Reference for original Leray definition of a sheaf

Question

Leray originally defined sheaves over closed sets. Is there any easily readable (i.e. obtainable through the Internet and written in English) reference that explicitly states the definition using closed sets?

Answer 1

The original two papers by Leray from 1946 and 1950:

• Jean Leray, L’anneau d’homologie d’une représentation. Comptes rendus hebdomadaires des séances de l’Académie des Sciences 222 (1946), 1366–1368. PDF.

• Jean Leray, L’anneau spectral et l’anneau filtré d’homologie d’un espace localement compact et d’une application continue, Journal de Mathématiques Pures et Appliquées. Neuvième Série, 29 (1950), 1–80, 81–139.

In particular, the first paper contains the following definition of a sheaf:

A sheaf $B$ of modules (or rings) over a topological space $E$ is given by the following data:

• an assignment to every closed subset $F$ of $E$ of a module (or ring) $B_F$, which is $0$ if $F$ is empty;

• an assignment to every pair of closed subsets $f⊂F$ of a homomorphism $B_F→B_f$, which maps an element $b_F∈B_F$ to its intersection $b_F.f$ with $f$; if $f'⊂f⊂F$ and $b_F∈B_F$, we have $(b_F.f).f'=b_F.f'$.

A sheaf $B$ is normal if it satisfies the following properties:

• if $b_F∈B_F$, there is a closed neighborhood $V$ of $F$ and an element $b_V$ of $B_V$ such that $b_F=b_V . F$;

• if $b_F∈B_F$, $f⊂F$, and $b_F.f=0$, then there is a closed neighborhood $v$ of $f$ in $F$ such that $b_F.v=0$.