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;

• 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 particular.

That seems to be roughly what is suggested in Lawvere. Of course one could play with this further and consider further refinement such as

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

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

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

The category-theoretic formalization of these notions as proposed by Bill Lawvere is disussed in print for instance in

• Bill Lawvere, Categorical refinement of a Hegelian principle, section 1 of Bill Lawvere, Tools for the advancement of objective Logic: Closed categories and toposes, in John Macnamara, Gonzalo Reyes, the logical foundations of cognition, Oxford University Press (1994)

See also an email comment recorded here.

For discussion of “particular” and related in philosophy see also

• Wikipedia, Particular, Abstract particular