For posting (in reduced form)

For posting (in reduced form)

[Marta Bunge]

Dear fellow category theorists,

Our book "Synthetic Differential Topology" (SDT) is set to appear in April
2018, published by Cambridge University Press. The SDT book can be used for
a one-semester course or seminar whose only prerequisite is some basic
category theory. Those interested in it for this purpose can request CUP
for a pre-publication. The details of it are below.

http://www.cambridge.org/ca/academic/subjects/mathematics/
logic-categories-and-sets/synthetic-differential-topology?format=PB#
b7i2wy1rx8lmYJBP.97



A brief description of the contents of the SDT book follows. In a first
part all notions of topos theory and SDG ("Synthetic Differential
Geometry") that are needed in the sequel are discussed, with emphasis on
the logico-geometric notions. The second part deals with the
Ambrose-Palais-Singer theorem on connections and sprays as well as with the
calculus of variations, both of them as illustrations of the uses of SDG
for classical differential geometry. In a third part, following a
discussion of some basic topological structures (intrinsic, euclidean,
weak), the basic axioms and postulates of SDT are introduced. Whereas in a
model (E, R) of SDG, it is the jets of R-valued mappings on finite powers
of R that are represented by (algebraic) tiny objects, if (E, R) is in
addition a model of SDT, then it is the germs of R-valued mappings on
finite powers of R that are represented by (logical) tiny objects. The
fourth part consists of an SDT version of the stability theory of smooth
mappings including Mather's theorem (with and without the preparation
theorem) and Morse theory. The construction of the Dubuc topos G of
germ-determined ideals of smooth mappings is then recalled in the fifth
part. In the sixth part it is shown that G is not only a well adapted model
of SDG but also one such of SDT. It follows from this that several
classical theorems of differential topology can be recovered in greater
generality and in a conceptually simpler manner than their classical c
ounterparts. There is a comprehensive list of references (both classical
and categorical) and an index.


With best regards,

Marta Bunge


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