ’topological’ aspects of the thing we think of when talking (outside mathematics). ’Infinity groupoid’ involves a term ’groupoid’ which we understand as an algebraic idea involving some multiplication or composition. It, initially, has very little to do with any spatial idea. Using ’space’ to mean ’infinity groupoid’ could easily end up being seen as another case of the obscurantism, of which mathematics is accused when it uses everyday terms for deep concepts. (I think that other sciences are worse than we are at this but that is another point.) An ‘Infinity groupoid’ is not spatial in any really accessible sense. On the other hand, sheaves do encode some geometric flavoured stuff, and Grothendieck topologies certainly have some spatial intuition behind them. Stacks to some people are spaces with local automorphisms of their points, and so on, and when working with stacks the analogy with topological spaces for defining new concepts is one of the central tools.
I disagree with an early claim. I would argue that ’space’ has a meaning when used without adjective. It is its everyday meaning as in ’outerspace’, ’space enough to swing a cat’, ’wide open spaces’ and so on. For fun, try replacing ’space’ in those context by infinity groupoid. Doh!)
Surely infinity-groupoids do have some spatial idea behind them, available whenever one would like to view them that way: they consist of points, and ways to get from one point to another (1-d blobs), and ways of connecting those (2-d blobs, like discs), and ways of connecting those (3-d blobs, like balls) and what have you. It’s an abstract view of space, yes, but then again, so are topological spaces, and sheaves and Grothendieck topologies and affine spaces and what have you.
Sure, this happens to coincide with an algebraic idea about multiplication or composition. Again, same for everything else; one might think of locales as some formalization of some aspects of the everyday concept of space, or as a purely algebraic concept (for an algebraist might well be interested in lattices with arbitrary joins and finitary meets, the latter distributing over the former, without caring at all about “space”). That’s the fun of it, that you can view everything in multiple ways!
Sridhar, one might argue that that spatial “intuition” for $\infty$-groupoids is actually misleading and should be avoided. What you call “ways to get from one point to another” are actually “ways in which one point is the same as another”, which is really quite a different thing. Thinking of $\infty$-groupoids via CW presentations (which is at least similar to what you describe) also tends to lead to confusion between discs and morphisms.
Regarding your second paragraph, there is a reason why we have separate words “frame” and “locale” for the algebraic and topological objects. (-:
Well, one needn’t think of things in such a way as that there is any difference between “ways to get from one point to another” and “ways in which one point is the same as another”; part of how I would think about it, in this context, is that a sameness of points is just a path between them that absolutely everything respects (i.e., is functorial with respect to). An $\infty$-groupoid is like an abstract space, at such a level of abstraction as that we have no notion of whether two points are the “same” or “distinct” other than the one given by asking whether there is or is not a path between them.
I should ask, though: What’s the potential erroneous confusion between discs and morphisms? (Perhaps I am falling prey to it!)
$S^2$ is built out of one 0-disc and one 2-disc only, but has nontrivial 3-morphisms, 4-morphisms, etc. — while $K(\mathbb{Z},2)$ has no nontrivial $k$-morphisms for $k\gt 2$, but requires infinitely many discs of all dimensions to build.
Sridhar, there may be spatial intuition in how we think of these $n$-categorical things but that does not mean that they should be called ’spaces’. An $\infty$-groupoid does encode some information about the space, but this is not even all the homotopy information as it encodes the WEAK homotopy type of the space not the homotopy type.
To suggest that we should avoid equating ∞\infty-groupoids with spaces by calling them spaces is not to say that we should avoid thinking spatially about them. (There are lots of slips that people make in their speak that are not important most of the time. One frequent in homotopy theory is to speak of a space as a CW-complex. No space is exactly a CW-complex if you look at the definitions as a CW-complex is a space together with instructions on how it is made. We thus have a CW-complex is a space but a space, strictly speaking, cannot be a CW-complex. In fact any CW-complex probably determines other smaller $\infty$-groupoid models of its homotopy type than the usual onebut that is another question.) The key idea is that one system models aspects of anothers and the distinction between ’models’ and ’is’ is important.
I’ve jumped into a thread which has already, I suppose, largely run its course, as noted in post #38, so while I clearly have thoughts on the reasonability of calling $\infty$-groupoids “spaces” (that it’s not any less reasonable than already using the word “space” in different contexts for topological spaces, affine spaces, Euclidean spaces, metric spaces, etc., and indeed, that myriad other examples in math uncontroversially reuse the name of a concept from ordinary language in reference to multiple different formalizations according to context), those aren’t what I care to defend right now.
