nLab
dichotomy between nice objects and nice categories

Idea

It is a general phenomenon that

• Nice objects tend to live in non-nice categories.

• Nice categories tend to contain non-nice objects.

Here we take a category to be the “nicer” the more limits, colimits etc. it admits.

On the other hand, a “nice object” is, loosely speaking, an object in some context which has more special properties than the generic object in that context will have. For instance manifolds are nice objects in the context of generalized smooth spaces.

Clearly, the more extra properties one imposes, the less likely it is that these are preserved under limits and colimits. For instance

• not all quotients $X/G$ of a manifold $X$ by the action of a group $G$ are again manifolds (this is a colimit which fails to exist);

• not all fiber products $Y{\times }_{X}Y$ of surjections $Y\to X$ of manifolds are again manifolds (this is a limit that fails to exist).

Formalization

The notion of “nice objects” can be formalized to some degree for instance in terms of Isbell self-duality as described in

• W. Lawvere, Taking categories seriously (pdf)