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By default, all terms are understood in the infinity sense (“category” means “(∞,1)-category”, etc.)

Recall that the morphism $X \to Y$ is an effective epimorphism if the Čech diagram $$ ... \to X \times_Y X \to X \to Y$$

is a colimit (if we restrict ourselves to (1, 1)-categories, the written part of the diagram is final and the colimit is reduced to it, but this is not our case).

The category of derived smooth loci is, by definition, the dual category to the category of $C^\infty\mathrm{Ring}$-objects in the category $\mathrm{Type}$ (one of the synonyms of $\mathrm{Anima}, \infty\text{-}\mathrm{Groupoid}$). Representing types in the standard way as localizations of simplicial sets and using the Dold-Kan correspondence, it is the dual category to differential graded $C^\infty$-algebras (localized in quasi-isomorphisms). See more about this in recent excellent works, which generally give an overview of the current state of derivative differential geometry:

Which ones are known...

  1. ..necessary..
  2. ..sufficient..
  3. ..equivalent..

..conditions for a morphism of differential graded $C^\infty$-algebras to be an effective monomorphism?

I don't know anything about this outside of the standard example: covering open subloci (to corresponding localizations in one element) is an effective epimorphism (see e.g. Notation 3.1.31 in the Steffens paper above). But I would like to be able to check whether given covering by closed subspaces (that is, corresponding to surjections of $C^\infty$-rings) is an effective epimorphism.

Any information on similar questions in the case of derived algebraic geometry would also be helpful (where such questions are probably simpler).

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    $\begingroup$ For an algebraic analogue, the morphism from $\mathrm{SL}_2$ to $\mathbb{A}^2$ sending a matrix to its first row is an effective epimorphism in affine schemes, but not in schemes or stacks. Derived affine $C^{\infty}$-spaces should be a bit better behaved because gluing works better, but algebraic settings would always take the colimit in sheaves. It's also worth noting that the simplicial/dg equivalence for $C^{\infty}$-rings is much more subtle than Dold-Kan (or even Eilenberg-Zilber, because the algebraic structure isn't operadic). $\endgroup$
    – Jon Pridham
    Commented May 3 at 15:50
  • $\begingroup$ @JonPridham: However, in complete analogy to the Dold–Kan correspondence, the normalized chains functor does induce a Quillen equivalence from simplicial C^∞-rings to dg-C^∞-rings, as shown in Theorem 1.1 of arxiv.org/abs/2303.12699. $\endgroup$
    – Dmitri Pavlov
    Commented May 3 at 18:00
  • $\begingroup$ @DmitriPavlov Indeed, or see Nuiten's 2017 thesis; my point was just that equivalence isn't a formal consequence of Dold-Kan. $\endgroup$
    – Jon Pridham
    Commented May 3 at 20:33
  • $\begingroup$ Hi @JonPridham, could you elaborate on why that map is/is not an effective epimorphism? Do you know if it's an effective epimorphism of (non-derived) affine schemes? This old MSE answer about the question was inconclusive math.stackexchange.com/questions/2403350/equalizers-in-cring/… $\endgroup$
    – Brendan Murphy
    Commented Jul 16 at 23:07
  • $\begingroup$ I guess the idea is that an effective epimorphism of derived schemes should be an effective epi on $\pi_0$, and then should be a surjection on points (which this isn't)? And if I've calculated right then the map $\mathrm{SL}_2 \to \mathbb{A}^2 \setminus \{0\}$ is a universal effective epimorphism of schemes, because these schemes are smooth and surjective + surjective on tangent spaces. Then after taking global section this (maybe) implies that $k[(\mathbb{A}^2 \setminus \{0\})] \to k[\mathrm{SL}_2]$ is an effective monomorphism? $\endgroup$
    – Brendan Murphy
    Commented Jul 17 at 0:41

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