The concept of smallness in homotopy theory generalizes the concept of compactness from classical topology. However, there are two possible generalizations of this notion: one is used in the model category theory, whether the other one is used in the realm of triangulated categories. The relation between these two concepts remained subtle for a long time. Mark Hovey has shown in his book on Model categories that the smallness in a stable finitely generated category implies smallness in its homotopy category. Recently Rosicky generalized this result to combinatorial model categories.
In this talk we will exhibit an example of a model category Quillen equivalent to the category of spaces with the following property: every homotopy type has a countably small representative. In particular, smallness in this model category does not translate into smallness in the homotopy category. Our example stems out of the work on enriched Brown representability. Connections with homotopy calculus and orthogonal calculus will also be discussed.