- General remark
This is the common abstract for a series of
three lectures to be given at the Montreal Categories Seminar during
February 2001. They are based on a recent paper of Marta Bunge and
Stephen Lack, ``A Van Kampen theorem for toposes'', written during my
sabbatical stay in Sydney and available for downloading in
http://www.math.mcgill.ca/~bunge (as well as in Steve's homepage).
The three lectures are entitled: (1) Extensive 2-categories and Top,
(2) A Van Kampen theorem for toposes and applications, and (3) Locally
constant objects in a topos, but the separation between the three themes
will not necessarily be as clear cut as the titles indicate.
- Rationale.
The Van Kampen theorems are useful
tools for calculating fundamental groups, for instance, knot groups. It
was observed by Fox (Lefschetz Festschrift '57) that the coverings version of
the VK theorems, combined with his own theory of complete spreads in
general, and of branched coverings in particular, allowed for a topological
(rather than just combinatorial) interpretation the known knot invariants of
Seifert, Alexander and others. The notion of a Grothendieck topos is a far reaching generalization of the
notion of a topological space and (therefore) it is of interest to develop
some algebraic topology in this context. What has been done so far is part
of program with many ramifications.
- Background.
Bunge (CT'91) gave a construction of the (coverings) fundamental group
P 1cov(E) of a connnected and locally connected
unpointed S-topos E and proved all the correct properties for it.
In Bunge-Moerdijk (JPAA'97), the above was shown to be equivalent (in the
locally paths simply connected case, as in topology) with the paths version
given by Moerdijk-Wraith (Trans. AMS'86). Another line of research was initiated in Bunge-Funk (JPAA'96) , where
we defined complete spreads over an S-topos E, and reproduced in
this context several results in Fox '57 (op. cit.). An additional interest of this
notion is its surprising connection with the S-valued distributions on E
in the sense of Lawvere (Aarhus Workshop'83). The more special notion of a
branched covering of E was analyzed independently in Bunge-Niefield
(JPAA'00) and in Funk(TAC'00).
- Motivation.
A motivation for the Bunge-Lack work
was specifically to provide a coverings version of the Van Kampen theorems
in the context of Top S, in order to apply it to knot groups
and branched coverings in topos theory and by means of the coverings
fundamental group of E. In view of an existing (coverings version of a Van Kampen theorem of Brown
and Janelidze (J.Alg'90) in the context of extensive categories, discussed
extensively (!) in Carboni-Lack-Walters (JPAA'93), we decided to extend
(and improve) their result to suit our purposes.
- Results
We denote by Top S
the 2-category of toposes bounded over a given base topos S.
We begin by stating a notion of extensive 2-category and by proving that
TopS is 2-extensive. We then state and prove a suitable
version of the Van Kampen theorem in the context of extensive 2-categories.
This is then applied to the notion of covering morphism that arises from
considering locally constant objects in an S-topos E. We observe
that this agrees with the notion employed in Bunge'91 (op.cit.) in the
locally connected case, but not with that of Barr-Diaconescu
(Cahiers TGDC '91), although the two are equivalent in either the case of a
connected cover (e.g., a torsor) or else if S is assumed to be
Set. For a fixed (complement i: E/Y >--->E of a ``knot'' in E, we introduce a
2-category BCY(E) of branched coverings over E with
non-singular part i: E/Y >--->E and prove that (at least in the locally simply
connected case) it is (bi)equivalent to (the connected atomic topos)
P1cov(E/Y), as we expected.