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Is there a fully faithful functor from the category of schemes to the category of topological spaces and continuous maps (or some other sufficiently topological objects, e.g. smooth manifolds and diffeomorphisms or metric spaces and continuous maps)? At least a faithful functor? We do not require the functor to interact well with the forgetful functor.

What if take a field $k$ with no non-trivial automorphisms and restrict to $k$-schemes and $k$-morphisms?

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  • $\begingroup$ If there exists/does not exists such a functor, what is the next thing you are interested to in? $\endgroup$
    – Praphulla Koushik
    Commented Mar 26, 2019 at 11:53
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    $\begingroup$ The category of schemes has a faithful functor to the category of sets (see mathoverflow.net/a/160768) and therefore also a faithful functor to topological spaces. $\endgroup$
    – Reid Barton
    Commented Mar 26, 2019 at 16:59
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    $\begingroup$ There's some related work by Trnkova, Pultr, and others -- keyword "universal categories". According to the introduction of Universal concrete categories and functors, every concrete category fully embeds into the category of topological spaces and open maps; this is apparently shown in Combinatorial, algebraic and topological representations of groups, semigroups and categories. $\endgroup$
    – Tim Campion
    Commented Mar 26, 2019 at 17:07
  • $\begingroup$ @ReidBarton let $a(X)$ be the cardinality of the underlying topological space of a scheme $X$ and let $b(X)$ be the cardinality of the disjoint union of the stalks of the structure sheaf of $X$ at all points. The construction in the linked answer satisfies $|F(X)|=a(X)+2^{b(X)}$. Is it possible to find a faithful functor $G$ such that say $|G(X)|\leq a(X)+b(X)$? $\endgroup$
    – user74900
    Commented Mar 26, 2019 at 20:10

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