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I have read the categorical definition of embedding, which is that of regular monomorphism. In the case of the category of locally ringed space, is it true that closed subspaces are defined by closed ...

Let $M,N$ be a manifold and consider the presheaf of spaces $\textrm{Emb}(-, N)$ on the open sets of $M$. Classical embedding calculus produces a goodwillie tower $$ \ldots \rightarrow T_{k+1} \textrm{...