## Motivation

There is presumably no single and widely accepted formal definition of **structured sets = sets plus structure** based on sets as primitive objects, but several approaches are around. See e.g. structures (model theory), echelons (Bourbaki), frames (Moschovakis). I don't want to discuss these approaches, but for simplicity's and specifity's sake I want to pick out one especially simple definition of rather generic structured sets —

**graphs**(which among other things are able to interpret

*any*structured set).

When we start with the „graph“ of **sets** $U = \langle V,\in\rangle$, sets as primitive objects are „defined“ just by

$$\text{Set}(X) :\equiv X\ \epsilon\ V$$

with $\epsilon$ indicating class membership. Graphs (structured sets), then, are - rather sophisticatedly - defined by

$$\text{Graph}(X) :\equiv (\exists S,R\ \epsilon\ V)\ R \subseteq S^2 \wedge X = \langle S,R\rangle$$

To complete the picture we define

$$\text{Relation}(X) :\equiv (\exists S\ \epsilon\ V)\ X \subseteq S^2$$

Thus, a graph is a set *plus* a relation over it, usually written as $G = \langle V,E\rangle$.

## Interlude

Oppose this set-based picture to the category $\mathsf{Graph} = \langle \mathcal{O},\mathcal{M},\dots\rangle$ with $\mathcal{O}$ the class of all graphs as primitive objects, related by graph homomorphisms $\mathcal{M}$, etc. Thus graphs as primitive objects are defined by

$$\text{Graph}(X) :\equiv X\ \epsilon\ \mathcal{O}$$

To define „set“ as a now *derived* concept one might try to resolve the above „equation“ informally to obtain **sets = structured sets minus structure**. Thus, sets don't have any structure (anymore), so morphisms as structure-

*preserving*functions don't have to preserve any structure (anymore), so

*every*function from a set to any other graph (= structured set) is a morphism in $\mathcal{M}$. According to the standard definition of graph homomorphism, the graphs being sets are exactly the edgeless graphs (which complies with intuition).

Translating this into categorical terms we obtain:

$$\text{Set}(X) :\equiv (\forall\ Y\ \epsilon\ \mathcal{O})(\exists\ f\ \epsilon\ \mathcal{M})\ f: X \rightarrow Y$$

Making use of categorical terminology we can equivalently say: Let $\mathcal{C}/$ be the quotient category of $\mathcal{C}$ which identifies all morphisms (if present) from $A$ to $B$ as one. Then:

$$\text{Set}(X) :\equiv X\ \text{is an initial object in } \mathsf{Graph}/$$

## Question(s)

I am aware that I didn't even mention the category $\mathsf{Set}$ of sets and the notion of a concrete category (even though $\mathsf{Graph}$ *is* a concrete category).

What I'd like to learn is - among other things - how the above definition of being a *set* relates to the usual notions:

Which concrete categories don't have sets?

Which non-concretizable categories

*do*have sets?

maps_amongthings are operationally more significant than internal structure, so, in that sense, I am not convinced that I care whether something is a set or not... nevertheless, and interesting line of thought. $\endgroup$5more comments