category theory

# Contents

## Idea

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

• “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 $T Mod(E)$ 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.

## Examples

### Groups

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.

## 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)