Note on The 60th birthday of toposes, April 21.

Note on The 60th birthday of toposes, April 21.

[Colin McLarty]

Let me try out some new research on the specific origin of the idea of
(Grothendieck) Topos.

At the start of 1958 Grothendieck believed the correct Weil cohomology of
a scheme S would be the derived functor cohomology of some category Ab(S)
of sheaves of Abelian groups on S---and Ab(S) was likely to be the category
of all Abelian group objects in some category of sheaves of sets on S.  He
had no name yet for such conjectural categories of sheaves of sets but for
later reference I will call this category T(S).  It would require some new
notion of sheaves of sets more general than the existing notion using
topological spaces.  He had no concrete idea what this new notion of sheaf
might be.  His experience in Kansas in 1955 suggested it  was likely to
come from some new notion of espace etale over a scheme. and it should have
some good exactness properties.

On April 21, 1958 Grothendieck went to the Seminaire Chevalley to hear
Serre describe isotrivial fiber bundles on a variety V, which are bundles
that become trivial when pulled back to some surjective family of finite,
unramified maps to V.  Serre showed this gave the expected one-dimensional
Weil cohomology groups of V.  Grothendieck immediately told Serre it would
work in all dimensions--which Serre found ``very optimistic'' at the time.

All of this is well documented in familiar sources.

Grothendieck's 1973 topos lectures in Buffalo show that during Serre's talk
Grothendieck saw the Weil espaces etales over a scheme S should be patched
together from Serre's unramified maps to S.  That is, the sought category
T(S) of sheaves of sets on S should be generated by the unramified maps to
S, and should have arbitrary colimits of these, and finite limits.  He
first sought to axiomatize this roughly the way his Tohoku paper had
axiomatized sheaves of groups.

A few changes came soon: He realized finite unramified maps should be
replaced by flat unramified maps (just as topological espaces etales are
trivial locally in the fiber and not on the base).  He abandoned the
axiomatic approach to T(S) as too vague and shifted to construction by
sites.  And for sites he began treating sheaves as a kind of functor,
rather than as espaces etales.  Only after the notion ot site developed
would Giraud give his topos axioms, and no one has yet really taken a
precise notion of espace etale much  beyond the topological case.

But April 21 1958 was the birth of topos theory.  The term topos came
later.  (Lots of people are not named on the day they are born.)

As of the summer of 1973 Grothendieck's stated preferred definition of
topos was still: a category wth arbitrary colimits, finite limits, and a
small generating set.  He says over and over this is not quite adequate for
proofs.  He says proofs require the notion of site, or else the Giraud
axioms, but he calls the vaguer idea more intuitive and says that is the
way to think about a topos.

Also in 1973 Grothendieck says the objects in any topos should be seen as
espaces etales over the terminal object of the topos, in a generalized
sense that includes saying any orbit of a group action lies ``etale'' over
a fixed point.  Today, it is not obvious that this can work well in
general.  In SGA 4 Grothendieck had already given evidence that it would
work for petit topos but not gros topos.  But still in 1973 he did say it.
And it works perfectly for petit etale toposes---as long as we generalize
the notion  of espace etale to include a Galois orbit (in an extension
field) lying over a single point (in the ground field).


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