Definition of a variety in category theory

Question

I want to define a variety in category-theoretic terms.

First of all, we define a functorial scheme as in the first definition from "Two functorial definitions of schemes". A variety over a field $k$ is normally defined as a reduced, irreducible and separated scheme of finite type over $k$.

First, I speak about irreducibility (this is from A functorial definition of a projective curve). We define an affine space $\Bbb A^n_k$ as $k$-scheme $\Bbb A^n_k$ such that $\Bbb A^n_k(R)=R^n$ (as in this paper from the arXiv). We consider the coproduct of two of $\Bbb A^n_k$ (in the functor category), that is $\Bbb A^n_k \sqcup \Bbb A^n_k$ (i.e. disjoin union). A $k$-scheme $X$ is said to be connected if there doesn't exists an epimorphism $X \rightarrow \Bbb A^n_k \sqcup \Bbb A^n_k$, and then a scheme $X$ is said to irreducible if for every closed immersion (i.e. proper monomorphism by the valuative criterion for properness) $Z \rightarrow X$ we have $X\setminus Z$ is connected.

Second, a separated scheme is as usual defined (for example, the nlab definition). Let $f:X \rightarrow Y$ be a morphism of schemes. Write $\Delta:X \rightarrow X \times_Y X$ for the diagonal morphism. The morphism $f$ is called separated if $Δ(X)$ is a closed subspace of $X\times_YX$. A scheme $X$ is called separated if the terminal morphism $X \rightarrow \operatorname{Spec}(\mathbb{Z})$ is separated.

Third, a scheme of finite type over $k$ defined as a scheme with a finite cover of ring spectra (functors $\operatorname{Spec}(R)$) of finitely generated $k$-algebras $R$ (from version 2 of this MSE post). Here $\operatorname{Spec}(R)$ is the functor as following:

$$\begin{array}{rrcl} \operatorname{Spec}(R): & \textbf{Ring} & \to & \textbf{Set} \\ & A & \mapsto & \operatorname{Hom}_{\textbf{Ring}}(R,A) \end{array}$$

from "The definition of an affine scheme using a functor" and "Two functorial definitions of schemes" again.

Finally, we want to define a reduced scheme. However the definition of a reduced scheme is the following: the local rings are reduced rings. Because we functorially define a scheme, we don't have directly local rings.

My questions are: Can we define a reduced scheme in category-theoretic terms? And is this definition except for reduced correct?

Thanks in advance!

Answer 1

Your definition of a reduced scheme is correct and it may be verified in the category-theoretic language you ask for. The key idea is the reduction morphism: given any scheme $X$, there is a unique scheme $X_{red}$ called the reduction of $X$ which admits a surjective closed immersion $X_{red}\to X$. If $X$ is reduced, then this is an isomorphism and any surjective closed immersion is isomorphic to this map. If $X$ is not reduced, then this is not an isomorphism and there exists non-isomorphic closed immersions. This is a category-theoretic property (you've covered closed immersions in your post, and surjectivity is categorical by my answer to your previous question), so you can detect whether a scheme is reduced in a category-theoretic way.