Definitions
Transfors between 2-categories
Morphisms in 2-categories
Structures in 2-categories
Limits in 2-categories
Structures on 2-categories
Background
Basic concepts
equivalences in/of -categories
Universal constructions
Local presentation
Theorems
Extra stuff, structure, properties
Models
category object in an (∞,1)-category, groupoid object
(directed enhancement of homotopy type theory with types behaving like -categories)
In any category, a cospan is a diagram like this:
A cospan in the category is the same as a span in the opposite category . So, all general facts about cospans in are general facts about spans in , and the reader may turn to the entry on spans to learn more.
A cospan that admits a cone is called a quadrable cospan.
In simplicial type theory, binary parametric observational type theory, and other type theoretic approaches to synthetic -category theory, a cospan in a synthetic -category can be defined as the following:
a record consisting of objects (elements of types) , , and and morphisms (elements of hom types) and
a function from the (2,0)-horn type to , where is defined in terms of the directed interval as the type of pairs of elements such that or .
The walking cospan or (2,0)-horn category is the category which consists of three objects and two morphisms and .
The category is also called the (2,0)-horn preorder or the (2,0)-horn poset because can be shown to be a poset.
Every cospan in a category is represented by a functor from the walking cospan to .
In type theoretic approaches to synthetic -category theory, the (2,0)-horn type defined above plays the role of the walking cospan.
Following the notion of a right quotient of two elements in a monoid, we can define in category the notion of a right quotient of a cospan in a category.
A right quotient of a cospan and is a morphism such that is the unique composite of and , .
Given a morphism , the cospan is always right divisible with right quotient . A section is a right quotient of the span .
Cospans in a category with small colimits form a bicategory whose objects are the objects of , whose morphisms are cospans between two objects, and whose 2-morphisms are commuting diagrams of the form
The category of cospans from to is naturally a category enriched in : for
two parallel cospans in , the -object of morphisms between them is the pullback
formed in analogy to the enriched hom of pointed objects. The initial object of is the coproduct of and .
If has a terminal object, , then cospans from to itself are bi-pointed objects in .
the boolean domain ; i.e. the walking pair of objects
the directed interval category ; i.e. the walking morphism
the (2,0)-horn category ; i.e. the walking cospan
the (2,1)-horn category ; i.e. the walking composable pair
the (2,2)-horn category ; i.e. the walking span
the 2-simplex category ; i.e. the walking commutative triangle
Adj; i.e. the walking adjunction
the syntactic category of a theory ; i.e. the walking -model
Topological cospans and their role as models for cobordisms are discussed in
Last revised on June 1, 2025 at 11:44:27. See the history of this page for a list of all contributions to it.