  Let M and N be smooth manifolds without boundary. Immersion theory suggests
that an understanding of the space of smooth embeddings emb(M,N) should come
from an analysis of the cofunctor V |--> emb(V,N) from the poset O of open
subsets of M to spaces. We therefore abstract some of the properties of this
cofunctor, and develop a suitable calculus of such cofunctors, Goodwillie
style, with Taylor series and so on. The terms of the Taylor series for the
cofunctor V |--> emb(V,N) are explicitly determined. In a sequel to this paper,
we introduce the concept of an analytic cofunctor from O to spaces, and show
that the Taylor series of an analytic cofunctor F converges to F. Deep excision
theorems due to Goodwillie and Goodwillie-Klein imply that the cofunctor V |-->
emb(V,N) is analytic when dim(N)-dim(M) > 2.
