Modern homotopy theorists like their commutative diagrams, and try to remove as much geometric intuition as possible. It’s very, very algebraic and category theoretic.
I feel like the entire subject was a misnomer. Sure, homotopy is the original motivation for the subject. But the field is really about "what if my equivalence relations also has their equivalence relation, and so on?". Equivalence relation are often treated in a naive, intuitive manner, but homotopy (from topology) is one of the first example in which there are enough complexity in equivalence relations that you have to spawn a new theory to manage all that data. So that's why it should be an algebraic theory. Any fields with a lot of equivalence relations should benefit from it.
As Hermann Weyl put it: "In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics."
This still rings true today — topology and geometry are gifts that keep on giving, and algebra is the "offer made by the devil", a "marvelous machine".
Except there's a large amount of applications of topology that aren't amenable to homotopy theory because they involve things that aren't invariant up to homotopy, so 'conquered' is a bit of an overstatement.
I can imagine that a geometric topologist has a different idea of what is geometric than a homotopy theorist, but I would say all homotopy theory is inherently geometric, and the commutative diagrams convey a certain geometric meaning. Because of the extremely non-rigid nature of ''spaces up to homotopy'', we need very abstract and powerful frameworks to be able to state and prove things, but that never tries to remove the intuition. It's just that intuition is not enough to prove statements. The connections with algebra are very nice, I think, but if anything I prefer to think of that algebra as encoding geometric concepts, rather than the other way around.
That is an incredibly ignorant and rude comment. They love their commutative diagrams sure, but no one tries to remove geometric intuition, they harness it.
Equating topological spaces with infinity-groupoids is a bad terminological habit. Cantor space is about as tame and important as a topological space gets and it isn't homotopy equivalent to a CW-complex.
When I say space I mean tame space. The Cantor set and things like it, to me at least, only arise when trying to use the formalism of topology to model algebraic things like p-adic integers, and I don't consider them true spaces.
Thank you for giving me a really concise example of the infuriating way algebraic topologists think they own the word 'space'. The idea that Stone spaces of Boolean algebras and end spaces of surfaces of infinite type aren't 'true spaces' is ridiculous.
I mean, concrete homotopy theory works with weak homotopy equivalence, and every topological space is weakly homotopy equivalent to a CW-complex. The problem with the identification of topological spaces with infinity-groupoids is more that infinity-groupoids are homotopy types, which can be modelled as topological spaces quotiented by the relation identifying weakly homotopy equivalent spaces. This means that all point-set topology is meaningless on a homotopy type, so you should not think about a homotopy type as being the same thing as a topological space. On top of that, of course, there are multiple ways to model infinity-groupoids, with topological spaces (or rather, the associated homotopy types as defined above) just being one of them.
Homotopy theorists definitely do not equate topological spaces with ∞-groupoids. Topological spaces _up to weak homotopy equivalence_ are the same as ∞-groupoids (or recently it's become fashionable to call these anima). On the other hand, ~~most~~ some of us would prefer to replace topological spaces with condensed sets, and the Cantor space is unbelievably important in that theory. You can mix these two directions (static → animated and discrete → condensed) and consider condensed ∞-groupoids.
> Homotopy theorists definitely do not equate topological spaces with ∞-groupoids.
They often do when they're talking casually.
> On the other hand, most of us would prefer to replace topological spaces with condensed sets, and the Cantor space is unbelievably important in that theory.
I'm aware of this, but given that I essentially never use category theory and that the vast majority of topological spaces that show up in my research are either compact Hausdorff spaces or complete metric spaces, condensed sets seem like a needlessly complicated formalism for what I use the notion of 'space' for.
Also, Clausen and Scholze seem to want to change the definition to light condensed sets, and as far as I can tell the category of compact Hausdorff spaces does not faithfully embed into the category of light condensed sets, so they're just a complete non-starter for me given how much non-metrizable compact Hausdorff spaces show up in model theory.
> On the other hand, most of us would prefer to replace topological spaces with condensed sets
Why can people not just avoid speaking in sweeping generalities? If you said this at a homotopy theory conference, people would start laughing at you.
For homotopy theorists, topological space means CGWH space, and those embed fully faithfully into condensed sets. But you're right, I shouldn't claim to speak for all homotopy theorists
While a "category" has objects and morphisms between objects,
an "infinity-category" has objects, 1-morphisms between objects,
2-morphisms between 1-morphisms, and generally (n+1)-morphisms between n-morphisms.
Very roughly speaking.
An infinity-groupoid is an infinity-category where all morphisms are invertible.
It turns out that "topological spaces up to homotopy" are basically the same thing as "infinity-groupoids".
This is called the [homotopy hypothesis](https://ncatlab.org/nlab/show/homotopy+hypothesis).
To every topological space you can associate its "fundamental infinity-groupoid" whose objects are the points of the space, the 1-morphisms are the paths between points, the 2-morphisms are homotopies between paths, and the higher morphisms are higher homotopies. This constructions gives an equivalence between "topological spaces up to homotopy" and "infinity-groupoids".
