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 or abstract universal (_abstraktes Allgemeines_)

• the category $T Mod(E)$ of $T$-models/algebras in any context $E$ is a concrete general or concrete universal

• 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

See also an email comment recorded here.

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

The terminology is inspired by

for instance

EL§61 If we are to believe the Critical philosophy, thought is subjective, and its ultimate and invincible mode is abstract universality or formal identity. Thought is thus set in opposition to Truth, which is no abstraction, but concrete universality. In this highest mode of thought, which is entitled Reason, the Categories are left out of account. The extreme theory on the opposite side holds thought to be an act of the particular only, and on that ground declares it incapable of apprehending the Truth. This is the Intuitional theory.

§71 It is only after profounder acquaintance with the other sciences that logic ceases to be for subjective spirit a merely abstract universal and reveals itself as the universal which embraces within itself the wealth of the particular

§1323 This universal Notion, which we have now to consider here, contains the three moments: universality, particularity and individuality.

§1337b When people talk of the determinate Notion, what is usually meant is merely such an abstract universal.

§1599 Such a universal which merely subsumes, is an abstraction which only becomes concrete in something else, in the particular. End, on the contrary, is the concrete universal, which possesses in its own self the moment of particularity

PS§456b This common element is either any one particular side of the object raised to the form of universality, such as, for example, in the rose, the red colour; or the concrete universal, the genus, for example, in the rose, the plant;

See also in and around the section The genus

Survey of thes Hegelian ideas includes

• John Grier Hibben, Eric v.d. Luft, Hegel’s Shorter Logic: An Introduction and Commentary

where on p. 143 it says about the Shorter Logic:

Particularity and individuality are related as “abstract” and “concrete”, respectively. The particular is the “abstract individual”. The individual is the “concrete particular”. The universal is their union, and may be either “abstract” or “concrete”. The so-called “concrete universal” is Hegel’s gold standard for conceptual thought $[$$]$.

as well as

• Robert Stern, Hegel, british idealism, and the curious case of the concrete universal, British Journal for the History of Philosophy 15(1) 2007: 115 – 153 (pdf)