# Definition of a variety in category theory

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:

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

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?

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.