In mathematics, a Segal space is a
simplicial space
In mathematics, a simplicial space is a simplicial object in the category of topological spaces. In other words, it is a contravariant functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors ...
satisfying some
pullback
In mathematics, a pullback is either of two different, but related processes: precomposition and fiber-product. Its dual is a pushforward.
Precomposition
Precomposition with a function probably provides the most elementary notion of pullback: in ...
conditions, making it look like a
homotopical version of a
category
Category, plural categories, may refer to:
Philosophy and general uses
* Categorization, categories in cognitive science, information science and generally
*Category of being
* ''Categories'' (Aristotle)
*Category (Kant)
*Categories (Peirce)
* ...
. More precisely, a
simplicial set
In mathematics, a simplicial set is an object composed of ''simplices'' in a specific way. Simplicial sets are higher-dimensional generalizations of directed graphs, partially ordered sets and categories. Formally, a simplicial set may be defined a ...
, considered as a simplicial discrete space, satisfies the Segal conditions iff it is the
nerve
A nerve is an enclosed, cable-like bundle of nerve fibers (called axons) in the peripheral nervous system.
A nerve transmits electrical impulses. It is the basic unit of the peripheral nervous system. A nerve provides a common pathway for the e ...
of a category. The condition for Segal spaces is a homotopical version of this.
Complete Segal spaces were introduced by as models for
(∞, 1)-categories.
References
*
External links
*
*{{nlab, id=complete+Segal+space, title=Complete Segal space
Category theory
Simplicial sets