nLab constructible sheaf

Idea

A sheaf on an étale site is constructible if its restriction to a suitable decomposition into constructible subsets is a locally constant sheaf.

Conceptual definition

A conceptual definition of a constructible sheaf can be given using topos theory.

Given a poset $P$, a $P$-stratification of a topos or (∞,1)-topos $X$ is a geometric morphism $s_*\colon X\to Fun(P,\infty Grpd)$.

Now a constructible sheaf in $X$ relative to the $P$-stratification $s_*$ can be defined as a locally constant sheaf in $X$ internal to $s_*$. (See the linked article for a definition.)

See He for more details.

References

Original articles:

• Pierre Deligne, La conjecture de Weil. II, Inst. Hautes Études Sci. Publ. Math. 52 (1980) 137-252 [doi:10.1007/BF02684780, numdam:PMIHES_1980__52__137_0/]

• Torsten Ekedahl: On the adic formalism. In The Grothendieck Festschrift, Vol. II, Progr. Math. 87, Birkhäuser (2007) 197-218 [doi:10.1007/978-0-8176-4575-5_4]

• Alexander Beilinson: Constructible sheaves are holonomic, Selecta Mathematica 22 (2016) 1797-1819; [doi:10.1007/s00029-016-0260-z,

  arXiv:1505.06768]

  (arXiv version slightly updated after publication)

An introductory survey:

• Florian Klein, Gerrit Begher: Constructible Sheaves and their derived category [pdf]

A list of relevant definitions and facts:

• Günter Tamme, section II 9.3.2 in: Introduction to Étale Cohomology

• James Milne, section 17 of: Lectures on Étale Cohomology

• The Stacks Project: 44.62. Constructible sheaves

and with more on chain complexes of sheaves and abelian sheaf cohomology in

• Bhargav Bhatt, Peter Scholze, section 6.3 of: The pro-étale topology for schemes [arXiv:1309.1198]

with an eye towards the application in l-adic cohomology/the pro-étale topos.

A general definition in terms of (∞,1)-toposes:

• Li He: An internal description of constructible objects in an $\infty$-topos, [arXiv:2510.25248]

See also:

• Jacob Lurie: Constructible sheaves [pdf]