## Smooth Maps and the Category of Smooth Manifolds (Pt. II)

**Point of Post: **This is a continuation of this post.

Of course, most of the standard results about smooth functions in the Euclidean case transfer over to this general case. In particular:

**Theorem: ***Let be smooth manifolds and maps. Let , if is smooth at , and if is smooth at , then is smooth at .*

**Proof: **Let , and be charts at respectively with , . We know that is smooth, being the composition of smooth maps in Euclidean spaces. But, we see that this is equal to and thus it’s easy to see that the charts and is a good chart at and which has smooth coordinate representation.

**Theorem: ***Let be a smooth map between manifolds, then is continuous.*

**Proof: **Let be arbitrary, then we know that there is a neighborhood at and at with such that is smooth. That said, since smooth maps between Euclidean spaces are continuous we see that is continuous. Since and differ only by homeomorphisms, we see that is continuous on . Since was arbitrary the conclusion follows.

The last theorem says that (just like continuous functions [note that this was used in the last proof!]) smoothness is a local property. In particular:

**Theorem: ***Let be an open cover for the smooth manifold . Suppose that for each we have a smooth map (thinking about as an open submanifold of ) where is some smooth manifold. If and agree on , then there exists a unique smooth map such that .*

As a last note, we discuss the notion of local diffeomorphisms. In particular, if and are smooth manifolds and if is a map, we call it a *local diffeomorphism *if for every point there exists neighborhoods of and of so that is a diffeomorphism of onto .

**The Category of Smooth Manifolds and the Algebra of Smooth Functions**

Now that we have defined the appropriate notion of smooth maps between smooth manifolds we can discuss the category of smooth (finite dimensional) manifolds. In particular, we define to be the category whose objects are smooth (finite dimensional) manifolds and whose arrows are smooth maps between these manifolds. In fact though, it is actually more convenient for us to, at this current point, consider the category of topological manifolds, thought of as a full subcategory of . We shall make the convention that shall be denoted and shall be denoted .

It is easy to get rid of some of the simple properties of the category . Indeed, we claim that the product of topological manifolds with the usual projections is actually the categorical product in . Indeed, this is obvious since the product in stays within in . Now, we similarly claim that the product of smooth manifolds with the usual projections is the categorical product in . Indeed, let be three smooth manifolds and suppose that we have smooth maps , we need to produce a unique smooth map such that (where is the usual projection). It’s clear that such a smooth map is necessarily unique, since there is a unique *continuous *map which satisfies this property (via the fact that this definition of product is a categorical product in which is a supercategory of ). Thus, it suffices to show that there exists a smooth map which satisfies these properties. Unsurprisingly, we define . We know this map is continuous, and thus it suffices to show it’s smooth. To do this, let for any point , and (in , and and at and respectively) be charts such that And is smooth. We then see that and is a chart at and respectively such that . Moreover, we see that

which is clearly a smooth map between Euclidean spaces. The conclusion follows.

Sadly enough, the category does not, in general, have colimits. This is somewhat of a touchy subject, and so I instead refer you to this mathoverflow thread which gives a good explanation as for why this is true.

**References:**

[1] Lee, John M. *Introduction to Smooth Manifolds*. New York: Springer, 2003. Print.

[2] Lee, John M. *Introduction to Topological Manifolds*. New York: Springer, 2000. Print.

[3] Milnor, John Willard, and David W. . Weaver. *Topology from the Differentiable Viewpoint*. Charlottesville (Va.): University of Virginia, 1969. Print.

[4] Bredon, Glen E. *Topology and Geometry*. New York: Springer-Verlag, 1993. Print.

[…] Smooth Maps and the Category of Smooth Manifolds (Pt. III) Point of Post: This is a continuation of this post. […]

Pingback by Smooth Maps and the Category of Smooth Manifolds (Pt. III) « Abstract Nonsense | September 3, 2012 |