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 mappings.

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 ‘conjugation by $g$’ for some $g \in G$.

Representation theory

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.