[[!redirects directed loop graph object]] +-- {: .rightHandSide} +-- {: .toc .clickDown tabindex="0"} ### Context #### Category Theory +-- {: .hide} [[!include category theory - contents]] =-- #### Graph theory +-- {: .hide} [[!include graph theory - contents]] =-- #### Relations +-- {: .hide} [[!include relations - contents]] =-- =-- =-- # Contents * table of contents {: toc} ## Idea A **loop graph object** [[internalization|internal to]] a [[category]] with [[finite products]] is an [[object]] that behaves in that category like loop [[graphs]] do in [[Set]]. ## Definition ### In a category with finite products A **loop graph object** in a category $\mathcal{C}$ with [[finite products]] is a [[loop digraph object]] $(V, E, R:E \to V \times V)$ with a morphism $i:E \to E$ such that * $p_1 \circ R = p_2 \circ R \circ sym$ * $p_2 \circ R = p_1 \circ R \circ sym$ * $i \circ i = id_E$ where $p_1, p_2:V \times V \to V$ are the unique projection morphisms of the binary product. ### In a general category A **loop graph object** in a category $\mathcal{C}$ is a [[loop digraph object]] $(V, E, s:E \to V, t:E \to V)$ with a morphism $i:E \to E$ such that * $s \circ R = t \circ sym$ * $t \circ R = s \circ sym$ * $i \circ i = id_E$ ## Properties A loop graph object is equivalently an object with an [[internal relation|internal symmetric binary endorelation]]. ## Related concepts * [[graph]] * [[symmetric relation]] * [[loop digraph object]] * [[internal relation]] [[!redirects loop graph objects]]