An EI-category is a category in which every endomorphism is an isomorphism.
Similarly an EI $(\infty,1)$-category is an (โ,1)-category in which every endomorphism is an equivalence.
A poset
The path category of an acyclic quiver.
A group, $G$, understood as delooped to a pointed connected groupoid.
Let $\mathcal{S}$ be a set of subgroups of a group $G$. The following are all EI-categories (Webb08, p. 4078):
The transporter category $\mathcal{T}_{\mathcal{S}}$ has as its objects the members of $\mathcal{S}$, and morphisms $Hom(H,K) = N_G(H,K) = \{g \in G|{}^{g}H \subseteq K\}$.
The orbit category $\mathcal{O}_{\mathcal{S}}$ associated to $\mathcal{S}$ in which the objects are the coset spaces $G/H$ where $H \in S$ and the morphisms are the $G$-equivariant functions.
The Frobenius category $\mathcal{F}_{\mathcal{S}}$ has the elements of $\mathcal{S}$ as its objects, and $Hom_{\mathcal{F}_{\mathcal{S}}} = N_G(H,K)/C_G(H)$. The morphisms may be identified with the set of group homomorphisms $H \to K$ which are of the form โconjugation by $g$โ for some $g \in G$.
Given an EI-category, $C$, the set of isomorphism classes $[x]$ of objects $x \in C$ forms a partially ordered set under the relation
A finite EI-category contains finitely many morphisms.
The category algebra $k C$ of a finite EI-category, $C$, for a fixed base ring $k$ and has as basis the set of morphisms in $C$ with multiplication induced by composition of morphisms. It is thus a generalization of the group algebra of a finite group, the path algebra of a finite quiver without oriented cycles or the incidence algebra of a finite poset. There is a stratification of $k C$ of depth equal to the number of isomorphism classes in the category.
The category of modules over the category algebra $k C$ is equivalent to the category of $k$-linear representations of $C$, i.e., the functor category $Fun(C, Mod k)$.
Maybe the earliest explicit observation that in an orbit category, and its relatives, endomorphisms are automorphisms is in:
Other references are:
Peter Webb, Standard stratifications of EI categories and Alperinโs weight conjecture, (doi)
Karsten Dietrich, Representation Theory of EI-categories, (pdf)
Last revised on April 14, 2021 at 12:16:16. See the history of this page for a list of all contributions to it.