On this page we shall record those aspects of the theory of Frölicher spaces that are particularly categorical in nature.
Every Frölicher space is functorially the colimit of a diagram of manifolds. In fact, it is a colimit of a diagram in the full subcategory consisting of the single object .
Let be a Frölicher space. Let be the category whose objects are the elements of and the morphisms correspond to the smooth functions with . Note that for a fixed curve and a smooth function then from the definition of a Frölicher space there is a curve such that defines a morphism (take ).
Define a functor by sending each object to and sending each morphism to the corresponding smooth function. We claim that is the colimit of this functor. The morphism is simply (note that so is a morphism in ).
Now suppose that we have suitable morphisms . For each , there is a constant curve at (these are characterised by the fact that if is a morphism in then ). Consider . We shall show that this is a constant curve in . Let and examine . As the are compatible, . But as is constant, so and thus is constant.
Define by . This is a set map, let us show that it lifts to Frölicher spaces. To do this, we look at for a smooth curve . Let and let in . Then . Let be the constant function at . Then and so . Thus . Hence . As is a morphism in the category of Frölicher spaces with source , it is an element of . Hence takes smooth curves in to smooth curves in and so is a morphism of Frölicher spaces.
This also establishes as the colimit since we have the factorisation .