nLab EI-category




An EI-category is a category in which every endomorphism is an isomorphism, hence an automorphism.

Similarly an EI (,1)(\infty,1)-category is an (∞,1)-category in which every endomorphism is an equivalence.


Let 𝒮\mathcal{S} be a set of subgroups of a group GG. The following are all EI-categories (Webb08, p. 4078):


The transporter category 𝒯 𝒮\mathcal{T}_{\mathcal{S}} has as its objects the members of 𝒮\mathcal{S}, and morphisms Hom(H,K)=N G(H,K)={gG| gHK}Hom(H,K) = N_G(H,K) = \{g \in G|{}^{g}H \subseteq K\}.


The orbit category 𝒪 𝒮\mathcal{O}_{\mathcal{S}} associated to 𝒮\mathcal{S} in which the objects are the coset spaces G/HG/H where HSH \in S and the morphisms are the GG-equivariant functions.


More generally: the fundamental category of a GG-spaceategory#FundamentalCategoryOfAGSpace) is an EI-category.


The Frobenius category 𝒮\mathcal{F}_{\mathcal{S}} has the elements of 𝒮\mathcal{S} as its objects, and Hom 𝒮=N G(H,K)/C G(H)Hom_{\mathcal{F}_{\mathcal{S}}} = N_G(H,K)/C_G(H). The morphisms may be identified with the set of group homomorphisms HKH \to K which are of the form ‘conjugation by gg’ for some gGg \in G.



EI-categories may be seen as those categories satisfying a kind of Schröder–Bernstein theorem.


A category CC is EI if and only if every antiparallel pair XYX \rightleftarrows Y exhibits a pair of isomorphisms.


Assume that CC is EI, and let f:XY:gf \colon X \rightleftarrows Y : g be an antiparallel pair. Consider XfYgXfYX \xrightarrow{f} Y \xrightarrow{g} X \xrightarrow{f} Y. Since isomorphisms have the 2-out-of-6 property, and gfgf and fgfg are isomorphisms, ff and gg are also isomorphisms. Conversely, suppose that CC satisfies the assumption of the proposition. Let i:XXi \colon X \to X be an endomorphism. Then i:XX:ii \colon X \rightleftarrows X : i exhibits an antiparallel pair, so in particular ii is an isomorphism.

In particular, assuming excluded middle, the Schröder–Bernstein theorem states that Inj, the wide subcategory of Set spanned by monomorphisms, is an EI-category.

Partial ordering

Given an EI-category, CC, the set of isomorphism classes [x][x] of objects xCx \in C forms a partially ordered set under the relation

[x][y]AAif and only ifAAthere is a morphism xy [x] \leq [y] \phantom{AA} \text{if and only if} \phantom{AA} \text{there is a morphism } x \to y

Representation theory

A finite EI-category contains finitely many morphisms.

The category algebra kCk C of a finite EI-category, CC, for a fixed base ring kk and has as basis the set of morphisms in CC 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 kCk C of depth equal to the number of isomorphism classes in the category.

The category of modules over the category algebra kCk C is equivalent to the category of kk-linear representations of CC, i.e., the functor category Fun(C,Modk)Fun(C, Mod k).


Maybe the earliest explicit observation that in an orbit category, and its relatives, endomorphisms are automorphisms is in:

Discussion in the context of algebraic K-theory:

On the representation theory of EI-categories:

Last revised on February 1, 2024 at 17:12:10. See the history of this page for a list of all contributions to it.