@Z.M: The first bullet point seems to suggest that a noncommutative space can be defined as a noncommutative unital Frechet pre-C*-algebra satisfying the property there. C^∞-rings are commutative, the noncommutative structure is encoded by the coconnective differential graded structure, which corresponds to the ∞-stack structure. Many (most?) examples of noncommutative spaces in Connes-style noncommutative geometry seem to be groupoid algebras, with groupoids corresponding to stacks.

@Z.M: Presumably dropping the commutativity condition from Ćaćić's answer will provide one possible answer, although I'm not sure if Hochschild homology was developed in this specific context (it is certainly used in Connes-style noncommutative geometry in general). As a very different answer, if we adopt the point of view that “noncommutative spaces” are ∞-stacks on the site of smooth manifolds, then coconnective differential-graded C^∞-rings provide another solution, with appropriate Hochschild homology properties.

@TimCampion: Thanks for spotting this, added “pointed” in a few places to make it more clear.

@JamesEHanson: In the same vein, I am not aware of any practical applications of nonsober Alexandroff spaces, e.g., whether we would lose anything of value if we used Alexandroff locales instead of Alexandroff spaces. The statement about a full subcategory of Loc was referring to MLoc, not Top. I now moved it to the previous sentence to reduce ambiguity.

@JamesEHanson: A topological space can be defined as an epimorphism of locales D→L, where D is discrete, and a continuous map of topological spaces can then be defined as a commutative square. So point-set general topology is indeed a part of pointfree general topology. On a more practical side, although one can indeed construct a (rather elaborate) example of a nonsober Scott space, I am not aware of any practical applications of nonsober Scott spaces beyond counterexamples, e.g., whether we would lose anything of value if we used Scott locales instead of Scott spaces.

@PinakBanerjee: The indexing for n-groups is shifted by 1 compared to groupoids: a ordinary group (i.e., a 1-group) is a 0-truncated ∞-groupoid equipped with a group operation.

Which paper of Mycielski proves the stated fact? I've seen claims that his paper “Algebraic independence and measure” supposedly proves it, but upon a closer examination, the statement there is different: if X⊂R^2 has measure 0, then there is a nonempty perfect P⊂R such that (P⨯P)∩X⊂{(r,r)|r∈R}.

@Z.M: A natural setting would the category of C^∞-schemes. In fact, the case of C^∞-loci is already meaningful.

What exactly is your question here? You already answered your question about generating cofibrations correctly. If your question was meant to be contained in the clause “…this doesn't seem to agree with another characterization…”, then more details are necessary about this “other characterization”.

@Yang: A categorical equivalence f:X→Y is weakly equivalent to the identity map id_Y:Y→Y via the commutative square with horizontal maps X→Y and Y→Y. An identity map is a (co)cartesian fibration because it satisfies the lifting property (tautologically).

@James: This is Maharam's theorem, I added a reference to the answer.