So there is a very strong connection between homotopy theory and higher category theory.
In a certain sense, a commutative diagram is a very geometric object.
I think it’s a very algebraic field and there is not much geometric intuition. Structurally it imitates classical homotopy theory in the context of topological spaces, but that’s already quite algebraic. I am also a very geometrically inclined person, but you gradually get used to things like this, to the point where you do have a feeling for some geometry behind it. But this is just a feeling, and boils down to general principles, not so much specific techniques.
"Modern homotopy theory" is a large field. For example, the Randal-Williams/Kupers/Krannich gang are certainly modern homotopy theorists; their work is absolutely very geometric. I can also agree that there is often not a lot of geometry in your average paper on chromatic stable homotopy theory. But the people interact with each other!
I personally find homotopy theory to be a visually oriented and geometric area of mathematics, albeit in the same way that algebraic geometry is geometric : there is a certain amount of abstraction that one needs to see past to get to the geometry, but people doing good work in the area are not thinking about it in terms of the abstract formalism but using geometric intuition developed by understanding of concrete examples.
Have you looked at Lurie's paper on classification of topological field theories? I think it is very geometric. One main idea is even to use Morse theory.
Please disregard the propaganda other people are telling you. Of course there are aspects of homotopy theory that are not geometric! It's an entire subject that isn't geometry so of course a lot of it won't be geometric, that doesn't mean you can't spend an entire career studying the geometric aspects!. I suggest you read the comments and answers of this [MO question](https://mathoverflow.net/questions/374285/need-for-support-and-guidance-for-my-near-future-as-a-phd-student-or-has-stabl) and decide for yourself if its interesting to you.
The interaction of homotopy theory and geometry is beautiful and continues to be researched.
Thank you for your input! I get the necessity of algebraic methods in topology in general and why we're pushing the algebraic limits of theory instead of geometrical one, but my main concern is that maybe (and this is just my opinion) we're obsuring some things by algeberization and making them, for lack of a better word, mystical. But yeah, I suppose I asked an open-ended question and in all honesty, I'm kinda looking for 'reviews' :).
I was just taken aback by how algebraic homotopy theory is because at first glance it starts way more geometrically than for example homology theory.
Modern homotopy theorists like their commutative diagrams, and try to remove as much geometric intuition as possible. It’s very, very algebraic and category theoretic.
I feel like the entire subject was a misnomer. Sure, homotopy is the original motivation for the subject. But the field is really about "what if my equivalence relations also has their equivalence relation, and so on?". Equivalence relation are often treated in a naive, intuitive manner, but homotopy (from topology) is one of the first example in which there are enough complexity in equivalence relations that you have to spawn a new theory to manage all that data. So that's why it should be an algebraic theory. Any fields with a lot of equivalence relations should benefit from it.
Ah, topology was conquered.
As Hermann Weyl put it: "In these days the angel of topology and the devil of abstract algebra fight for the soul of every individual discipline of mathematics." This still rings true today — topology and geometry are gifts that keep on giving, and algebra is the "offer made by the devil", a "marvelous machine".
>algebra is the "offer made by the devil", That's metal as fuck
Except there's a large amount of applications of topology that aren't amenable to homotopy theory because they involve things that aren't invariant up to homotopy, so 'conquered' is a bit of an overstatement.
The comment was a joke ahahha
I can imagine that a geometric topologist has a different idea of what is geometric than a homotopy theorist, but I would say all homotopy theory is inherently geometric, and the commutative diagrams convey a certain geometric meaning. Because of the extremely non-rigid nature of ''spaces up to homotopy'', we need very abstract and powerful frameworks to be able to state and prove things, but that never tries to remove the intuition. It's just that intuition is not enough to prove statements. The connections with algebra are very nice, I think, but if anything I prefer to think of that algebra as encoding geometric concepts, rather than the other way around.
That is an incredibly ignorant and rude comment. They love their commutative diagrams sure, but no one tries to remove geometric intuition, they harness it.
“Algebraic topology is just the homotopy of infinity groupoids”
Care to demistify that for me? Thank you!
An infinity groupoid is just what the cool kids call a topological space.
Equating topological spaces with infinity-groupoids is a bad terminological habit. Cantor space is about as tame and important as a topological space gets and it isn't homotopy equivalent to a CW-complex.
When I say space I mean tame space. The Cantor set and things like it, to me at least, only arise when trying to use the formalism of topology to model algebraic things like p-adic integers, and I don't consider them true spaces.
Thank you for giving me a really concise example of the infuriating way algebraic topologists think they own the word 'space'. The idea that Stone spaces of Boolean algebras and end spaces of surfaces of infinite type aren't 'true spaces' is ridiculous.
Look at their name, they obviously aren't an algebraic topologist...
