About the relation between the categories $\text{Sch}$, $\text{LRS}$ and $\text{RS}$

Question

I've asked this question https://math.stackexchange.com/questions/1407451/about-the-relation-between-the-categories-textsch-textlrs-and-text on math.stackexchange , however I don't think I will receive any response there (because of the current activity in my post). Therefore I'm asking it here. If the question seems inconvenient because of the excess of questions, I can split this question into other ones (let me know).

I've been trying to find some useful categorical facts about the category of schemes, locally ringed spaces and ringed spaces (that I shall denote by $\text{Sch}$, $\text{LRS}$ and $\text{RS}$ respectively). The motivation is that I'm trying to compute some (co)limits explicitly in the category of schemes.

There's this excellent answer here https://math.stackexchange.com/questions/102973/on-limits-schemes-and-spec-functor , but I still have some doubts.

More precisely, I want to know about the following assertions:

1) Is the category of locally ringed spaces (co)complete? The answer is yes by prop 1.6 in Demazure and Gabriel's "Groupes Algébriques" (I didn't notice that they proved the general case and not just the case of filtered colimits when I posted this question, sorry)

In the answer cited above, the references implies the existence of cofiltered limits and filtered colimits, however as I understand the notion of filtered in these cases is restricted to the case where the index category is a poset.

2)Is the category of ringed spaces (co)complete?

3)What can be said about the underlying topological space of the (co)limit of locally ringed spaces? (Is it the (co)limit of the topological spaces?)

4)What can be said about the underlying topological space of the (co)limit of ringed spaces? (Is it the (co)limit of the topological spaces)

5)What can be said about the underlying topological space of the colimit of schemes? (Is it the (co)limit of the topological spaces)

Obviously, the underlying topological space of the pullback of schemes is not the pullback of the topological spaces (for instance, $\text{Spec} (\mathbb{C}) \times_{\text{Spec} (\mathbb{R})}\text{Spec} (\mathbb{C}) \cong \text{Spec} (\mathbb{C}\times\mathbb{C})$ by $a \otimes z \mapsto (az, a\overline{z})$), but the case of push outs seems to be true.

6) Are (co)limits preserved under the inclusions $\text{Sch} \hookrightarrow\text{LRS} \hookrightarrow \text{RS}$?

The inclusion $\text{LRS} \hookrightarrow \text{RS}$ preserves colimits since it's a left adjoint (see below)

7) For each forgetful functor $U : \mathcal{C} \rightarrow \mathcal{D}$, where $\mathcal{C}$ and $\mathcal{D}$ are equal to $\text{Sch}$, $\text{LRS}$ and $\text{RS}$ (for all possible coherent substitutions), are there adjoint functors?

8) For each inclusion $\mathcal{C} \hookrightarrow \mathcal{D}$, where $\mathcal{C}$ and $\mathcal{D}$ are equal to $\text{Sch}$, $\text{LRS}$ and $\text{RS}$ (for all possible coherent substitutions), are there adjoint functors?

According to http://arxiv.org/abs/1103.2139 [Cor. 6], the inclusion $\text{LRS} \hookrightarrow \text{RS}$ have a right adjoint given by localization of the terminal prime system.

Thanks in advance.

Answer 1

Edited to incorporate Marc's comments below:

1) The cocompleteness of LRS is Prop 1.6 in Demazure-Gabriel's Groupes algebriques. Completeness is proved in http://arxiv.org/abs/1103.2139, Corollary 5.

2) Yes. The category of ringed spaces is (co)fibered over the category of topological spaces (which is (co)complete) and has (co)complete fibers. EDIT: To answer the OPs question, I can't think of a reference offhand but it's not too bad to prove straight from the definitions. Take your diagram upstairs, form the (co)limit downstairs, choose (co)cartesian lifts of the projection/inclusion maps from/to your (co)limit, form the (co)limit in the fiber, and you're good.

4) The forgetful functor RS → Top preserves limits and colimits. See (2), in this situation (co)limits are constructed via their projections to the underlying topological space.

3) The proof of Demazure-Gabriel's Prop 1.6 shows that the inclusion $LRS\subset RS$ preserves colimits (even better: creates them).

5) See (6).

6) Colimits are not preserved by the inclusion $Sch \subset LRS$, see Emerton's comment here: Colimits of schemes. Colimits are preserved by $LRS \subset RS$ as stated above in (3). The inclusion $Sch \subset LRS$ preserves finite limits by results in section 5.1 of Demazure-Gabriel.