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EDIT: As mentioned below in the comments, take subcategory of $\mathbf{Vec}_k$ consisting of endomorphisms. Then $\text{End}_k(V)$ carries a natural ring structure.

My question is in multiple parts:

$\textit{1}.$ Given a category $\mathcal{C}$ enriched over the category $\mathbf{Comm}$ of commutative unital rings, can we obtain a related category $\text{Spec}\mathcal{C},$ together with a functor $F\colon\mathcal{C} \to \text{Spec}\mathcal{C}$ such that $$\text{Hom}_{\text{Spec}\mathcal{C}}(FA, FB) = \text{Spec}\left(\text{Hom}_\mathcal{C}(A,B)\right)$$ for every $A, B \in \mathcal{C}.$ Moreover, I would like that the pair $(\text{Spec}\mathcal{C}, F)$ to be universal with this property. Ideally, I would appreciate a reference request (asuuming that one exists). If this is impossible to do, why so?

$\textit{2.}$ If a category as described in $\textit{1}$ exists, is it useful? Specifically, have such objects been used in the literature? My idea in asking this question is that the geometric properies of the enrichment of $\text{Spec}C$ could be used to better understand the original cateogry $\mathcal{C}.$

I realize that this question is not entirely well-posed. This question arose through my recent study of $\infty$-categories and some derived algebraic geometry. In particular, I am intersted in the extensions of questions $\textit{1}$ and $\textit{2}$ replacing $\mathbf{Comm}$ with $\mathbf{sComm},$ the category of simplicial (commutative, unital) rings. My hope is that if the answer is in the affirmative, then derived algebraic geometry could be used to analyze geometric properties of categories enriched over $\mathbf{sComm}.$

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    $\begingroup$ How do those $k$-linear maps carry a ring structure? $\endgroup$ – Tobias Kildetoft May 9 '18 at 18:23
  • $\begingroup$ You are right, the example I provided was wrong. I was thinking of the endomorphisms of a vector space. $\endgroup$ – Aurel May 9 '18 at 18:31
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    $\begingroup$ I assume Aurel meant to say that $Hom_k (V,V)$ has a natural ring structure. This isn't an enrichment over Ring, but if you take the fully faithful subcategory of $Vect_k$ which has only endomorphisms, then it is enriched over Ring. $\endgroup$ – Tbraz May 9 '18 at 18:33
  • $\begingroup$ @Tbraz Right, but once we do that, we might as well not work with it as a category, since we just have a bunch of distinct objects with no morphisms between them. $\endgroup$ – Tobias Kildetoft May 9 '18 at 18:38
  • $\begingroup$ We can also consider a vector space $V$ together with $\text{End}_k(V)$ as a one object category. I am having trouble answer question $\textit{1}$ even in this case. $\endgroup$ – Aurel May 9 '18 at 18:50
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You're confused about a few things, I think.

First of all, $\text{Vect}$ is not enriched over rings, not even if you only consider the subcategory consisting of endomorphisms. While it's true that $\text{End}(V)$ has a ring structure, in order for this to constitute an enrichment over the category of rings (with the product as the symmetric monoidal structure), composition $\circ : \text{End}(V) \times \text{End}(V) \to \text{End}(V)$ would need to be a ring homomorphism, which it's not as soon as $V$ has dimension greater than $1$.

More generally, if $C$ is any category enriched over a symmetric monoidal category $M$, and $c \in C$ is an object, then $\text{End}(c)$ must be a monoid in $M$. And you can check that if $M$ itself is the category of rings, equipped with the product symmetric monoidal structure, then a monoid in $M$ is a commutative ring, by a special case of the Eckmann-Hilton argument.

In any case, your question was about commutative rings, so I don't know why you brought $\text{Vect}$ into it either way.

The relationship between commutative rings and affine schemes is contravariant, so I don't understand your first question, which seems to be asking for something covariant. If you consider a category $C$ enriched over affine schemes and consider all of its hom-objects as commutative rings, you've changed the direction of composition from something going from $2$ objects to $1$ object to something going from $1$ object to $2$ objects. In other words you no longer have a category but a "cocategory."

And in any case I don't know any particularly natural examples of categories enriched over either commutative rings or affine schemes, except for some mildly degenerate examples of the latter. For example, the category of f.g. projective modules over a commutative ring $k$ is enriched over affine schemes over $k$; the functor of points of the affine scheme $\text{Hom}(P, Q)$ sends a commutative $k$-algebra $R$ to $R \otimes_k \text{Hom}(P, Q)$. But all of these affine schemes are easy-to-understand commutative group schemes, so not very interesting from the point of view of algebraic geometry.

There are various senses in which one might try to take the "spectrum of a category," but this isn't one of them. Instead the input is a category which itself is a commutative ring in some sense, e.g. some kind of symmetric monoidal category where the symmetric monoidal structure categorifies a ring product; see the nLab for some details.

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  • $\begingroup$ Thank you! I indeed was confused in bringing Vec into this, but your answer clarified my confusion. Incidentally, I made a typo in question one which you noted, I should be looking for something contravariant. This does not patch the lack of examples however, which should be an indication that this is not a particularly relevant question, at least not presently. In any case, thanks for a great answer and for the nLab link! $\endgroup$ – Aurel May 11 '18 at 14:20

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