# What's the Kan's theorem about geometric realization functor in algebraic geometry?

Denote $\mathbf{LRS}$ the category of locally ringed spaces (where the stalks of morphisms are local morphisms), and $\mathbf{SRng}$ a small category of (commutative unital) rings. For $X\in\mathbf{LRS}$, there is a functor $S_X\colon\mathbf{SRng}\to\mathbf{Set}$ specified by $S_X(A)=\mathbf{LRS}(\operatorname{Spec}A,X)$, which induces a functor $S\colon\mathbf{LRS}\to\mathbf{Funct}(\mathbf{SRng},\mathbf{Set})$.

(Notation: Given a category $\mathcal C$, $\mathcal C(X,Y)$ is the set of morphisms between $X,Y\in\mathcal C$. We assume that all categories are locally small)

The theorem of existence of geometric realization, claims that $S$ has a left adjoint, called the geometric realization functor. In Demazure & Gabriel's Introduction to Algebraic Geometry and Algebraic Groups, they claim that it is a particular case of a well-known theorem of Kan. However, they sketch a proof for this special case.

I wonder what is the well-known theorem of Kan in question? The proof they sketch works along the line that every functor $\mathbf{SRng}\to\mathbf{Set}$ is a colimit of representable functors (via category of elements), and for representable functors $\mathbf{SRng}(A,-)$, they just define the geometric realization as $\operatorname{Spec}A$.

• I don't know, but I would guess it has something to do with Kan extensions. I'm more commenting because I don't even understand what the point of this theorem is--do you mind enlightening me as to why its useful in the context of the book? For example, suppose that I give you a functor on rings which is non-representable in the category of schemes. What then does this geometric realization look like?? – Alex Youcis Nov 20 '16 at 14:47
• @AlexYoucis I'm just trying to understand what this geometric realization is (In fact I wanted to post a question asking for any application for this realization other than to construct the underlying locally ringed space, but I gave up because I doubted that it's too trivial). What they then prove seems that the point-set of geometric realization is "determined by field-valued points", and they have some theorem characterizing the geometric realization. I'm just halfway to understand these. – Yai0Phah Nov 20 '16 at 15:05
• @AlexYoucis I don't know whether you have some examples of non-representable functors in this case, and I don't know whether functors (as presheaves over the opposite category of rings) are better if one assumes in addition that presheaves are sheaves w.r.t. the gros Zariski site. – Yai0Phah Nov 20 '16 at 15:13
• So, my assumption is that this is just Demazure using fancy language. All of these statements are trivial for schemes, and probably extend to locally ringed spaces, but I don't know why you would want to (thus my question)--maybe someone will come along an explain. As for your second question, I'm thinking, for examples, of presheaves that are not sheaves. Namely, there are sheaves for the bit Zariski site which are not etale sheaves. What does this geometric realization look like then? This must imply that things like the etale topology don't extend to a subcanonical topology on LRS. – Alex Youcis Nov 20 '16 at 15:15
• I'm just hesitant for the following reason which I say in good faith. I remember reading the beginning of that book and being convinced that they were trying to develop a scheme theory without schemes. How much that's useful when talking just about schemes (vs. say, algebraic spaces) is not clear, but I would certainly not think it's very intuitive/helpful. Good luck--hopefully someone comes and answers this! – Alex Youcis Nov 20 '16 at 15:17

Let $\cal C$ be small and $\cal D$ be cocomplete, and let $\text{Spec}\colon \mathcal C \to \cal D$ be a functor. Then there is an adjunction $$\text{Lan}_y\text{Spec} \dashv \text{Lan}_\text{Spec} y$$ between the left Kan extension of Spec along the yoneda embedding, and the left Kan extension of the yoneda embedding along Spec.
(the functor $\text{Lan}_\text{Spec} y$ coincides with $\hom(\text{Spec}(-),=)$)