nLab EI-category

Definition

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

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

Examples

• A poset

• The path category of an acyclic quiver.

• A group, $G$, understood as delooped to a pointed connected groupoid.

• A one-way category, which is a category in which every endomorphism is an identity, is trivially an EI-category.

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

Example

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

Example

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.

Example

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

Example

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 $g \in G$.

Properties

General

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

Proposition

A category $C$ is EI if and only if every antiparallel pair $X \rightleftarrows Y$ exhibits a pair of isomorphisms.

Proof

Assume that $C$ is EI, and let $f \colon X \rightleftarrows Y : g$ be an antiparallel pair. Consider $X \xrightarrow{f} Y \xrightarrow{g} X \xrightarrow{f} Y$. Since isomorphisms have the 2-out-of-6 property, and $gf$ and $fg$ are isomorphisms, $f$ and $g$ are also isomorphisms. Conversely, suppose that $C$ satisfies the assumption of the proposition. Let $i \colon X \to X$ be an endomorphism. Then $i \colon X \rightleftarrows X : i$ exhibits an antiparallel pair, so in particular $i$ 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, $C$, the set of isomorphism classes $[x]$ of objects $x \in C$ forms a partially ordered set under the relation

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

Representation theory

A finite EI-category contains finitely many morphisms.

The category algebra $k C$ 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 $Fun(C, Mod k)$.

Related concepts

• inverse EI (∞,1)-category

References

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

• Tammo tom Dieck, p. 73 of: Transformation Groups, de Gruyter 1987 (doi:10.1515/9783110858372)

Discussion in the context of algebraic K-theory:

• Wolfgang Lück, Transformation Groups and Algebraic K-Theory, Lecture Notes in Mathematics 1408 (Springer 1989) (doi:10.1007/BFb0083681)

On the representation theory of EI-categories:

• Peter Webb, Standard stratifications of EI categories and Alperin’s weight conjecture, Journal of Algebra

  320 12 (2008) 4073-4091 (doi:10.1016/j.jalgebra.2006.03.052)

• Karsten Dietrich, Representation Theory of EI-categories, 2010 (urn:nbn:de:hbz:466-20100701014, pdf, pdf)

• Liping Li, A generalized Koszul theory and its application, Transactions of the American Mathematical Society 366 2 (2014) 931-977 (arXiv:1109.5760, jstor:23812975)

• Ergün Yalçın, Projective resolutions over EI-categories, 2012 (hdl:11693/15472)