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The well-known correspondence between vector bundles with flat connection on a smooth complex algebraic variety $X$ and complex representations of $\pi_1(X^{an})$, the fundamental group of the analytification. This extends to the famous Riemann-Hilbert correspondence between regular holonomic $D_X$-modules on $X$ and perverse sheaves on $X^{an}$. Is there any way in which one can view perverse sheaves as representations of some topological object associated to $X^{an}$?

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    $\begingroup$ I don't think so because my understanding is that there are no "canonical" way to take the tensor product in the category of perverse sheaves. $\endgroup$
    – Nicolas Hemelsoet
    Commented Feb 10 at 15:16
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    $\begingroup$ Not in a helpful way, I don't think. One can define a topological object as the category of functors from perverse sheaves to vector spaces and then perverse sheaves give representations of this category in the sense of functors to vector spaces, and probably one can put suitable finiteness conditions on the functors and representations to make all representations arise from perverse sheaves. But this probably isn't useful for much. $\endgroup$
    – Will Sawin
    Commented Feb 10 at 16:15
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    $\begingroup$ Or one can use exodromy to define perverse sheaves on a stratified space to be representations of a certain exit path category in the derived category of vector spaces and then define perverse sheaves to be the heart of a certain t-structure on such representations. This could be a useful perspective but isn't really more enlightening than the usual definition of perverse sheaves. $\endgroup$
    – Will Sawin
    Commented Feb 10 at 16:17
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    $\begingroup$ MathOverflow tip: although not explicitly mentioned in the help centre, it is considered good practice to include at least one top-level tag, such as ag.algebraic-geometry or rt.representation-theory. $\endgroup$
    – R. van Dobben de Bruyn
    Commented Feb 10 at 20:49

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You're not going to get representations of a group (or groupoid, or even category), even in simple examples. On the other hand, it is probably true for formal reasons that perverse sheaves for a fixed stratification (by algebraic subvarieties) are representations of some finitely generated (non-commutative) algebra. In many easy examples, you can actually write down a quiver with relations whose representations give the category you want.

Example. Perverse sheaves on $\mathbf C$ for the stratification $U \amalg Z$ with $U = \mathbf C \setminus \{0\}$ and $Z = \{0\}$ are equivalent to representations of the quiver $u\!:\bullet \rightleftarrows \bullet :\!v$ such that the action of $vu+1$ is invertible.

This example, and a generalisation to perverse sheaves on $\mathbf C^n$ for the product stratification, is due to Galligo, Granger, and Maisonobe [GGM85].

There are many similar results in the literature for other stratified spaces, e.g. hyperplane arrangements or (recently) symmetric powers of $\mathbf C^2$. Representation theorists seem to really like these descriptions.

My understanding is that experts believe something similar to be true in general, related to the exodromy theorem of Treumann [Tre09], Lurie [HA, Appendix A], and Barwick, Glasman, and Haine [BGH18]. I am not aware of a method to compute this quiver in general, nor is it clear to me what you could use it for. That said, I believe that there are people thinking about these questions as we speak (I may or may not turn out to be one of them).

References.

[BGH18] C. Barwick, S. Glasman, P. Haine, Exodromy. Preprint/book in preparation, 2018.

[GGM85] A. Galligo, M. Granger, Ph. Maisonobe, ${\mathcal D}$-modules et faisceaux pervers dont le support singulier est un croisement normal. Ann. Inst. Fourier 35.1 (1985), p. 1-48.

[HA] J. Lurie, Higher algebra. Preprint/book in preparation, version from 2017.

[Tre09] D. Treumann, Exit paths and constructible stacks. Compos. Math. 145.6 (2009), p. 1504-1532.

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