This may be a vague quesion.
I am confused between the basepointed case and non-basepointed case in algebraic topology. Is there any convenience in base pointed case? For example, it leads to the definition of smash product, which is left adjoint to the functor $Map(X,-)$ (here maps are preserving base point). In the non-basepointed case, the left adjoint will be the product. Is there any thing more behind it?
Thank you so much!
Without a specified basepoint, it makes it difficult to define the fundamental group or higher homotopy groups. In particular, loops that are freely homotopic needn't be distinct in $\pi_1(X,x_0)$; in fact, elements of $\pi_1(X,x_0)$ are freely homotopic iff they're conjugate in the group. So if we forget our basepoint, we lose our group structure. Higher (basepointed) homotopy groups are abelian, so this issue is no longer an issue; two maps $S^n \to X$ (that both preserve a chosen basepoint in $S^n$ and $X$) are basepoint-homotopic iff they're freely homotpic. But the most natural way of defining a sum of two elements - collapsing the equator of $S^n$ and composing with the map $f_1 \wedge f_2: S^n \wedge S^n \to X$ - requires that we've chosen a basepoint of $S^n$ in the first place.
As Ronnie Brown comments above, there's certainly a way to work with more than one basepoint, or without a basepoint at all; the MO question he links is helpful for this perspective.