# Span (category theory)

﻿
Span (category theory)

A span, in category theory, is a generalization of the notion of relation between two objects of a category. When the category has all pullbacks (and satisfies a small number of other conditions), spans can be considered as morphisms in a category of fractions.

## Formal definition

A span is a diagram of type $\Lambda = (-1 \leftarrow 0 \rightarrow +1),$ i.e., a diagram of the form $Y \leftarrow X \rightarrow Z$.

That is, let Λ be the category (-1 ← 0 → +1). Then a span in a category C is a functor S:Λ → C. This means that a span consists of three objects X, Y and Z of C and morphisms f:X → Y and g:X → Z: it is two maps with common domain.

The colimit of a span is a pushout.

## Examples

• If R is a relation between sets X and Y (i.e. a subset of X × Y), then XRY is a span, where the maps are the projection maps $X \times Y \overset{\pi_X}{\to} X, X \times Y \overset{\pi_Y}{\to} Y$.
• Any object yields the trivial span A = A = A; formally, the diagram AAA, where the maps are the identity.
• More generally, let $\phi\colon A \to B$ be a morphism in some category. There is a trivial span A = AB; formally, the diagram AAB, where the left map is the identity on A, and the right map is the given map φ.
• If M is a model category, with W the set of weak equivalences, then the spans of the form $X \leftarrow Y \rightarrow Z,$

where the left morphism is in W, can be considered a generalised morphism (i.e., where one "inverts the weak equivalences"). Note that this is not the usual point of view taken when dealing with model categories.

## Cospans

A cospan K in a category C is a functor K:Λop → C; equivalently, a contravariant functor from Λ to C. That is, a diagram of type $\Lambda = (-1 \rightarrow 0 \leftarrow +1),$ i.e., a diagram of the form $Y \rightarrow X \leftarrow Z$.

Thus it consists of three objects X, Y and Z of C and morphisms f:Y → X and g:Z → X: it is two maps with common codomain.

The limit of a cospan is a pullback.

An example of a cospan is a cobordism W between two manifolds M and N, where the two maps are the inclusions into W. Note that while cobordisms are cospans, the category of cobordisms is not a "cospan category": it is not the category of all cospans in "the category of manifolds with inclusions on the boundary", but rather a subcategory thereof, as the requirement that M and N form a partition of the boundary of W is a global constraint.

Wikimedia Foundation. 2010.

### Look at other dictionaries:

• Diagram (category theory) — In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms. An indexed family of sets is a collection of sets …   Wikipedia

• Allegory (category theory) — In mathematics, in the subject of category theory, an allegory is a category that has some of the structure of the category of sets and binary relations between them. Allegories can be used as an abstraction of categories of relations, and in… …   Wikipedia

• Pushout (category theory) — In category theory, a branch of mathematics, a pushout (also called a fibered coproduct or fibered sum or cocartesian square or amalgamed sum) is the colimit of a diagram consisting of two morphisms f : Z → X and g : Z → Y with a common …   Wikipedia

• Span — may refer to length or space:* span (length), the width of a human hand * span (architecture) ** For powerlines, the distance between two pylons ** For aerial tramways, the distance between two supporting structures ** For a bridge, the distance… …   Wikipedia

• Theory of mind — is the ability to attribute mental states beliefs, intents, desires, pretending, knowledge, etc. to oneself and others and to understand that others have beliefs, desires and intentions that are different from one s own. Though there are… …   Wikipedia

• Interference theory — Contents 1 History 2 Proactive Interference 2.1 Proactive Interference with Single and Multiple Lists 2.2 Proactive Interference and Context …   Wikipedia

• Tight span — If a set of points in the plane, with the Manhattan metric, has a connected orthogonal convex hull, then that hull coincides with the tight span of the points. In metric geometry, the metric envelope or tight span of a metric space M is an… …   Wikipedia

• life span — 1. the longest period over which the life of any organism or species may extend, according to the available biological knowledge concerning it. 2. the longevity of an individual. [1915 20] * * * Time between birth and death. It ranges from a… …   Universalium

• Dedicated Portfolio Theory — Dedicated Portfolio Theory, in finance, deals with the characteristics and features of a portfolio built to generate a predictable stream of future cash inflows. This is achieved by purchasing bonds and/or other fixed income securities (such as… …   Wikipedia

• List of mathematics articles (S) — NOTOC S S duality S matrix S plane S transform S unit S.O.S. Mathematics SA subgroup Saccheri quadrilateral Sacks spiral Sacred geometry Saddle node bifurcation Saddle point Saddle surface Sadleirian Professor of Pure Mathematics Safe prime Safe… …   Wikipedia