World Library  
Flag as Inappropriate
Email this Article

Span (category theory)

Article Id: WHEBN0010318351
Reproduction Date:

Title: Span (category theory)  
Author: World Heritage Encyclopedia
Language: English
Subject: Pushout (category theory), Amalgamation property
Collection:
Publisher: World Heritage Encyclopedia
Publication
Date:
 

Span (category theory)

In category theory, a span, roof or correspondence 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 and 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^\text{op} = (-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.

See also

References

  • Template:Nlab
This article was sourced from Creative Commons Attribution-ShareAlike License; additional terms may apply. World Heritage Encyclopedia content is assembled from numerous content providers, Open Access Publishing, and in compliance with The Fair Access to Science and Technology Research Act (FASTR), Wikimedia Foundation, Inc., Public Library of Science, The Encyclopedia of Life, Open Book Publishers (OBP), PubMed, U.S. National Library of Medicine, National Center for Biotechnology Information, U.S. National Library of Medicine, National Institutes of Health (NIH), U.S. Department of Health & Human Services, and USA.gov, which sources content from all federal, state, local, tribal, and territorial government publication portals (.gov, .mil, .edu). Funding for USA.gov and content contributors is made possible from the U.S. Congress, E-Government Act of 2002.
 
Crowd sourced content that is contributed to World Heritage Encyclopedia is peer reviewed and edited by our editorial staff to ensure quality scholarly research articles.
 
By using this site, you agree to the Terms of Use and Privacy Policy. World Heritage Encyclopedia™ is a registered trademark of the World Public Library Association, a non-profit organization.
 


Copyright © World Library Foundation. All rights reserved. eBooks from Project Gutenberg are sponsored by the World Library Foundation,
a 501c(4) Member's Support Non-Profit Organization, and is NOT affiliated with any governmental agency or department.