Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
One analog in (∞,1)-category theory of epimorphism in category theory. Beware that there are other notions such as effective epimorphism in an -category and, more generally, the concept of n-epimorphism.
For an (∞,1)-category, a 1-morphism in is an epimorphism if for all objects the induced morphism on hom -groupoids
is a monomorphism in ∞Grpd.
(terminal epimorphisms of -groupoids)
For , if the terminal map is an epimorphism in the sense of Def. then
In particular, this means that in order for the delooping groupoid of a discrete group (i.e. the Eilenberg-MacLane space ) to be such that is epi, the group must be perfect.
(terminal epimorphisms of 1-groupoids) In contrast to Ex. , in the (2,1)-category of 1-groupoids the maps
are always fully faithful, for all , without further conditions on , notably so for non-trivial abelian groups . Explicitly, for the case (to which the general situation reduces by extensivity), we have the coproduct decomposition
which plays a central role in discussion such as of inertia orbifolds, equivariant principal bundles, equivariant K-theory and other aspects of equivariant cohomology.
The point is that a natural transformation out of into a 1-groupoid has only a single component, corresponding to the point , but a pseudonatural transformation out of into a 2-groupoid (and higher) has in addition a component for each element of , which are not reflected on .
A morphism between connected spaces is an epimorphism iff is formed via a Quillen plus construction from a perfect normal subgroup of the fundamental group .
More generally, a map of spaces is an epi iff all its fibers are acyclic spaces in the sense that their suspensions are contractible.
A generalization of this to epimorphisms and acyclic spaces in -toposes is discussed in Hoyois 19.
George Raptis, Some characterizations of acyclic maps, Journal of Homotopy and Related Structures volume 14, pages 773–785 (2019) 2017 (arxiv:1711.08898, doi:10.1007/s40062-019-00231-6)
Marc Hoyois, On Quillen’s plus construction, 2019 (pdf, pdf)
Oren Ben-Bassat, D. Mukherjee, Analytification, localization and homotopy epimorphisms, Bulletin des Sciences Mathématiques 176 (2022) 103129 arXiv:2111.04184 doi
Last revised on November 13, 2023 at 12:25:09. See the history of this page for a list of all contributions to it.