nLab dichotomy between nice objects and nice categories

Idea

If the decision to let every commutative ring define a scheme gives standing to bizarre schemes, allowing it gives a category of schemes with nice properties (Deligne)

In the context of category theory, it is a general phenomenon that

• Nice objects tend to form 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, and additionally the more good properties like exactness properties, being a topos, etc. it has.

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. Fields are nice objects in the context of commutative rings.

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).

• it is often false that the coproduct of fields (in the category of commutative rings) is again a field.

Some nPOV rules of thumb

Without making any sweeping judgments, a general nPOV heuristic is that faced with a choice between working with a non-nice category of nice objects and a larger nice category of non-nice objects, it is usually preferable to switch attention to the nice category. For example, instead of working with the category of smooth manifolds, work instead with an ambient category with better properties, such as a category of diffeological spaces or a model of synthetic differential geometry, in which the category of manifolds is fully embedded. Moerdijk-Reyes can be read largely as an implementation of that philosophy.

On the other hand, another methodological heuristic is to move from a nice object to a nice category attached to it. For instance, the category of modules over a field is about as well behaved as it could possibly be, and similarly one often contemplates the category of vector bundles over a smooth compact manifold.

Formalization

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

References

• Pierre Deligne, Quelques idées maîtresses de l’œuvre de A. Grothendieck, Matériaux pour l'histoire des mathématiques au XX$^e$ siecle (Nice, 1996), Societé Mathématique de France, 1998, pp. 11–19.

• Ieke Moerdijk and Gonzalo Reyes, Models for Smooth Infinitesimal Analysis

• William Lawvere, Taking categories seriously (pdf)