symmetric monoidal (∞,1)-category of spectra
Let be a (commutative) ring. Then a pointed -module is an -module equipped with the further choice of an element . A homomorphism of pointed modules is a homomorphism of the underlying modules which preserves the choice of the extra points.
If is a field, then pointed -modules are also called pointed -vector spaces.
If is the integers then pointed -modules are also called pointed abelian groups.
Equivalently, a pointed -module (Def. ) is an -module equipped with an -linear map (which is fixed by its image of ). Since is the tensor unit for the tensor product of modules in the monoidal category of -modules and -linear maps, this means that pointed -modules are equivalently pointed objects in the monoidal-categorical sense (in addition to their underlying sets being pointed objects in the set-theoretic sense). Accordingly, the category of pointed modules is equivalently the coslice category of under (the underlying -module of) .
Last revised on August 7, 2023 at 14:20:16. See the history of this page for a list of all contributions to it.