So if you have a pointed space (X,x), a non-pointed space Y, and a continuous map f: X -> Y, then there is a unique point y in Y such that f is a pointed continuous map (X,x) -> (Y,y) (namely if we take y = f(x)). This phenomenon feels similar to taking a Kleisli extension of a morphism A -> T(B) for some monad T, but among other differences it goes the other direction.
Let U: Topₚ -> Top be the forgetful functor from the category of pointed topological spaces to the category of topological spaces. For every f ∈ Top(U(X),Y) there is a unique Z ∈ Obj(Topₚ) and g ∈ Topₚ(X,Z) such that U(Z) = Y and U(g) = f.
This is used pretty frequently in algebraic topology (‘let X,Y be spaces, x ∈ X, and let f: X -> Y be continuous, then f*: π₁(X,x) -> π₁(Y,f(x)), etc.’). I wonder if there’s other scenarios that have the same categorical structure?
isn’t this just like. a cofibration.