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Is there a way to define perverse sheaves categorically/geometrically? Definitions like the following from lectures by Sophie Morel:

The category of perverse sheaves on $X$ is $\mathrm{Perv}(X,F):=D^{\leq 0} \cap D^{\geq 0}$. We write $^p\mathrm{H}^k \colon D_c^b(X,F) \to \mathrm{Perv}(X,F)$ for the cohomology functors given by the $\mathrm{t}$-structure.

are well and good, but to me, it feels like they come with too many "luggages", and therefore make me feel like I don't have as good of an intuition of derived categories as I should.

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    $\begingroup$ Although this question's got upvotes already, so obviously some folks like it (and I can see the appeal!), it seems to me that observing that a definition carries 'luggage' doesn't really constitute a well defined question. I can see that the visible geometry is squeezed out of the definition, but what's not categorical about it? $\endgroup$
    – LSpice
    Commented Dec 30, 2019 at 4:28
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    $\begingroup$ @LSpice I wasn't sure about the terminology (hence the slash). Do you have a suggestion ? $\endgroup$
    – Dat Minh Ha
    Commented Dec 30, 2019 at 4:31
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    $\begingroup$ I don't—actually, although I am sympathetic to the urge to ask this question, I think that it should be more precisely formed (so as, ideally, to have a well defined answer, and probably a single one) before asking. $\endgroup$
    – LSpice
    Commented Dec 30, 2019 at 4:52

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An $\infty$-categorical perspective is given here https://arxiv.org/abs/1507.03913 and a triangulated expansion of those ideas is here https://arxiv.org/abs/1806.00883

More or less, perverse sheaves are the heart of a certain $t$-structure that you build "gluing along a perversity datum".

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If you're looking for a more geometric interpretation of perverse sheaves, you might be interested in MacPherson's 1990 lecture notes "Intersection Homology and Perverse Sheaves." As far as I know, I don't think these were ever officially published, but you can find them on my web site at: http://faculty.tcu.edu/gfriedman/notes/ih.pdf

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After some searching I've found the notes An illustrated guide to perverse sheaves, by Geordie Williamson, which is a beautifully illustrated (and sort of topologically oriented) introduction to perverse sheaves.

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