Re: terminology in definitions of limits

[Paul Taylor <pt09@PaulTaylor.EU>]

Peter E observed that
> each definition of a limit which I've seen contains something
> I would describe as a "probe object" or "test object"
although I am not sure whether his question is about the name for
this (for which either of his suggestions is reasonable), or what.

Limits are, of course, examples of right adjoints, and the situation
that Peter describes is a case of the adjoint correspondence

   (considered as a trivial diagram) test object  ----->  diagram
   ==============================================================
                            test object ------>  limit of diagram

So the left adjoint is a "forgetful" functor,  which takes the test
object and considers it as a trivial diagram,  ie with identities
as edges.

Giving the test object a "name" in the sense of an English word
is not such a big deal.

However, I would argue that it is important to give it a "name"
in the sense of using a particular letter uniformly for it.

For this purpose, I propose the Greek letter capital Gamma.

The reason for this choice is that the same role is played in
symbolic logic by the "context",  ie the collection of parameters,
along with their types and hypotheses,  that occurs in any
mathematical statement.   In type theory, the letter Gamma is
traditionally and uniformly used for this purpose.

(Can some type or proof theorist tell me who introduced or
established this convention?)

Indeed, I use this convention both for this test object and for
other parts of the anatomy of an adjunction systematically throughout
my book, "Practical Foundations of Mathematics"  (CUP, 1999).

In so far as there was a previous convention in category theory for
the name of this object, it was "U".  This came from sheaf theory,
where, by the Yoneda lemma, we need only consider maps from
hom(-,U), where U belongs to the base category.  This category was
primordially the lattice of open subsets of a topological space,
so the convention came from that of using "U" for an open set.
I believe that German-speaking authors were responsible for this,
though I don't know what German word it was that began with U.

Speaking of sheaf theory,  when and to whom was it first apparent
that the category of sheaves depends only on the lattice of open
sets, and not on the points of a topological space?

Paul Taylor
www.PaulTaylor.EU
pt09 @ PaulTaylor.EU