The theory of subanalytic sets is an excellent tool in various 

analytic-geometric contexts. We axiomatize the notion of "behaving like 

the category of subanalytic sets" by introducing the notion of 

analytic-geometric category. The objects of such a category share many of 

the hereditary and geometric finiteness properties of subanalytic sets. 

Proofs of the more difficult results of this nature, like the 

Whitney-stratifiability of sets and maps in such a category, often involve 

the use of charts to reduce to the case of subsets of R^n. For subsets of 

R^n, the theory of o-minimal structures on the real field, an abstraction 

of the theory of semialgebraic sets (and the subject of Math 949 in Sp08) 

provides an elegant and efficient setting in which to work.



Certain fairly natural sets---like {(x,x^r):x>0} for positive irrational 

r, and {(x,e^{-1/x}): x>0}---are not subanalytic (at the origin) in R^2. 

Because there are o-minimal structures on the real field which include 

these sets, we now have analytic-geometric categories which include these 

sets among its objects.