Endotrivial categories

A thin category is a category $C$ such that for each object $a,b$ we have that $|Hom(a,b)| \leq 1$. We can generalize this condition by relaxing this axiom in a number of ways, including by requiring that $|Hom(a,b)| \leq 1$ only when $a$ and $b$ are equal. The result is the category of endotrivial categories. These categories have interesting order-theoretic properties.

Definition. a category $C$ is endotrivial provided that for each object $a \in Ob(C)$ we have that $|Hom(a,a)|$ = 1.

Partially ordered endotrivial categories:
As an introduction to endotrivial categories, we shall first consider the case of a partially ordered endotrivial category.

Definition. a endotrivial category is partially ordered provided that its object preorder is antisymmetric.

These categories have the following interesting characterisation which describes their role in category theory.

Proposition. a category $C$ is partially ordered endotrivial iff it has the property that for each symmetric pair of morphisms if they go in opposite directions $f: A \to B$ and $g: B \to A$ then $f = g$.

Proof. suppose that all opposite morphisms are equal. Then the category is endotrivial because if $f: A \to A$ and $g: A \to A$ are two endomorphisms then they are opposite because their respective sources equal the others targets and vice versa. So they must be equal, which implies that all endo hom classes are trivial. Further, suppose that $C$ is not antisymmetric then there exists $A \subseteq B$ and $B \subseteq A$ which means there are opposite edges $f: A \to B$ and $g : B \to A$ which violates the non-existence of opposite edges, so $C$ is partially ordered. $\square$

These categories are the strongly directed categories, in the sense that all their arrows are going the same direction along some partial order. As a direct generalisation of thin categories, they are one of the classes of categories that are most like partial orders.

Example 1. any partial order such as Young's lattice $\mathbb{Y}$ or the subobject lattice of a category $Sub(C)$ is a partially ordered endotrivial category.

Example 2. the index category $T_2^*$ of the elementary presheaf topos of quivers $Quiv$ is a partially ordered endotrivial category, as is any other index category for a nary quiver topos.

The general theory:
In the same way that partial orders are the basis of the entire theory of thin categories, the strongly directed categories are the basis of the whole theory of endotrivial categories.

Theorem. let $C$ be an endotrivial category and let $a,b$ be two objects in $Ob(C)$ that are related to one another in the object preorder, so that there exists $f: A \to B$ and $g : B \to A$. Then $a$ and $b$ are isomorphic.

Proof. $f : A \to B$ and $g : B \to A$ are composable so there exists composites $g \circ f : A \to A$ and $f \circ g : B \to B$, but in an endotrivial category all endomorphisms are equal so this means that $g \circ f = 1_A$ and $f \circ g = 1_B$ which implies that $f$ and $g$ are opposite isomorphisms. Therefore, $A$ and $B$ are isomorphic. $\square$

These allows us to characterize the skeletal endotrivial categories, which demonstrates that they are strongly directed. Thusly, endotrivial categories can be considered by their skeletal counterparts in the same way that thin categories are considered in terms of partial orders.

Corollary. the skeletal endotrivial categories are precisely the partially ordered endotrivial categories

The endotrivial categories can be considered to be generalisations of thin categories, but they could equally be considered to be the antithesis of monoids. This theorem provides their general theory.