I mainly just wanted to make the weaker point that, whether or not one was willing to call $\infty$-groupoids “spaces”, one should not go all the way to denying that there is any spatial idea behind them (or, rather, I wanted to say that there was no principled reason to consider them clearly devoid of spatial idea while the whole menagerie of topological spaces, affine spaces, Euclidean spaces, metric spaces, etc., are all considered to clearly have spatial idea), as seemed to be implied in #52 (though I may have interpreted it too aggressively).
[Re: #56, if I understand correctly what you mean by “nontrivial”, I don’t see this particular point as giving us reason to think $\infty$-groupoids are nothing like familiar “spaces”, since this particular point might just as well be considered a fact about familiar topological spaces: that $S^2$ is a 2-dimensional surface, yet has maps into it from the 3-sphere which are nontrivial (do not extend into maps from the 4-ball), etc. But perhaps you were just warning against forgetting the fact that, as I would put it, the free $\infty$-groupoid generated by some discs may contain yet other discs in addition to those used as generators, and even ones of greater dimension altogether.]
Sridhar: There is , of course, no reason why when considering spatial aspects of infinity groupoid one cannot start by saying so and inform the reader that you will be using ’space’ to refer to infinity groupds for the duration of the article/chapter/book/thesis or whatever. It would seem however strange to, for instance, take a groupoid, consider it as an infinity groupoid and then say it is to be called a space. The initial intuition behind a groupoid includes a composition and there is no composition in our usual intuition of a space, so there will be a mismatch. A groupoid gives via the nerve construction a simplicial set that is a Kan complex. (The composition is now encoded in the Kan filler condition, of course.) If you take the geometric realisation of that nerve then the composition is hidden by the spatial structure to quite a large extent. That being said looking at a result about groupoids through a Kan complex lens can reveal different aspects and perhaps new results.
My feeling is that we should keep the rich language that refers to spaces (of various types), of infinity groupoids, as models for various structure which may come from a (topological) spatial context, but may reflect some more combinatorial or algebraic origin, etc. and then study what the linkages are between the concepts. They are different concepts to start with so need different names and then clarification of where the real differences are. as to the spatial ideas (perhaps I would prefer ’intuitions’) involved with infinity groupoid, my ’fear’ is that not making the distinctions can lead to worse proofs, that is, arguments that hide what is going on because they are built from a non-optimal intuition of the objects involved.
At the G+ thread I have expressed some sympathy for the idea that ∞-groupoids can be regarded as A notion of space, which analogously to “topological space”, “smooth space”, etc. we might call a “homotopy space”, with any of those adjectives being omittable given a local convention.
Ah, great. That’s exactly how I feel, so there’s no longer any need for me to talk about how I feel. :)
I’d call them “stage”.
I like the idea of using “homotopy space” to disambiguate from other cases.
I think you’ve got the causality backwards. Such things involve the word “space” because “space” is used to mean ∞-groupoid. If we called ∞-groupoids “phyles” (to take my most recent suggestion on G+, just for concreteness), then we would talk of A_∞-phyles rather than A_∞-spaces.
I never meant to imply any causality relationships in that statement. My only point was that the word “space” is extensively used in the existing literature to refer to homotopical spaces.
Moreover, it’s also very useful to be able to talk about “spaces” in a generic nonspecific sense to refer to “things with some kind of geometric structure”.
What kind of geometric structure do vector, affine, and projective spaces carry?
A more practical problem is that one object can have both geometric space-structure and ∞-groupoid structure. It’s common practice when adding (say) a topology to something that you add the word “topological” in front, e.g. “topological group” or “topological vector space”. But if you want to add a topology to an ∞-groupoid and your word for ∞-groupoid is “space”, then when you do this you get a… “topological space”?
I think the word “stack” (in its modern usage, meaning ∞-stack) is an appropriate name for the model-independent notion of a homotopy sheaf of spaces on a site. We have topological stacks, differentiable stacks, algebraic stacks, complex analytic stacks, etc.
What kind of geometric structure do vector, affine, and projective spaces carry?
Lots! They are the basis for a lot of the geometry that is then generalised to manifolds. Of course, in those situations geometric structure is to be interpreted in a more elementary way.
What kind of geometric structure do vector, affine, and projective spaces carry?
That this kind of question is being posed shows how dangerously deceptive the use of “space” in much modern homotopy theory is: people forget what a space actually is.
To clarify my remark about vector/affine/projective spaces, there certainly is a forgetful functor from the category of vector spaces over a field F to schemes over Spec(F), which can play the role of the underlying “space” of V. However, I doubt that such an interpretation will find support among many mathematicians.
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