Smash product
Encyclopedia
In mathematics
, the smash product of two pointed space
s (i.e. topological space
s with distinguished basepoints) X and Y is the quotient
of the product space X × Y under the identifications (x, y0) ∼ (x0, y) for all x ∈ X and y ∈ Y. The smash product is usually denoted X ∧ Y. The smash product depends on the choice of basepoints (unless both X and Y are homogeneous
).
One can think of X and Y as sitting inside X × Y as the subspaces X × {y0} and {x0} × Y. These subspaces intersect at a single point: (x0, y0), the basepoint of X × Y. So the union of these subspaces can be identified with the wedge sum
X ∨ Y. The smash product is then the quotient
The smash product has important applications in homotopy theory, a branch of algebraic topology
. In homotopy theory, one often works with a different category
of spaces than the category of all topological spaces. In some of these categories the definition of the smash product must be modified slightly. For example, the smash product of two CW complex
es is a CW complex if one uses the product of CW complexes in the definition rather than the product topology. Similar modifications are necessary in other categories.
s
However, for the naive category of pointed spaces, this fails. See the following discussion on MathOverflow.
These isomorphisms make the appropriate category of pointed spaces into a symmetric monoidal category with the smash product as the monoidal product and the pointed 0-sphere (a two-point discrete space) as the unit object. One can therefore think of the smash product as a kind of tensor product
in an appropriate category of pointed spaces.
make the analogy between the tensor product and the smash product more precise. In the category of R-modules
over a commutative ring
R, the tensor functor (– ⊗R A) is left adjoint to the internal Hom functor
Hom(A,–) so that:
In the category of pointed spaces, the smash product plays the role of the tensor product. In particular, if A is locally compact Hausdorff then we have an adjunction
where Hom(A,Y) is the space of based continuous maps together with the compact-open topology
.
In particular, taking A to be the unit circle
S1, we see that the suspension functor Σ is left adjoint to the loop space functor Ω.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the smash product of two pointed space
Pointed space
In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i.e. a continuous map f : X → Y such that f = y0...
s (i.e. topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
s with distinguished basepoints) X and Y is the quotient
Quotient space
In topology and related areas of mathematics, a quotient space is, intuitively speaking, the result of identifying or "gluing together" certain points of a given space. The points to be identified are specified by an equivalence relation...
of the product space X × Y under the identifications (x, y0) ∼ (x0, y) for all x ∈ X and y ∈ Y. The smash product is usually denoted X ∧ Y. The smash product depends on the choice of basepoints (unless both X and Y are homogeneous
Homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a non-empty manifold or topological space X on which G acts continuously by symmetry in a transitive way. A special case of this is when the topological group,...
).
One can think of X and Y as sitting inside X × Y as the subspaces X × {y0} and {x0} × Y. These subspaces intersect at a single point: (x0, y0), the basepoint of X × Y. So the union of these subspaces can be identified with the wedge sum
Wedge sum
In topology, the wedge sum is a "one-point union" of a family of topological spaces. Specifically, if X and Y are pointed spaces the wedge sum of X and Y is the quotient of the disjoint union of X and Y by the identification x0 ∼ y0:X\vee Y = \;/ \sim,\,where ∼ is the...
X ∨ Y. The smash product is then the quotient
The smash product has important applications in homotopy theory, a branch of algebraic topology
Algebraic topology
Algebraic topology is a branch of mathematics which uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify up to homotopy equivalence.Although algebraic topology...
. In homotopy theory, one often works with a different category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
of spaces than the category of all topological spaces. In some of these categories the definition of the smash product must be modified slightly. For example, the smash product of two CW complex
CW complex
In topology, a CW complex is a type of topological space introduced by J. H. C. Whitehead to meet the needs of homotopy theory. This class of spaces is broader and has some better categorical properties than simplicial complexes, but still retains a combinatorial naturethat allows for...
es is a CW complex if one uses the product of CW complexes in the definition rather than the product topology. Similar modifications are necessary in other categories.
Examples
- The smash product of any pointed space X with a 0-sphere is homeomorphic to X.
- The smash product of two circles is a quotient of the torusTorusIn geometry, a torus is a surface of revolution generated by revolving a circle in three dimensional space about an axis coplanar with the circle...
homeomorphic to the 2-sphere. - More generally, the smash product of two spheres Sm and Sn is homeomorphic to the sphere Sm+n.
- The smash product of a space X with a circle is homeomorphic to the reduced suspension of X:
- The k-fold iterated reduced suspension of X is homeomorphic to the smash product of X and a k-sphere
- In domain theoryDomain theoryDomain theory is a branch of mathematics that studies special kinds of partially ordered sets commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational...
, taking the product of two domains (so that the product is strict on its arguments).
As a symmetric monoidal product
For any pointed spaces X, Y, and Z in an appropriate "convenient" category (e.g. that of compactly generated spaces) there are natural (basepoint preserving) homeomorphismHomeomorphism
In the mathematical field of topology, a homeomorphism or topological isomorphism or bicontinuous function is a continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are...
s
However, for the naive category of pointed spaces, this fails. See the following discussion on MathOverflow.
These isomorphisms make the appropriate category of pointed spaces into a symmetric monoidal category with the smash product as the monoidal product and the pointed 0-sphere (a two-point discrete space) as the unit object. One can therefore think of the smash product as a kind of tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
in an appropriate category of pointed spaces.
Adjoint relationship
Adjoint functorsAdjoint functors
In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...
make the analogy between the tensor product and the smash product more precise. In the category of R-modules
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
over a commutative ring
Commutative ring
In ring theory, a branch of abstract algebra, a commutative ring is a ring in which the multiplication operation is commutative. The study of commutative rings is called commutative algebra....
R, the tensor functor (– ⊗R A) is left adjoint to the internal Hom functor
Hom functor
In mathematics, specifically in category theory, hom-sets, i.e. sets of morphisms between objects, give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics.-Formal...
Hom(A,–) so that:
In the category of pointed spaces, the smash product plays the role of the tensor product. In particular, if A is locally compact Hausdorff then we have an adjunction
where Hom(A,Y) is the space of based continuous maps together with the compact-open topology
Compact-open topology
In mathematics, the compact-open topology is a topology defined on the set of continuous maps between two topological spaces. The compact-open topology is one of the commonly-used topologies on function spaces, and is applied in homotopy theory and functional analysis...
.
In particular, taking A to be the unit circle
Unit circle
In mathematics, a unit circle is a circle with a radius of one. Frequently, especially in trigonometry, "the" unit circle is the circle of radius one centered at the origin in the Cartesian coordinate system in the Euclidean plane...
S1, we see that the suspension functor Σ is left adjoint to the loop space functor Ω.