- Source: Category of manifolds
In mathematics, the category of manifolds, often denoted Manp, is the category whose objects are manifolds of smoothness class Cp and whose morphisms are p-times continuously differentiable maps. This is a category because the composition of two Cp maps is again continuous and of class Cp.
One is often interested only in Cp-manifolds modeled on spaces in a fixed category A, and the category of such manifolds is denoted Manp(A). Similarly, the category of Cp-manifolds modeled on a fixed space E is denoted Manp(E).
One may also speak of the category of smooth manifolds, Man∞, or the category of analytic manifolds, Manω.
Manp is a concrete category
Like many categories, the category Manp is a concrete category, meaning its objects are sets with additional structure (i.e. a topology and an equivalence class of atlases of charts defining a Cp-differentiable structure) and its morphisms are functions preserving this structure. There is a natural forgetful functor
U : Manp → Top
to the category of topological spaces which assigns to each manifold the underlying topological space and to each p-times continuously differentiable function the underlying continuous function of topological spaces. Similarly, there is a natural forgetful functor
U′ : Manp → Set
to the category of sets which assigns to each manifold the underlying set and to each p-times continuously differentiable function the underlying function.
Pointed manifolds and the tangent space functor
It is often convenient or necessary to work with the category of manifolds along with a distinguished point: Man•p analogous to Top• - the category of pointed spaces. The objects of Man•p are pairs
(
M
,
p
0
)
,
{\displaystyle (M,p_{0}),}
where
M
{\displaystyle M}
is a
C
p
{\displaystyle C^{p}}
manifold along with a basepoint
p
0
∈
M
,
{\displaystyle p_{0}\in M,}
and its morphisms are basepoint-preserving p-times continuously differentiable maps: e.g.
F
:
(
M
,
p
0
)
→
(
N
,
q
0
)
,
{\displaystyle F:(M,p_{0})\to (N,q_{0}),}
such that
F
(
p
0
)
=
q
0
.
{\displaystyle F(p_{0})=q_{0}.}
The category of pointed manifolds is an example of a comma category - Man•p is exactly
(
{
∙
}
↓
M
a
n
p
)
,
{\displaystyle \scriptstyle {(\{\bullet \}\downarrow \mathbf {Man^{p}} )},}
where
{
∙
}
{\displaystyle \{\bullet \}}
represents an arbitrary singleton set, and the
↓
{\displaystyle \downarrow }
represents a map from that singleton to an element of Manp, picking out a basepoint.
The tangent space construction can be viewed as a functor from Man•p to VectR as follows: given pointed manifolds
(
M
,
p
0
)
{\displaystyle (M,p_{0})}
and
(
N
,
F
(
p
0
)
)
,
{\displaystyle (N,F(p_{0})),}
with a
C
p
{\displaystyle C^{p}}
map
F
:
(
M
,
p
0
)
→
(
N
,
F
(
p
0
)
)
{\displaystyle F:(M,p_{0})\to (N,F(p_{0}))}
between them, we can assign the vector spaces
T
p
0
M
{\displaystyle T_{p_{0}}M}
and
T
F
(
p
0
)
N
,
{\displaystyle T_{F(p_{0})}N,}
with a linear map between them given by the pushforward (differential):
F
∗
,
p
:
T
p
0
M
→
T
F
(
p
0
)
N
.
{\displaystyle F_{*,p}:T_{p_{0}}M\to T_{F(p_{0})}N.}
This construction is a genuine functor because the pushforward of the identity map
1
M
:
M
→
M
{\displaystyle \mathbb {1} _{M}:M\to M}
is the vector space isomorphism
(
1
M
)
∗
,
p
0
:
T
p
0
M
→
T
p
0
M
,
{\displaystyle (\mathbb {1} _{M})_{*,p_{0}}:T_{p_{0}}M\to T_{p_{0}}M,}
and the chain rule ensures that
(
f
∘
g
)
∗
,
p
0
=
f
∗
,
g
(
p
0
)
∘
g
∗
,
p
0
.
{\displaystyle (f\circ g)_{*,p_{0}}=f_{*,g(p_{0})}\circ g_{*,p_{0}}.}
References
Lang, Serge (2012) [1972]. Differential manifolds. Springer. ISBN 978-1-4684-0265-0.
Tu, Loring W. (2011). An introduction to manifolds (2nd ed.). New York: Springer. ISBN 9781441974006. OCLC 682907530.