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From: Victoria_Beth_Berdon@cup.portal.com
Newsgroups: sci.math,comp.theory
Subject: Re: Intro Category Theory?
Message-ID: <33013@cup.portal.com>
Date: 20 Aug 90 08:43:31 GMT
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August 19, 1990

Michael --

Thank you for your response, and examples of topoi.  I need some definitional 
clarification.  You said:
>One kind of topos comes from a fixed set T in the following way: the topos is 
>the category of pairs (S,f: S --> T) of sets endowed with a structured map to 
>T.  You could think of this as a category fibred over T.

I need to understand *fibred*.  In Goldblatt ("Topoi: The Categorial Analysis 
of Logic"), p.89-90, he develops the concept as follows: 

(paraphrasing)  Start with a collection of pairwise disjoint sets, indexed by 
integers, i in I, such that the sets are A-sub-i (Sorry, no fancy character 
set).  Let A be the union of all A-sub-i's.  Then there's a map, p:A --> I.  If
 
x is in A, then there's exactly one A-sub-i s.t. x is an element of A-sub-i.  
Here Goldblatt says, "We put p(x)=i."  (I'm not sure what he means -- we write 
it so?) "Thus all the members of A-sub-i get mapped to i", etc.  And we can 
"re-capture A-sub-i as the inverse image under p of {i}.  The set A-sub-i is 
called the stalk, or the fibre over i.  The members of A-sub-i are the germs 
at i.  ANd the whole structure is called a bundle of sets over the base space. 

Is this your understanding of a fibre?


You asked for a definition of an elementary topos.  Let me quote from a couple 
of sources:

Goldblatt, p. 84:
Definition.  An elementary topos is a category E s.t.
        (1) E is finitely complete,
        (2) E is finitely co-complete,
        (3) E has exponentiation,
        (4) E has a subobject classifier.
Since (1) and (3) constitute "Cartesian closed", so (1) can be replaced by 
      (1') E has a terminal object and pull-backs,
and (2) can be replaced by,
      (2') E has an initial object 0, and pushouts.  
Goldblatt says this is from the definition by Lawvere and Tierney "in terms of 
which they started topos theory in 1969.

Hatcher (*The Logical Foundations of MAthematics*, Pergamon Press, 1982) 
p.279) I add:
            
The category in question is a category of sets, CS.  Let M be any model for the
 
category of sets which is complete.  "By 'complete' we mean that, in addition 
to satisfying the axioms of CS, M has the further property that sums and 
products exist in M on any indexing set..." -- I guess I think of this as 
closure under + and * (is this ok to do?)

(I suppose initial/terminal objects, pull-back, pushout need definition.  
Later?)

From J.L. Bell (*Category Theory and the Foundations of MAthematics*, 
Brit.J.Phil.Sci. 32 (1981)), p.357:
"A topos is a category E  which has the following features in common with the 
category Set of sets.

        (1) E has a terminal object and all finite products.

        (2) There is in E a "truth-value" object sigma which plays the same 
        role in E as the truth-value set 2={0,1} plays in Set, i.e., for each
        object X there is a natural correspondence between subobjects of X and 
        arrows from X to sigma ('characteristic functions' on X)

        (3) For each object X of E there is a 'power object' PX in E which 
       plays the formal role of a power set of X in E.

These conditions can all be formulated in the first-order language of category 
theory: hence the use of the term 'elementary'."

(I guess this definition itself opens up a a need for a whole list of other 
underlying category theoretic definitions.  At least for me. )

                                                           
From J.L. Bell (*From Absolute to Local Mathematics*, Synthese 69 (1986)) 
another wording:

"To arrive at the concept of a topos, we start with the familiar  category S 
of sets whose objects are all sets (in a given model M of set theory) and 
whose arrows are all mappings (in M) between sets in M.  ...S has the 
following properties. 

(i)  There is a 'terminal object' 1 such that, for any object X, there is a 
unique arrow X --> 1 (for 1 we may take any one-element set, in particular 
{0}).

(ii)  Any pair of objects A,B has a Cartesian product AxB.

(iii) For any pair of objects A,B one can form the 'exponential' object B^A of 
all mappings A --> B.

(iv)  There is an 'object of truth values' sigma such that for each object X 
there is a natural correspondence between subobjects (subsets) of X and arrows 
X --> sigma.  (For sigma we may take the set {0,1}; arrows X --> sigma are 
then the characteristic functions on X, and the exponential object sigma^X 
corresponds to the power set of X.)

All four of the above conditions can be formulated in purely category-
theoretic (arrows only) language: a (small) category satisfying them is called 
a topos."

As Bell puts it, a topos is a more general model of set theory.  p.415: "A 
theory T may be regarded as a generalized set theory and a topos which is a 
model of T as a local universe of discourse within which the mathematical 
assertions made by T are true and the constructions sanctioned by T can be 
carried out."

I realize that in my attempt to provide a definition, I  have introduced more 
questions than I can handle at this time.  But this is a start, no?

I'm embarrassed, but I must ask, what is the reference to Avernus?  
(Humility seems to be the word of the day.  No, make that month.)

-- Victoria

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