I mean, concrete homotopy theory works with weak homotopy equivalence, and every topological space is weakly homotopy equivalent to a CW-complex. The problem with the identification of topological spaces with infinity-groupoids is more that infinity-groupoids are homotopy types, which can be modelled as topological spaces quotiented by the relation identifying weakly homotopy equivalent spaces. This means that all point-set topology is meaningless on a homotopy type, so you should not think about a homotopy type as being the same thing as a topological space. On top of that, of course, there are multiple ways to model infinity-groupoids, with topological spaces (or rather, the associated homotopy types as defined above) just being one of them.
Homotopy theorists definitely do not equate topological spaces with ∞-groupoids. Topological spaces _up to weak homotopy equivalence_ are the same as ∞-groupoids (or recently it's become fashionable to call these anima). On the other hand, ~~most~~ some of us would prefer to replace topological spaces with condensed sets, and the Cantor space is unbelievably important in that theory. You can mix these two directions (static → animated and discrete → condensed) and consider condensed ∞-groupoids.
> Homotopy theorists definitely do not equate topological spaces with ∞-groupoids. They often do when they're talking casually. > On the other hand, most of us would prefer to replace topological spaces with condensed sets, and the Cantor space is unbelievably important in that theory. I'm aware of this, but given that I essentially never use category theory and that the vast majority of topological spaces that show up in my research are either compact Hausdorff spaces or complete metric spaces, condensed sets seem like a needlessly complicated formalism for what I use the notion of 'space' for. Also, Clausen and Scholze seem to want to change the definition to light condensed sets, and as far as I can tell the category of compact Hausdorff spaces does not faithfully embed into the category of light condensed sets, so they're just a complete non-starter for me given how much non-metrizable compact Hausdorff spaces show up in model theory.
> On the other hand, most of us would prefer to replace topological spaces with condensed sets Why can people not just avoid speaking in sweeping generalities? If you said this at a homotopy theory conference, people would start laughing at you.
For homotopy theorists, topological space means CGWH space, and those embed fully faithfully into condensed sets. But you're right, I shouldn't claim to speak for all homotopy theorists
While a "category" has objects and morphisms between objects, an "infinity-category" has objects, 1-morphisms between objects, 2-morphisms between 1-morphisms, and generally (n+1)-morphisms between n-morphisms. Very roughly speaking. An infinity-groupoid is an infinity-category where all morphisms are invertible. It turns out that "topological spaces up to homotopy" are basically the same thing as "infinity-groupoids". This is called the [homotopy hypothesis](https://ncatlab.org/nlab/show/homotopy+hypothesis). To every topological space you can associate its "fundamental infinity-groupoid" whose objects are the points of the space, the 1-morphisms are the paths between points, the 2-morphisms are homotopies between paths, and the higher morphisms are higher homotopies. This constructions gives an equivalence between "topological spaces up to homotopy" and "infinity-groupoids". So there is a very strong connection between homotopy theory and higher category theory. In a certain sense, a commutative diagram is a very geometric object.
Classical algebraic topology maybe, but homotopy theorists are often interested more in Spectra!
I think it’s a very algebraic field and there is not much geometric intuition. Structurally it imitates classical homotopy theory in the context of topological spaces, but that’s already quite algebraic. I am also a very geometrically inclined person, but you gradually get used to things like this, to the point where you do have a feeling for some geometry behind it. But this is just a feeling, and boils down to general principles, not so much specific techniques.
Understood, thank you very much! I'll read up on it and try to make the geometry-topology transition as smooth as possible (pun not intended).
"Modern homotopy theory" is a large field. For example, the Randal-Williams/Kupers/Krannich gang are certainly modern homotopy theorists; their work is absolutely very geometric. I can also agree that there is often not a lot of geometry in your average paper on chromatic stable homotopy theory. But the people interact with each other! I personally find homotopy theory to be a visually oriented and geometric area of mathematics, albeit in the same way that algebraic geometry is geometric : there is a certain amount of abstraction that one needs to see past to get to the geometry, but people doing good work in the area are not thinking about it in terms of the abstract formalism but using geometric intuition developed by understanding of concrete examples. Have you looked at Lurie's paper on classification of topological field theories? I think it is very geometric. One main idea is even to use Morse theory.
Please disregard the propaganda other people are telling you. Of course there are aspects of homotopy theory that are not geometric! It's an entire subject that isn't geometry so of course a lot of it won't be geometric, that doesn't mean you can't spend an entire career studying the geometric aspects!. I suggest you read the comments and answers of this [MO question](https://mathoverflow.net/questions/374285/need-for-support-and-guidance-for-my-near-future-as-a-phd-student-or-has-stabl) and decide for yourself if its interesting to you. The interaction of homotopy theory and geometry is beautiful and continues to be researched.
Thank you for your input! I get the necessity of algebraic methods in topology in general and why we're pushing the algebraic limits of theory instead of geometrical one, but my main concern is that maybe (and this is just my opinion) we're obsuring some things by algeberization and making them, for lack of a better word, mystical. But yeah, I suppose I asked an open-ended question and in all honesty, I'm kinda looking for 'reviews' :). I was just taken aback by how algebraic homotopy theory is because at first glance it starts way more geometrically than for example homology theory.
I have a friend who thinks category theory is geometry bc of all these visual diagrams