Treating the Connected Sum (and other constructions) as a Push-out  It is easy to see (and presumably well-known) that the operation of attaching manifolds along a submanifold (as defined Characteristic Classes of a Fibered Sum) can be expressed as a push-out in the Smooth category, so long as the construction includes collars.  The connected sum follows as a special case where the submanifold is a point.  Furthermore, Thom's "shperical modifications" can also be treated as a connected sum.  Basic facts about these operations (well-defined under isotopy of the embeddings, associative, etc.) can be proven as corollaries of theorems about push-outs.
I've done some work on my own developing this idea, but I'm wondering if there's some theory already in place that could help me save some time or point me in better directions.  I haven't come across anything in my searches.  What I'm looking for specifically are references which treat these operations as pushouts, and develop them as such.  (What I am NOT looking for are modifications of the smooth category which make it closed under pushouts)
 A: I don't think it is true that this kind of construction is a pushout in the smooth category.  Consider the function $f:\mathbb{R}\to\mathbb{R}$ given by $f(x)=|x|$.  This is smooth on $(-\infty,0]$ and on $[0,\infty)$.  (There are some subtleties about how to define smoothness for functions on manifolds with boundary, but all possible variants are easily seen to be satisfied in this case.) If $\mathbb{R}$ were the pushout of $(-\infty,0]$ and $[0,\infty)$ along $\{0\}$ then $f$ would be smooth on $\mathbb{R}$, which is false.
A: In Chapter 3 of Chris Schommer-Pries's PhD thesis, surfaces with corners (more generally, manifolds with faces) are equipped with extra structure called a "halation", with precisely the goal of making the gluing operation a pushout. The topic of making the gluing operation into a pushout in the smooth category is discussed (and motivated) in some detail, quite lucidly.
Added: Alternatively, you could work in the microlinear category, where the problems we are discussing do not arise, and the naive gluing operation is a pushout. See e.g Basic Concepts of Synthetic Differential Geometry by Rene Lavendhomme.
Added 2: References for the "just add collars" approach are:

*

*T. Kerler and V. Lyubashenko, Non-Semisimple Topological Quantum Field Theories for 3-Manifolds with Corners, Lecture Notes in Math. 1765, Springer-Verlag, Berlin (2001).

*J. Morton, Extended TQFT's and Quantum Gravity, Ph.D Thesis (University of California, Riverside).


A: In the link the OP provided, the definition includes a collar, so the example given by Neil is not really a counter-example, since it does not satisfies the hypothesis. (This is more a comment than an answer, but I'm not allowed to comment yet, sorry.)
