In mathematics, especially in algebraic topology, the mapping space between two spaces is the space of all the (continuous) maps between them. Viewing the set of all the maps as a space is useful because that allows for topological considerations. For example, a curve h : I → Map ( X , Y ) {\displaystyle h:I\to \operatorname {Map} (X,Y)} in the mapping space is exactly a homotopy between the starting point and the end point. From the category theory point of view, a mapping space provides the internal Hom (i.e., hom that is also an object) in the category of spaces.
Topologies
A mapping space can be equipped with several topologies. A common one is the compact-open topology or the k-ification of it. Typically, there is then the adjoint relation
Map ( X × Y , Z ) ≃ Map ( X , Map ( Y , Z ) ) {\displaystyle \operatorname {Map} (X\times Y,Z)\simeq \operatorname {Map} (X,\operatorname {Map} (Y,Z))}
and thus Map {\displaystyle \operatorname {Map} } is an analog of the Hom functor. (For pathological spaces, this relation may fail.) Here is another common one. We have:
Map ( X , Y ) ↪ X × Y {\displaystyle \operatorname {Map} (X,Y)\hookrightarrow X\times Y}
given by f ↦ Γ f = {\displaystyle f\mapsto \Gamma _{f}=} the graph of f {\displaystyle f} . Then we can give Map ( X , Y ) {\displaystyle \operatorname {Map} (X,Y)} the Whitney topology (also called the fine topology or the strong topology) where a basic open set consists of those g {\displaystyle g} such that Γ g ⊂ U {\displaystyle \Gamma _{g}\subset U} for some open subset U ⊂ X × Y {\displaystyle U\subset X\times Y} . The compact-open topology does not handle a behavior at infinity well and so sometimes the Whitney topology is used instead. If X {\displaystyle X} is a paracompact and Y {\displaystyle Y} is a metric space, then the Whitney topology has a basic open set of the form
B ( f , ϵ ) := { g ∣ d ( g ( x ) , f ( x ) ) < ϵ ( x ) } {\displaystyle B(f,\epsilon ):=\{g\mid d(g(x),f(x))<\epsilon (x)\}}
for some f ∈ Map ( X , Y ) {\displaystyle f\in \operatorname {Map} (X,Y)} and some continuous function ϵ : X → R > 0 {\displaystyle \epsilon :X\to \mathbb {R} _{>0}} . If, moreover, Y {\displaystyle Y} is complete, then we have the following important fact:
Let Q ⊂ Map ( X , Y ) {\displaystyle Q\subset \operatorname {Map} (X,Y)} be a subset such that every uniform limit of a sequence in Q {\displaystyle Q} , if any, is in Q {\displaystyle Q} . Then Q {\displaystyle Q} is a Baire space. This is proved by the same way Baire's category theorem is proved except we use the above family-version of a ball.
Smooth mappings
For manifolds M , N {\displaystyle M,N} , there is the subset C r ( M , N ) ⊂ Map ( M , N ) {\displaystyle {\mathcal {C}}^{r}(M,N)\subset \operatorname {Map} (M,N)} that consists of all the C r {\displaystyle {\mathcal {C}}^{r}} -smooth maps from M {\displaystyle M} to N {\displaystyle N} . It can be equipped with the weak or strong topology. A basic approximation theorem says that C W s ( M , N ) {\displaystyle {\mathcal {C}}_{W}^{s}(M,N)} is dense in C S r ( M , N ) {\displaystyle {\mathcal {C}}_{S}^{r}(M,N)} for 1 ≤ s ≤ ∞ , 0 ≤ r < s {\displaystyle 1\leq s\leq \infty ,0\leq r<s} . See also: Grauert's approximation theorem
Homotopy type of a mapping space A basic result here is a theorem of Milnor which says that the mapping space Map ( X , Y ) {\displaystyle \operatorname {Map} (X,Y)} has the homotopy type of a CW-complex if X {\displaystyle X} is a compact Hausdorff space and Y {\displaystyle Y} has the homotopy type of a CW-complex.
References
Hirsch, Morris (1997). Differential Topology. Springer. ISBN 0-387-90148-5. Milnor, John (1959). "On spaces having the homotopy type of CW-complex". Transactions of the American Mathematical Society. 90 (2): 272–280. doi:10.2307/1993204. JSTOR 1993204. Wall, C. T. C. (4 July 2016). Differential Topology. Cambridge University Press. ISBN 9781107153523. M. Golubitsky and V. Guillemin, Stable mappings and their singularities, x+209 pp, Springer Graduate Texts 14, Springer-Verlag, 1973.