nLab
abstract general, concrete general and concrete particular

Idea

The category theoretic notions of

• category and object

on the one hand and of

• theory and model

on the other have been suggested (Lawvere) to usefully formalize, respectively, the heuristic notions

• “general” and “particular”

as well as

• “abstract” and “concrete”, respectively.

We have:

• a (syntactic category of a) Lawvere theory $T$ (or the equivalent in any doctrine) $T$ is an abstract general;

• a (generating) object in $T$ is an abstract particular ;

• the category $T\mathrm{Mod}(E)$ of $T$-models/algebras in any context $E$ is a concrete general;

• an object of any $T\mathrm{Mod}(E)$ is a concrete particular.

(In (Lawvere) the item “abstract particular” is apparently not used.)

Examples

Groups

The syntactic category $T_{\mathrm{Grp}}$ of the theory of groups is the “general abstract” of groups. Its essentially unique generating object is the abstract particular group.

The category $T_{\mathrm{Grp}}\mathrm{Mod}(\mathrm{Set})=$ Grp of all groups is the concrete general of groups.

An object in there is some group: a concrete particular.

References

• Bill Lawvere, various lectures