To install click the Add extension button. That's it.

The source code for the WIKI 2 extension is being checked by specialists of the Mozilla Foundation, Google, and Apple. You could also do it yourself at any point in time.

4,5
Kelly Slayton
Congratulations on this excellent venture… what a great idea!
Alexander Grigorievskiy
I use WIKI 2 every day and almost forgot how the original Wikipedia looks like.
Live Statistics
English Articles
Improved in 24 Hours
Added in 24 Hours
Languages
Recent
Show all languages
What we do. Every page goes through several hundred of perfecting techniques; in live mode. Quite the same Wikipedia. Just better.
.
Leo
Newton
Brights
Milds

Span (category theory)

From Wikipedia, the free encyclopedia

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.

The notion of a span is due to Nobuo Yoneda (1954) and Jean Bénabou (1967).

YouTube Encyclopedic

  • 1/3
    Views:
    13 980
    2 669 862
    2 818
  • The Continuum Hypothesis and the search for Mathematical Infinity, W. Hugh Woodin
  • Michio Kaku: Is God a Mathematician?
  • Diffusion of innovations

Transcription

Formal definition

A span is a diagram of type i.e., a diagram of the form .

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 and .
  • Any object yields the trivial span AAA, where the maps are the identity.
  • More generally, let be a morphism in some category. There is a trivial span 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 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 i.e., a diagram of the form .

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.

The category nCob of finite-dimensional cobordisms is a dagger compact category. More generally, the category Span(C) of spans on any category C with finite limits is also dagger compact.

See also

References

  • span at the nLab
  • Yoneda, Nobuo (1954). "On the homology theory of modules". J. Fac. Sci. Univ. Tokyo Sect. I. 7: 193–227.
  • Bénabou, Jean (1967). "Introduction to Bicategories". Reports of the Midwest Category Seminar. Lecture Notes in Mathematics. Vol. 47. Springer. pp. 1–77. doi:10.1007/BFb0074299. ISBN 978-3-540-35545-8.
This page was last edited on 4 March 2024, at 19:13
Basis of this page is in Wikipedia. Text is available under the CC BY-SA 3.0 Unported License. Non-text media are available under their specified licenses. Wikipedia® is a registered trademark of the Wikimedia Foundation, Inc. WIKI 2 is an independent company and has no affiliation with Wikimedia Foundation.