74 min
- Source: 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.
The notion of a span is due to Nobuo Yoneda (1954) and Jean Bénabou (1967).
Formal definition
A span is a diagram of type
Λ
=
(
−
1
←
0
→
+
1
)
,
{\displaystyle \Lambda =(-1\leftarrow 0\rightarrow +1),}
i.e., a diagram of the form
Y
←
X
→
Z
{\displaystyle 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 X ← R → Y is a span, where the maps are the projection maps
X
×
Y
→
π
X
X
{\displaystyle X\times Y{\overset {\pi _{X}}{\to }}X}
and
X
×
Y
→
π
Y
Y
{\displaystyle X\times Y{\overset {\pi _{Y}}{\to }}Y}
.
Any object yields the trivial span A ← A → A, where the maps are the identity.
More generally, let
ϕ
:
A
→
B
{\displaystyle \phi \colon A\to B}
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
X
←
Y
→
Z
,
{\displaystyle 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
Λ
op
=
(
−
1
→
0
←
+
1
)
,
{\displaystyle \Lambda ^{\text{op}}=(-1\rightarrow 0\leftarrow +1),}
i.e., a diagram of the form
Y
→
X
←
Z
{\displaystyle 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.
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
Kata Kunci Pencarian:
- Homoseksualitas
- Span (category theory)
- Span
- Pushout (category theory)
- Category of sets
- Cone (category theory)
- Coproduct
- Diagram (category theory)
- Group representation
- Completions in category theory
- Span of control
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