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EI-category

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Definition

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

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

Examples

Let ๐’ฎ\mathcal{S} be a set of subgroups of a group GG. The following are all EI-categories (Webb08, p. 4078):

  • The transporter category ๐’ฏ ๐’ฎ\mathcal{T}_{\mathcal{S}} has as its objects the members of ๐’ฎ\mathcal{S}, and morphisms Hom(H,K)=N G(H,K)={gโˆˆG| gHโŠ†K}Hom(H,K) = N_G(H,K) = \{g \in G|{}^{g}H \subseteq K\}.

  • The orbit category ๐’ช ๐’ฎ\mathcal{O}_{\mathcal{S}} associated to ๐’ฎ\mathcal{S} in which the objects are the coset spaces G/HG/H where HโˆˆSH \in S and the morphisms are the GG-equivariant functions.

  • The Frobenius category โ„ฑ ๐’ฎ\mathcal{F}_{\mathcal{S}} has the elements of ๐’ฎ\mathcal{S} as its objects, and Hom โ„ฑ ๐’ฎ=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 Hโ†’KH \to K which are of the form โ€˜conjugation by ggโ€™ for some gโˆˆGg \in G.

Properties

General

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

[x]โ‰ค[y]AAif and only ifAAthere is a morphismxโ†’y [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).

References

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 Alperinโ€™s weight conjecture, (doi)

  • Karsten Dietrich, Representation Theory of EI-categories, (pdf)

Last revised on May 10, 2021 at 18:07:41. See the history of this page for a list of all contributions to it.