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!

  • $\begingroup$ I've improved your formatting a bit. Please put all math inside math mode, and use the relevant math typesetting commands like \Delta and \times instead of copying those characters from elsewhere. Presentation matters and will help people take your questions more seriously. $\endgroup$
    – KReiser
    Jun 27, 2020 at 22:21
  • $\begingroup$ Thank you for the improving and the useful advice.I've never asked a long question before, so it's very helpful. $\endgroup$
    – undertate
    Jun 28, 2020 at 3:40

1 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.

  • $\begingroup$ Thanks for the answer.It is very interesting to note that variety is defined in categorical theory, as well as the projection curve. $\endgroup$
    – undertate
    Jun 28, 2020 at 3:49

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