# What are “Lazard” sheaves?

Early in Categories, Allegories, by Freyd and Scedrov (p.12, in the section on basic examples) there appears the following example:

Let $\mathcal{LH}$ be the category whose objects are topological spaces and whose morphisms are continuous maps that are LOCAL HOMEOMORPHISMS: … $\mathcal{LH}/Y$ is the category of LAZARD SHEAVES over Y…. Its objects may be viewed as continuously $Y$-indexed families of pairwise-disjoint sets, its maps are $Y$-indexed families of functions which, in concert, are continuous, …

On the following page, the authors opine:

Lazard sheaves and functor categories provide the two most important families of examples in Geometric Logic.

"Lazard sheaves" are not defined anywhere (except, I think, implicitly as the objects of $\mathcal{LH}/Y$) and I cannot find this terminology used anywhere else. Google Book Search yields this one book and no others.

What are Lazard sheaves? The same as just regular sheaves? Who's Lazard? Are Freyd and Scedrov using nonstandard terminology?

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Another term is "étale space". There's an equivalence between the category of étale spaces over $X$ and the category of sheaves on $X$. I have also wondered why they call them "Lazard" sheaves. –  Hurkyl May 31 '12 at 17:10
I do not know the answer, but I have asked the person I know who purports to be algebraist/computer scientist Daniel Lazard. It might also be Michel Lazard, but I know even less about that mathematician. –  rschwieb May 31 '12 at 18:00
Apparently it was Michel Lazard who introduced the formulation of sheaves in terms of étale spaces. In the notes to his Séminaire, Faisceaux sur un espace topologique. I., Séminaire Henri Cartan, 3 (1950-1951), Exposé No. 14, Henri Cartan deviates from his previous approach and attributes the new definition to Lazard. See also the "Prologue" to Mac Lane Moerdijk, page 1. –  t.b. May 31 '12 at 18:05
@t.b.If you don't post that as an answer, then I will. –  MJD May 31 '12 at 18:18
@Mark: I can do that if you don't get a better answer, say, within the next day or two. But I'm curious if somebody more conversant with the topic at hand than me can contribute something more substantial. If you're in a hurry, feel free to do so :) –  t.b. May 31 '12 at 18:20

Yes, judging by the Universal Measure of Standard Terminology™, Freyd and Ščedrov use non-standard terminology: Googling for "Lazard sheaves" currently gives me four hits: two of them lead to Freyd and Ščedrov's book, the other two lead to the present thread.

The word “Cartan–Lazard sheaf” appears on page 129 of Dieudonné's A History of Algebraic and Differential Topology, 1900 – 1960, Springer 2009 edition. In this book, Dieudonné distinguishes between Leray's definition and the Cartan–Lazard definition of sheaves and gives details on their relation in §7B The Concept of Sheaf, page 123.

As I mentioned in a comment,

[...] it was [very likely] Michel Lazard who introduced the formulation of sheaves in terms of étalé spaces. In the notes to his Séminaire, Faisceaux sur un espace topologique. I., Séminaire Henri Cartan, 3 (1950-1951), Exposé No. 14, Henri Cartan deviates from his earlier approach and attributes the new definition to Lazard. See also the “Prologue” of Mac Lane–Moerdijk, Sheaves in Geometry and Logic on page 1.

The history of sheaves is covered in many books. Apart from the already mentioned section of Dieudonné, two nice accounts are:

Freyd and Ščedrov define Lazard sheaves over a topological space $Y$ via the slice category $\mathscr{LH}/Y$, and as was already pointed out, the resulting notion is better known under the name espace étalé or étalé space, see the nLab entry and the Wikipedia page on sheaves.

An étalé space $(X,p)$ over a topological space $Y$ thus is a local homeomorphism $p\colon X \to Y$ and a morphism $f\colon(X,p) \to (X',p')$ is a local homeomorphism $f\colon X \to X'$ such that $p'f = p$. It is easy to check that it suffices for $f$ to be continuous, the local homeomorphism property is a consequence of continuity.

These are the same as the usual sheaves in that there is an equivalence of categories between étalé spaces and the usual sheaves. This is detailed in (1.37.) of Freyd and Ščedrov.

Added: There is some remaining uncertainty as to whether it really was Michel Lazard (as opposed to another Lazard) who introduced the notion.

Henri Cartan only attributes the concept to a certain Lazard (it was customary among Bourbakistes to mention other mathematicians by last name only). Unfortunately, the notes for the talks 12–17 of the first Séminaire Cartan (1948–1949) seem to be lost from the public records. The table of contents contains the following note:

Houzel speaks of M. Lazard in the last paragraph on page 12. Incidentally, it is also mentioned there that Godement coined the term espace étalé. See also the footnotes on page 110 of Krömer's book.

M. Lazard participated in the Séminaire Cartan in the early fifties, see his éxposé on Algèbres Affines from the eighth seminar (1955–1956).

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Unfortunately, I was unable to find biographical information on Michel Lazard. –  t.b. Jun 1 '12 at 6:32