# So what exactly are perverse sheaves anyway?

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.

• 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? Dec 30 '19 at 4:28
• @LSpice I wasn't sure about the terminology (hence the slash). Do you have a suggestion ? Dec 30 '19 at 4:31
• 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. Dec 30 '19 at 4:52

## 3 Answers

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".

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

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.