Category of Pointed Sets

The category of pointed sets is a category $\mathrm{Sets}{.}_{\bullet}$ where

• The objects are Pointed H-Sets
• The morphisms are functions that preserve the designated base point

Properties

• The category of pointed sets is equivalent to the Coslice Category $1/\mathrm{Sets}{.}$.

• The Unit Type with obvious choice of point is a Zero Object

• The forgetful functor $U:\mathrm{Sets}{.}_{\bullet}\to\mathrm{Sets}{.}$ has a Left Adjoint that freely adds a point: the resulting Monad is the Maybe Monad.

• The category of pointed sets has Indexed Coproducts given by the Wedge Sum of Pointed Sets.

• The category of pointed sets has Indexed Cartesian Products, where the basepoint of $\prod_{i:I}X_{i}$ is given by the function $\lambda i.\;\mathrm{pt}_{i}$.

• The category of pointed sets is equipped with a Symmetric Monoidal tensor product, known as the Smash Product of Pointed Sets.