The category theoretic notions of
on the one hand and of
on the other have been suggested (Lawvere) to usefully formalize, respectively, the heuristic notions
as well as
We have:
a (syntactic category of a) Lawvere theory $T$ (or the equivalent in any doctrine) $T$ is an abstract general;
the category $T Mod(E)$ of $T$-models/algebras in any context $E$ is a concrete general;
an object of any $T Mod(E)$ 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 $T Mod(E)$ is a concrete particular.
The syntactic category $T_{Grp}$ of the theory of groups is the “general abstract” of groups. Its essentially unique generating object is the abstract particular group.
The category $T_{Grp} Mod(Set) =$ Grp of all groups is the concrete general of groups.
An object in there is some group: a concrete particular.
The category-theoretic formalization of these notions as proposed by Bill Lawvere is disussed in print for instance in
See also an email comment recorded here.
For discussion of “particular” and related in philosophy see also