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Let $k$ be a field. Does the fully faithful inclusion from $k$-algebras to dg-$k$-agebras concentrated in cohomological degrees $\leq 0$ $$\operatorname{Alg}_k\hookrightarrow\operatorname{dgAlg}_k^{\leq 0}$$ satisfy a simple universal property? Or, in the same vein, can we obtain the $\infty$-category of dg-$k$-algebras by applying some formal categorical procedure to the category of $k$-algebras? Note that $\operatorname{Alg}_k$ and $\operatorname{dgAlg}_k^{\leq 0}$ are the categories of algebra objects in the symmetric monoidal ($\infty$-)categories of vector spaces over $k$ and chain complexes over $k$, respectively.

Here is a first attempt which is probably too naive. Consider the Yoneda embedding of $\operatorname{Alg}_k$ into $\operatorname{Fun}((\operatorname{Alg}_k)^{\mathrm{op}},\mathrm{Set})$. Now further compose with the full faithful inclusion of $\mathrm{Set}$ into $\mathcal S$ (the $\infty$-category of spaces). We thus have an embedding $$\operatorname{Alg}_k\hookrightarrow\operatorname{Fun}((\operatorname{Alg}_k)^{\mathrm{op}},\mathrm{Set})\hookrightarrow\operatorname{Fun}((\operatorname{Alg}_k)^{\mathrm{op}},\mathcal S).$$ Now consider the full subcategory of $\operatorname{Fun}((\operatorname{Alg}_k)^{\mathrm{op}},\mathcal S)$ spanned by colimits of (images of) objects of $\operatorname{Alg}_k$. Geometrically, we are considering affine $k$-schemes and formally adjoining (homotopy) limits. Is this full subcategory equivalent to $\operatorname{dgAlg}_k^{\leq 0}$? Obviously there is going to be an enrichment issue: instead of $\mathcal S$, we should consider chain complexes over $k$, or instead of (dg-)$k$-algebras we should consider rings and $E_\infty$ ring spaces. Is it true modulo this issue?

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    $\begingroup$ $\mathrm{Alg}_k$ is already closed under limits and homotopy limits (the inclusion functor is right Quillen), so that won't get you anything more. Also, you can't replace spaces with chain complexes, as ring homomorphisms can't be added. However, the infinity category of dg algebras is equivalent to the infinity category of simplicial algebras (Quillen), which can be characterised in a similar fashion to what you envisage, but using Yoneda rather than co-Yoneda and asking for functors to preserve limits - see for instance arXiv:2105.07888v1 section 4.2. $\endgroup$
    – Jon Pridham
    Commented Jan 29, 2022 at 16:25
  • $\begingroup$ @JonPridham: Argh, yes of course I meant Yoneda instead of co-Yoneda (I was thinking of co-Yoneda of affine schemes). I will edit. $\endgroup$
    – John Pardon
    Commented Jan 29, 2022 at 16:41
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    $\begingroup$ The relationship between simplicial commutative rings (or "animated commutative rings" as Clausen and Scholze would say) and $E_\infty$ rings is something which people (not me!) understand well. See here for example. Simplicial commutative rings over $k$ have a universal property: they are obtained by universally adjoining homotopy sifted colimits (equivalently, homotopy colimits of simplicial objects and filtered colimits) to the subcategory of free objects. $\endgroup$
    – Tim Campion
    Commented Jan 30, 2022 at 20:33

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