Let $\Bbbk$ be a field, $X$ affine scheme of finite type over $\Bbbk$. Let $\mathcal C_X$ be the category of closed embeddings of $X$ into (say affine) *smooth* $Y$'s of finite type over $\Bbbk$, morphisms being closed embeddings of $Y$'s (forming commutative triangle). We have a functor $\mathcal C_X^{\mathrm{op}}\to \text{Vect}_\Bbbk$ by $[X\hookrightarrow Y] \mapsto\Omega^1(Y)$ You can also post-compose it with the embedding $\text{Vect}_\Bbbk\to\text{Ch}_\Bbbk$ to the $\infty$-category of chain complexes.

I would like to claim that *if you take the direct limit over $\mathcal C^{\mathrm{op}}_X$ of both versions of this functor (the vector-space and the chain-complex ones), what you get is the module of Kähler differentials, resp. the Illusie cotangent complex, of $X$.* (There is a natural map in one direction and I claim it's an isomorphism of $\Bbbk$-vector spaces, resp. of objects of the derived category of $\text{Vect}_\Bbbk$.)

Are such statements known? Or is there a reference for something similar?

(I think I have an argument at least for the first version, but I couldn't find any references.)

**Edit:** (some additional thoghts)

Clearly the first statment follows from the second, since the functor $H_0$ from non-negatively (homologically) graded complexes to vector spaces preserves colimits.

The cotangent complex is defined as a similar colimit for some chosen "cosimplicial resolution" of $X$ by smooth affine varietis. The colimit in question can thus be thought of as "taking all such reslutions at once", but right now I can't formalize this.

It looks like instead of closed embeddings, one could take all maps to affines and (hope to) get the same answer.

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