FOM: Toposy-turvey

[Charles Silver]

Solomon Feferman said:

> >[T]he notion of topos is a relatively sophisticated mathematical
> >notion which assumes understanding of the notion of category and that in
> >turn assumes understanding of notions of collection and function.

Colin Mclarty answered:

> 	To me, understanding the notion of category is the best way
> known today of understanding the notions of "collection and function".
> 
> 	To be precise: Pace Feferman's theory, the first thing you
> need to know about collections and functions is that a pair of functions
> may have a composite, and this composition is associative when it is 
> defined. And now I have completely stated the category axioms. This 
> seems to me not very sophisticated mathematics. The topos axioms are
> more sophisticated, but no more so than ZF or Feferman's theory of
> collections and functions.


	I want to raise some elementary concerns about the picture of
category theory under consideration.  First, I'll present a very basic,
perhaps very naive picture of set theory:
	
	Almost all beginning set theory textbooks get off the ground by
stating that the central concept is that of a "collection," an
"assemblage," an "ensemble," etc.  On these accounts, the beginning notion
is one that we are fairly familiar with in real life, via counting and so
on. "Set," as I understand it, is a technical term pertaining to set
theory.  The idea is (or one of them, anyway) that a somewhat familiar,
but imprecise concept ("collection") is being captured or merely defined
by the precise axioms of set theory.  Thus, the ordinary concept of a
collection is turned into the precise one of a set.  To say exactly what
sets are, various axioms are provided.  And, there doesn't seem to be much
disagreement about the cumulative hierarchy. Further on, what properties
constitute a set are in some dispute.  And, there is dispute about whether
the notion of "set" is such as to determine various properties, such as
CH, etc.

	I won't go on with the above, naive picture.  I just wanted to
provide a background for some comments and questions.  According to the
above, "set" is a foundational concept that derives from the ordinary
concept of "collection" that we're familiar with in counting, and so on. 
Fundamental to the notion of a collection is the concept of something
being an element or not of that collection.  As I understand it, this is
all that is needed to start to mold (informal) collections into (formal)
sets.  To me, this is an appealing picture.  There are aspects of the
picture I presented above that I feel a bit uncomfortable with, but I'd
like to skip my doubts about the set-theoretic account I sketched and
ask whether someone partial to category theory can explain the foundations
of category theory at the same elementary level that I explained the
foundations of set theory.

	I have followed Colin Mclarty's and others' very interesting and
informative explanations about toposes, but I still don't grasp some of
the basic ideas.  For example, if category theory takes as basic the
notion of "function," could someone please explain in a very basic way how
the category-theoretic notion of "function" derives from the ordinary
notion of "function," on a par with the way "set" derives from
"collection"?  Let me call the technical notion of function in category
theory `Function_C', and I'll stick with `function' for our ordinary
notion (vague as that notion is).  I can see that it is pretty easy to
give a *formal* definition of Function_C. The formal stuff doesn't bother
me. What I don't see is how Function_C relates to function.  That is, how
does the formal definition arise from the ordinary notion?  In set theory,
I can see that much (but not all) of the formal account of sets arises
from the ordinary account of collection.  From the postings on category
theory so far, I don't have this understanding for category theory.  I
realize that in asking these elementary questions I may be straining the
patience of Colin and others who have so patiently responded to a
multitude of challenges. But, I honestly don't get it.  If someone would
attempt to answer my elementary questions, I would greatly appreciate it.

	Thank you,

Charlie Silver
Smith College