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The smash product is itself a pointed space, with basepoint being the equivalence class of (x 0, y 0). The smash product is usually denoted X ∧ Y or 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 × {y 0} and {x 0} × Y.
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A sphere of influence (SOI) in astrodynamics and astronomy is the oblate spheroid-shaped region where a particular celestial body exerts the main gravitational influence on an orbiting object. This is usually used to describe the areas in the Solar System where planets dominate the orbits of surrounding objects such as moons , despite the ...
The smash product of spectra extends the smash product of CW complexes. It makes the stable homotopy category into a monoidal category; in other words it behaves like the (derived) tensor product of abelian groups. A major problem with the smash product is that obvious ways of defining it make it associative and commutative only up to homotopy.
The smash product of two pointed spaces is essentially the quotient of the direct product and the wedge sum. We would like to say that the smash product turns the category of pointed spaces into a symmetric monoidal category with the pointed 0-sphere as the unit object, but this is false for general spaces: the associativity condition might fail.
A further construction in A 1-homotopy theory is the category SH(S), which is obtained from the above unstable category by forcing the smash product with G m to become invertible. This process can be carried out either using model-categorical constructions using so-called G m -spectra or alternatively using infinity-categories.
For example, if E is the trivial bundle , then is and, writing + for B with a disjoint basepoint, () is the smash product of + and ; that is, the n-th reduced suspension of +. Alternatively, [ citation needed ] since B is paracompact, E can be given a Euclidean metric and then T ( E ) {\displaystyle T(E)} can be defined as the quotient of ...