An EI-category is a category in which every endomorphism is an isomorphism, hence an automorphism.
Similarly an EI -category is an (∞,1)-category in which every endomorphism is an equivalence.
A poset
The path category of an acyclic quiver.
A group, , 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 be a set of subgroups of a group . The following are all EI-categories (Webb08, p. 4078):
The transporter category has as its objects the members of , and morphisms .
The orbit category associated to in which the objects are the coset spaces where and the morphisms are the -equivariant functions.
More generally: the fundamental category of a -spaceategory#FundamentalCategoryOfAGSpace) is an EI-category.
The Frobenius category has the elements of as its objects, and . The morphisms may be identified with the set of group homomorphisms which are of the form ‘conjugation by ’ for some .
EI-categories may be seen as those categories satisfying a kind of Schröder–Bernstein theorem.
A category is EI if and only if every antiparallel pair exhibits a pair of isomorphisms.
Assume that is EI, and let be an antiparallel pair. Consider . Since isomorphisms have the 2-out-of-6 property, and and are isomorphisms, and are also isomorphisms. Conversely, suppose that satisfies the assumption of the proposition. Let be an endomorphism. Then exhibits an antiparallel pair, so in particular 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.
Given an EI-category, , the set of isomorphism classes of objects forms a partially ordered set under the relation
A finite EI-category contains finitely many morphisms.
The category algebra of a finite EI-category, , for a fixed base ring and has as basis the set of morphisms in 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 of depth equal to the number of isomorphism classes in the category.
The category of modules over the category algebra is equivalent to the category of -linear representations of , i.e., the functor category .
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:
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)
Last revised on February 1, 2024 at 17:12:10. See the history of this page for a list of all contributions to it.