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Stream: learning: questions

Topic: Geometrically realized topological spaces

Adittya Chaudhuri (Jun 05 2025 at 13:32):

We know that [[Segal condition]] gives us a way to check "when a simplicial set is a nerve of a small category".

Are there any standard conditions (like Segal conditions) to check "when a topological space is a geometric realization of some simplicial set?"

John Baez (Jun 05 2025 at 22:24):

When you say "is", do you mean "homeomorphic to", "homotopy equivalent to" or "weakly homotopy equivalent to"?

If you're looking for a sufficient condition for a topological space to be homeomorphic to the geometric realization of a simplicial set: I believe that any regular CW complex is homeomorphic to the geometric realization of a simplicial set because it can be subdivided into a simplical complex.

Kevin Carlson (Jun 05 2025 at 23:23):

Why do you want to know this? It's a more natural thing to ask for categories and simplicial sets because you're characterizing objects of a full reflective subcategory. It's less obviously useful to characterize objects in the image of some left adjoint that isn't fully faithful.

Adittya Chaudhuri (Jun 06 2025 at 06:39):

John Baez said:

If you're looking for a sufficient condition for a topological space to be homeomorphic to the geometric realization of a simplicial set: I believe that any regular CW complex is homeomorphic to the geometric realization of a simplicial set because it can be subdivided into a simplical complex.

Thank you. Yes, it makes sense. I hope if I want to create `a mental image of prototype of a space which is homeomorphic to a geometric realisation of a simplicial set then I may not loose much if I just build my intuition by treating them as regular CW complexes like loopless undirected simple graphs or closed 2-cell embeddings of surfaces, etc., Or am I possibly being misled??

Adittya Chaudhuri (Jun 06 2025 at 06:41):

John Baez said:

When you say "is", do you mean "homeomorphic to", "homotopy equivalent to" or "weakly homotopy equivalent to"?

At the moment I am not sure, I want to build my intuitions in each of the above three cases.

Adittya Chaudhuri (Jun 06 2025 at 07:18):

Kevin Carlson said:

Why do you want to know this? It's a more natural thing to ask for categories and simplicial sets because you're characterizing objects of a full reflective subcategory. It's less obviously useful to characterize objects in the image of some left adjoint that isn't fully faithful.

Thank you. I agree with your argument that it is more natural to ask about categories and simplicial sets. Although at the moment it is very premature to tell anything meaningful, my curiosity is partly because I am exploring the following idea:

In any compositional theory of open systems based on the theory of structured cospans, I think it is almost now standard to find an adjunction between $\mathsf{Set}$ and $C$, where $C$ is the finitely cocomplete category of our interest. Let $L \colon \mathsf{Set} \to C$ be the left adjoint. Now, I think the role of the left adjoint is to `choose certain points on an object $a$ in $C$' through which which the object $a$ can interact with other objects in $C$. Now, I think $L \colon \mathsf{Set} \to C$ is capable of playing such a role because the domain category is $\mathsf{Set}$. However, the general theorem (by Baez-Courser) does not put much restriction on the domain category (it just needs the domain category to be finitely cocomplete).

I am trying to explore the idea of appropriately internalising the choosing part of the above in the context of topological spaces. In particular, I was thinking about working with the ($|-|$-$Sing$)-adjunction between $\mathsf{sSet}$ and $\mathsf{Top}$, as both are nice categories, and the adjunction is standard.

John Baez (Jun 06 2025 at 08:16):

Adittya Chaudhuri said:

John Baez said:

When you say "is", do you mean "homeomorphic to", "homotopy equivalent to" or "weakly homotopy equivalent to"?

At the moment I am not sure, I want to build my intuitions in each of the above three cases.

Okay. They're very different!

1. If you're working up to homeomorphism, you need to know that geometric realizations of simplicial spaces are 'locally nice': you'll never get $$\math{Q}$$ with its usual topology, or the Cantor set, and I guess you'll never get a locally infinite-dimensional space like an infinite-dimensional Hilbert space. You can get any regular CW complex: that is, any space that can be built by starting with 0-balls, then attaching 1-balls along maps that are homeomorphisms of their boundaries, then attaching 2-balls along maps that are homeomorphisms of their boundaries, etc.

On MathOverflow you can see a claimed example of a CW complex that is not homeomorphic to a regular CW complex and thus (if I'm right) not homeomorphic to the geometric realization of a simplicial set. If you read the comments you'll see there's a false remark at the top of this answer, but then a more detailed example which sounds like it could be correct.

I don't know if the spaces homeomorphic to the geometric realizations of simplicial sets are precisely those homeomorphic to regular CW complexes. But it seems plausible.

2. If you're working up to homotopy equivalence, I believe every CW complex is homotopy equivalent to the geometric realization of a simplicial set. An infinite-dimensional Hilbert space certainly has this property, simply because it's contractible.

I don't know any general description, up to homeomorphism, of all spaces that are homotopy equivalent to the geometric realization of a simplicial set.

3. I believe every topological space is weakly homotopy equivalent to the geometric realization of a simplicial set. This is part of the motivation of [[weak homotopy equivalence]].

Adittya Chaudhuri (Jun 06 2025 at 09:37):

@John Baez Thank you very much!! This is very helpful. I think from the point of view of my curiosity, as I mentioned here #learning: questions > Geometrically realized topological spaces @ 💬 , at the moment, I want to focus my effort on (1)-the up to homeomorphism case, and for that I think I will first try to prove/find a reference about your conjecture:

I don't know if the spaces homeomorphic to the geometric realizations of simplicial sets are precisely those homeomorphic to regular CW complexes. But it seems plausible.

John Baez (Jun 06 2025 at 10:01):

The reference I gave earlier does most of the work needed.

Adittya Chaudhuri (Jun 06 2025 at 10:21):

Thanks very much!!

John Baez (Jun 06 2025 at 12:46):

I didn't see it claim that the geometric realization of a simplicial set is always homeomorphic to a regular CW complex. However, this should be the easy part.

Adittya Chaudhuri (Jun 06 2025 at 19:14):

Thanks a lot. I will try to complete the proof.