Given a manifold, one can associate a stack over the category of manifolds, which is a differential geometric stack. This gives a functor $\text{Man}\rightarrow \text{D.Stacks}$. This is an embedding.

Given a Scheme $S$, one can consider the category of schemes over $S$. Given an $S$-scheme $X$, one can associate a stack over the category $\text{Sch}/S$, which is an algebraic geometry stack. This gives a functor $\text{Sch}/S\rightarrow \text{A.Stacks}/S$. This is an embedding (I did not check but I am almost sure this is true)

I would like to collect what all notions of differential/algebraic geometry of manifolds/schemes are extended to the setup of differential/algebraic stacks.

I know some notions. For example notion of Cohomology of manifolds is extended to Cohomology of differentiable stacks. Similarly, Cohomology of schemes is extended to Cohomology of algebraic stacks.

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    $\begingroup$ Have you looked at the Stacks project? The chapters on algebraic stacks collect this. It's a long list. All the basic notions go over: separated, finite type, proper, smooth, dimension, normal, flat... You can extend quasicoherent sheaf cohomology by descent. $\endgroup$ – dorebell Jul 25 at 4:11
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    $\begingroup$ It's harder to extend etale cohomology, since you might not have enough etale maps from schemes (unless you're Deligne-Mumford). Nevertheless, one can make sense of a derived category of etale sheaves together with the various usual functorialities (including the Grothendieck-Lefschetz trace formula!). A much harder question is to extend notions related to algebraic cycles and intersection theory. This has been intensely studied via things like Gromov-Witten theory. (I've never seriously thought about differentiable stacks; in principle mostly everything should carry over) $\endgroup$ – dorebell Jul 25 at 4:13
  • $\begingroup$ @dorebell Can you please make your comment as an answer (when you are free), may be by adding one or two references.. I do not know about etale Cohomology. $\endgroup$ – Praphulla Koushik Jul 25 at 5:07

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