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It is easy to see (and presumably well-known) that the operation of attaching manifolds along a submanifold (as defined http://mathoverflow.net/questions/87857/characteristic-classes-of-a-fibered-sum) can be expressed as a push-out in the Smooth category, so long as the instruction 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)

2 Made it more clear what I am looking for

It is easy to see (and presumably well-known) that the operation of attaching manifolds along a submanifold (as defined http://mathoverflow.net/questions/87857/characteristic-classes-of-a-fibered-sum) can be expressed as a push-out in the Smooth category, so long as the instruction 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)

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# 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 http://mathoverflow.net/questions/87857/characteristic-classes-of-a-fibered-sum) can be expressed as a push-out in the Smooth category. 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.