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In section 19.2.6 of Lurie's "Spectral Algebraic Geometry," he constructs the smooth projective space, which represents the derived version of [the dual of] the usual functor of points interpretation of projective space. The functorial construction is as follows for convenience:

Construction 19.2.6.1: Fix a nonnegative integer $n\geq 0.$ For every connective $\mathbb{E}_\infty$-ring $A,$ let $X(A)$ denote the subcategory of $(\mathsf{Mod}_A)_{/A^{n+1}}$ whose morphisms are equivalences and whose objects are maps $f : L\to A^{n+1}$ with the following property:

  1. The map $f$ admits a left homotopy inverse (that is, exhibits $L$ as a direct summand of $A^{n+1}$).
  2. The $A$-module $L$ is projective of rank $1.$

We will regard the construction $A\mapsto X(A)$ as a functor $X : \mathsf{CAlg}^{\textrm{cn}}\to\mathcal{S}.$

My question is about the last sentence: regarding this construction as a functor. Only assigning an $\infty$-groupoid is not enough to describe a functor, and as I understand, even if we describe how $X$ acts on morphisms of $\mathbb{E}_\infty$-rings this is not enough to construct $X$ as a functor.

This is probably standard, but how do we actually confirm that $X$ is a functor of $\infty$-categories? Specifically, I want to understand what needs to be specified in order to define $X$ as a functor, as well as how those details can be verified in this instance.

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    $\begingroup$ I think that it decomposes into the followings: 1. The association $A\mapsto(\operatorname{Mod}_A)_{/A^{n+1}}$ upgrades to a functor $\operatorname{CAlg}\to\operatorname{Cat}_\infty$; 2. Composing with the maximal groupoid $\operatorname{Cat}_\infty\to\operatorname{An}$; 3. Selecting connected components which satisfy these two properties. By the way, Lurie's definition does not seem to literally coincide with usual functor of points, since in usual description, one considers quotients of $A^{n+1}$ being projective of rank 1 (equivalent up to taking dual, but not the same). $\endgroup$
    – Z. M
    Commented Oct 15 at 10:17
  • $\begingroup$ @Z.M oh, of course! That makes a lot of sense -- then we just need to check that the functor $A\mapsto (\mathsf{Mod}_A)_{/A^{n+1}}$ preserves the subcategories described. I suppose it's also fine to use the slice category over $A^{n+1}$ here since the pullback/base change along $A\to B$ will send $A^{n+1}$ to $B^{n+1}.$ However, I don't see why we need point 3. $\endgroup$
    – Stahl
    Commented Oct 15 at 17:08
  • $\begingroup$ As for the comment of quotients vs. subs, I'm aware of the distinction in convention (i.e., do we use $\operatorname{Sym}\mathcal{E}$ or $\operatorname{Sym}\mathcal{E}^{\vee}$ when defining $\mathbb{P}(\mathcal{E})$)-- good to mention just in case someone isn't familiar, of course. $\endgroup$
    – Stahl
    Commented Oct 15 at 17:10

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