Contractibility as uniqueness

What does it mean for something to exist uniquely? Classically, to say that a set A has a unique element means that there is an element x of A and any other element y of A equals x. When this assertion is applied to a space A, instead of a mere set, and interpreted in a continuous fashion, it encodes the statement that the space is contractible, i.e., that A is continuously deformable to a point. This talk will explore this notion of contractibility as uniqueness and its role in generalizing from ordinary categories to infinite-dimensional categories.