In algebraic topology, the nth symmetric product of a topological space consists of the unordered n-tuples of its elements. If one fixes a basepoint, there is a canonical way of embedding the lower-dimensional symmetric products into the higher-dimensional ones. That way, one can consider the colimit over the symmetric products, the infinite symmetric product. This construction can easily be extended to give a homotopy functor. From an algebraic point of view, the infinite symmetric product is the free commutative monoid generated by the space minus the basepoint, the basepoint yielding the identity element. That way, one can view it as the abelian version of the James reduced product. One of its essential applications is the Dold-Thom theorem, stating that the homotopy groups of the infinite symmetric product of a connected CW complex are the same as the reduced homology groups of that complex. That way, one can give a homotopical definition of homology.
Definition Let X be a topological space and n ≥ 1 a natural number. Define the nth symmetric product of X or the n-fold symmetric product of X as the space
SP n ( X ) = X n / S n . {\displaystyle \operatorname {SP} ^{n}(X)=X^{n}/S_{n}.}
Here, the symmetric group Sn acts on Xn by permuting the factors. Hence, the elements of SPn(X) are the unordered n-tuples of elements of X. Write [x1, ..., xn] for the point in SPn(X) defined by (x1, ..., xn) ∈ Xn. Note that one can define the nth symmetric product in any category where products and colimits exist. Namely, one then has canonical isomorphisms φ : X × Y → Y × X for any objects X and Y and can define the action of the transposition ( k k + 1 ) ∈ S n {\displaystyle (k\ k+1)\in S_{n}} on Xn as Id k − 1 × ϕ × Id n − k − 1 {\displaystyle \operatorname {Id} ^{k-1}\times \phi \times \operatorname {Id} ^{n-k-1}} , thereby inducing an action of the whole Sn on Xn. This means that one can consider symmetric products of objects like simplicial sets as well. Moreover, if the category is cartesian closed, the distributive law X × (Y ∐ Z) ≅ X × Y ∐ X × Z holds and therefore one gets
SP n ( X ⨿ Y ) = ∐ k = 0 n SP k ( X ) × SP n − k ( Y ) . {\displaystyle \operatorname {SP} ^{n}(X\amalg Y)=\coprod _{k=0}^{n}\operatorname {SP} ^{k}(X)\times \operatorname {SP} ^{n-k}(Y).}
If (X, e) is a based space, it is common to set SP0(X) = {e}. Further, Xn can then be embedded into Xn+1 by sending (x1, ..., xn) to (x1, ..., xn, e). This clearly induces an embedding of SPn(X) into SPn+1(X). Therefore, the infinite symmetric product can be defined as
SP ( X ) = colim SP n ( X ) . {\displaystyle \operatorname {SP} (X)=\operatorname {colim} \operatorname {SP} ^{n}(X).}
A definition avoiding category theoretic notions can be given by taking SP(X) to be the union of the increasing sequence of spaces SPn(X) equipped with the direct limit topology. This means that a subset of SP(X) is open if and only if all its intersections with the SPn(X) are open. We define the basepoint of SP(X) as [e]. That way, SP(X) becomes a based space as well. One can generalise this definition as well to pointed categories where products and colimits exist. Namely, in this case one has a canonical map Xn → Xn+1, induced by the identity Xn → Xn and the zero map Xn → X. So this results in a direct system of the symmetric products, too and one can therefore define its colimit as the infinite symmetric product.
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