Categorical Garside structures and simple arrows

Let $\mathcal{C}$ be a small category, meaning that its objects and arrows form sets. An automorphism of $\mathcal{C}$ means a functor $\varphi :\mathcal{C}\to \mathcal{C}$ admitting a functor $\psi :\mathcal{C}\to \mathcal{C}$ with $\psi \circ \varphi ={\mathrm{id}}_{\mathcal{C}}=\varphi \circ \psi$ as equalities on objects and arrows. Let $\varphi$ be such an automorphism, and let $\mathrm{\Delta }$ be a natural transformation from the identity functor to $\varphi$. Thus its component at $x$ is an arrow ${\mathrm{\Delta }}_x:x\to \varphi (x)$, and for every $f:x\to y$ one has $f{\mathrm{\Delta }}_y={\mathrm{\Delta }}_x\varphi (f)$ in traversal order. An arrow $f:x\to y$ is simple for this structure if there is an arrow $g:y\to \varphi (x)$ with $fg={\mathrm{\Delta }}_x$.

An atomic categorical Garside structure is such a triple $(\mathcal{C},\varphi ,\mathrm{\Delta })$ in which $\mathcal{C}$ is atomic and cancellative, every atom is simple, and, for every object $x$, both the outgoing left-divisibility order and the incoming right-divisibility order are lattices. A Garside category means a category equipped with such a structure. In the one-object case its arrows form a Garside monoid in this categorical convention. The structure has finite type if its set of simple arrows is finite.

Divisibility and atomicity in a category

Category, object, morphism, domain, codomain, identity, composition, and hom-collection

Class-sized category theory in ZFC: definable-class schemas, small and locally small categories, and why CAT is not formed

Covariant functor, identity functor, composite functor, and contravariant functor

Natural transformation and its components

Lattices, distributive lattices, and order ideals

Semigroup and monoid

The naturality equation has matching source and target on both sides. The condition $fg={\mathrm{\Delta }}_x$ determines the domain and codomain required of $g$, so simplicity is an unambiguous property of an arrow relative to the specified triple. Atomicity and cancellation make the two divisibility relations partial orders, as established in the divisibility definition; the lattice condition is consequently meaningful. Smallness ensures that the arrow collections used for the divisibility orders and the one-object monoid are sets. In a one-object category, composition and its identity give a monoid. The cited definition records the atomic variant adopted here; finite type is an additional condition, not an implicit assumption.