category of sets equipped with a symmetric binary relation

Unknown properties

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Special objects

• terminal object: $(\{\ast\},\{(\ast,\ast)\})$, which can be seen as undirected loop-graph with a single vertex and a loop
• initial object: $(\varnothing,\varnothing)$, which can be seen as the undirected loop-graph with no vertices and no directed edges
• products: The product of a family $(X_i,R_i)_{i \in I}$ is $(\prod_{i \in I} X_i, \prod_{i \in I} R_i)$, where $\prod_{i \in I} R_i \subseteq \prod_{i \in I} (X_i \times X_i) \cong \prod_i X_i \times \prod_i X_i$.
• coproducts: The coproduct of a family $(X_i,R_i)_{i \in I}$ is $(\coprod_{i \in I} X_i, \coprod_{i \in I} R_i)$, where $\coprod_{i \in I} R_i \subseteq \coprod_{i \in I} (X_i \times X_i) \subseteq \coprod_{i \in I} X_i \times \coprod_{i \in I} X_i$.

Special morphisms

• isomorphisms: bijective maps that preserve and reflect the relation
• monomorphisms: injective relation-preserving maps
• epimorphisms: surjective relation-preserving maps
• regular monomorphisms: injective maps that reflect and preserve the relation
• regular epimorphisms: morphisms $f : (X,R) \to (Y,S)$ such that $f : X \to Y$ is surjective and $S = \{(f(x),f(x')) : (x,x') \in R\}$

Indistinguishable categories

These categories in the database currently have exactly the same properties as the category of sets equipped with a symmetric binary relation. This indicates that the data may be incomplete or that a distinguishing property may be missing from the database.

• category of sets equipped with a binary relation

Comments

• The inclusion functor $\Bin_{\symm} \to \Bin$ has a left adjoint mapping $(X,R)$ to $(X,R \cup R^{\op})$.
• The inclusion functor $\Bin_{\symm} \to \Bin$ has a right adjoint mapping $(X,R)$ to $(X,R \cap R^{\op})$.
• Although the categories $\Bin_{\symm}$ and $\Bin$, or equivalently $\Graph_L$ and $\DiGraph$, agree on all properties currently recorded in the database, they are not equivalent categories. In both categories, the one-vertex graph $K_1$ without edges can be characterized up to isomorphism as the unique connected object that admits a morphism to every non-initial object. We can then count the connected objects up to isomorphism that admit exactly two morphisms from $K_1$ (i.e. have two vertices) and no morphism from the terminal object (i.e. have no loops). In $\Graph_L$, there is only one such graph, $\bullet \longleftrightarrow \bullet$, whereas in $\DiGraph$ there are two: $\bullet \longrightarrow \bullet$ and $\bullet \longleftrightarrow \bullet$.