{{Short description|Generalization of a notion in 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.
The notion of a span is due to Nobuo Yoneda (1954) and Jean Bénabou (1967).
== Formal definition == A span is a diagram of type <math>\Lambda = (-1 \leftarrow 0 \rightarrow +1),</math> i.e., a diagram of the form <math>Y \leftarrow X \rightarrow Z</math>.
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 ''X'' ← ''R'' → ''Y'' is a span, where the maps are the projection maps <math>X \times Y \overset{\pi_X}{\to} X</math> and <math>X \times Y \overset{\pi_Y}{\to} Y</math>. * Any object yields the trivial span ''A'' ← ''A'' → ''A,'' where the maps are the identity. * More generally, let <math>\phi\colon A \to B</math> be a morphism in some category. There is a trivial span ''A'' ← ''A'' → ''B'', 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 <math>X \leftarrow Y \rightarrow Z,</math> 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 : Λ<sup>op</sup> → '''C'''; equivalently, a ''contravariant'' functor from Λ to '''C'''. That is, a diagram of type <math>\Lambda^\text{op} = (-1 \rightarrow 0 \leftarrow +1),</math> i.e., a diagram of the form <math>Y \rightarrow X \leftarrow Z</math>.
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 == * Binary relation * Pullback (category theory) * Pushout (category theory) * Cobordism
==References== {{refbegin}} * {{nlab|id=span}} *{{cite journal |first=Nobuo |last=Yoneda |title=On the homology theory of modules |journal=J. Fac. Sci. Univ. Tokyo Sect. I |volume=7 |issue= |pages=193–227 |date=1954 |doi= |url=}} *{{cite conference |first=Jean |last=Bénabou |title=Introduction to Bicategories |book-title=Reports of the Midwest Category Seminar |publisher=Springer |series=Lecture Notes in Mathematics |date=1967 |isbn=978-3-540-35545-8 |pages=1–77 |volume=47 |doi=10.1007/BFb0074299}} {{refend}}
Category:Functors