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Say a point $x\in X$ is an immediate a strong specialization of a point $y$ if it is a specialization and there is no intermediate specialization, in the obvious sense. This property is categorical. To see this, note that it is equivalent to the existence of a morphism $T\to X$ where $T$ is a connected two-point scheme, sending the closed point $a$ to $x$ and the generic point $b$ to $y$ (note that $T$ is automatically local,  irreducible and one-dimensional), plus the condition that there is no intermediate point in the same sense.It
This notion is categorical: to see this, it remains to distinguish $b$ from $a$ categorically : we on a scheme $T$ as above. We may assume $T$ reduced, and then there is only one monomorphism $Y\to T$ from a one-point scheme $Y$ with image $b$ and infinitely many with image $a$. (Proof: we have $T=\mathrm{Spec}\,R$ where $R$ is a 1-dimensional local domain with fraction field $K$ and residue field $k$. First, a morphism $Y\to T$ with image $b$ must factor through $\mathrm {Spec} (K)$ Spec}\,(K)$ (which is open), hence must be the inclusion if it is a monomorphism. Second, take some$t\neq0$in the maximal ideal of$R$: then the closed immersions $\mathrm{Spec}\,(R/t^n)\to\mathrm{Spec}\,R$ ($n\geq1$) are distinct monomorphisms with image$a$.) Consequence: "$X$is irreducible of dimension$\leq d$" is categorical, because it is equivalent to "$X$has a point$y$such that every point$x$is obtained from$ y$by at most$d$immediate specializations". Similarly, "$X$is local of dimension$\leq d$" is categorical: replace "specialization" by "generalization" in the previous argument. As a consequenceMartin points out, all specializations are strong on a finite-dimensional locally noetherian scheme, the localization at a point is categoricalbut probably not in general. 3 added 886 characters in body Specialization of points on Immediate specializations: Say a scheme point$x\in X$is an immediate specialization of a point$y$if it is a specialization and there is no intermediate specialization, in the obvious sense. This property is categorical. To see this, note that every connected two-point scheme it is equivalent to the existence of a morphism$T\to X$where$T$is locala connected two-point scheme, irreducible and one-dimensional. To distinguish between sending the closed point$a$to$x$and the generic point$b$, b$ to $y$ (note that $T$ is automatically local,  irreducible and one-dimensional), plus the condition that there is no intermediate point in the same sense.  It remains to distinguish $b$ from $a$ categorically: we may assume $T$ reduced, and then there is only one monomorphism $Y\to T$ from a one-point scheme $Y$ with image $b$ and infinitely many with image $a$. This implies  the claim: (Proof: we have $x\in X$ T=\mathrm{Spec}\,R$where$R$is a specialization of 1-dimensional local domain with fraction field$y$iff there is K$ and residue field $k$. First, a morphism from some $Y\to T$ as above to with image $X$ sending b$must factor through$a$to \mathrm {Spec} (K)$ (which is open), hence must be the inclusion if it is a monomorphism. Second, take some $x$ and t\neq0$in the maximal ideal of$b$to R$: then the closed immersions $y$. \mathrm{Spec}\,(R/t^n)\to\mathrm{Spec}\,R$($n\geq1$) are distinct monomorphisms with image$a$.) Consequence: "$X$is irreducible of dimension$\leq d$" is categorical. Local schemes are categorical:$X$, because it is local with closed equivalent to "$X$has a point$x$iff y$ such that every point $x$ is a generalization obtained from $y$ by at most $d$ immediate specializations".
Similarly, "$X$ is local of dimension $x$. \leq d$" is categorical: replace "specialization" by "generalization" in the previous argument. As a consequence, localizing on a finite-dimensional scheme, the localization at a point is categoricalby the obvious universal property. 2 added 467 characters in body A scheme$X$is reduced if and only if the natural map $$\coprod_{x\in X}\mathrm{Spec} \kappa( x )\to X$$ is an epimorphism. So "reduced" is categorical, and so is$X\mapsto X_{red}$. [EDIT to answer Martin's question: If$X_{red}\subset X$is an epimorphism, then it is an isomorphism; this is true for any closed immersion$X_{0}\subset X$, because closed immersions are equalizers. Indeed, let$I$be the ideal sheaf, and let$p:Y\to X$be the spectrum of the symmetric algebra$A$of$I$. Then$p$has a natural section$s$deduced from the inclusion$I\subset\mathcal{O}_{x}$, and$X_0$is the equalizer of$s$and the zero section.] Specialization of points on a scheme$X$is categorical. To see this, note that every connected two-point scheme$T$is local, irreducible and one-dimensional. To distinguish between the closed point$a$and the generic point$b$, we may assume$T$reduced, and then there is only one monomorphism$Y\to T$from a one-point scheme$Y$with image$b$and infinitely many with image$a$. This implies the claim:$x\in X$is a specialization of$y$iff there is a morphism from some$T$as above to$X$sending$a$to$x$and$b$to$y$. Consequence: dimension is categorical. Local schemes are categorical:$X$is local with closed point$x$iff every point is a generalization of$x\$. As a consequence, localizing a scheme at a point is categorical by the obvious universal property.