In higher category theory, there are notions of biased and unbiased definitions of composition of $n$-morphisms (or, as a special case, tensor products of objects). In the biased framework, we define what it means to have a $2$-fold composition of $n$-morphisms as well as the $0$-fold composition (i.e., the identity $n$-morphism), and then we add in some sort of coherent associativity relating the various ways of composing $k$ $n$-morphisms for all $k$. In the unbiased framework, we define, for all $k$, the notion of a $k$-fold composition of morphisms, and then give compatible isomorphisms between a $k$-fold composition and the various composites of smaller-fold compositions.

The traditional definitions of mathematics lean strongly towards the biased framework. Perhaps the simplest example is a group (or monoid), which is usally defined as having a binary operation ($2$-fold product) and identity ($0$-fold product), with the condition that $(xy)z = x(yz)$. Of course, this definition is easier to write out than an unbiased one. In the unbiased world, we would define a group to have a $k$-fold product for all $k$ and then say that for any $k_1, k_2, \ldots, k_r \in \mathbb{N}$ such that $\sum_{i = 1}^r k_i = k$, and any $x_{11}, \ldots, x_{1k_1}, x_{21}, \ldots, x_{2k_2}, \ldots, x_{r1}, \ldots, x_{rk_r}$, the $k$-fold product of the $x_{ij}$ is equal to the $r$-fold product of the $k_i$-fold products of the $x_{ij}$. However, the relative simplicity of biased definitions seems to vanish as one moves up the $n$-categorical ladder, as more and more complicated coherence axioms are required for fully weak notions of composition of $n$-morphisms.

It seems that in many cases, an unbiased definition feels more natural. For instance, when defining tensor products of modules, we use a universal property, and this universal property definition works just as well for any number of modules as it does for two. Moreover, the universal property immediately provides the relevant maps from the $k$-fold tensor product to composites of smaller-fold tensor products, whereas I'm not sure of an obvious direct way to give an isomorphism $(X \otimes Y) \otimes Z \to X \otimes (Y \otimes Z)$ without using elements or at least implicitly going through the three-fold tensor product. 

Ross Street, in his <a href="http://www.ams.org/mathscinet-getitem?mr=2094071">review</a> of Leinster's <a href="http://front.math.ucdavis.edu/0305.5049"><i>Higher Operads, Higher Categories</i></a> (which is a good reference for biased and unbiased definitions), seems to imply that the difference between unbiased and biased notions is more technical than foundational. Is this the case, or are some concepts of tensor product or morphism really more suited to a biased or an unbiased interpretation? It seems, for instance, that the theory of Lie algebras of Lie groups is rather firmly planted in a biased definition of group, as it relates to the failure of $2$-fold products to commute. Is there a formulation of Lie algebras that is unbiased? Do biased or unbiased definitions better lend themselves to categorification?

EDIT: I should probably clarify what I mean by technical vs. foundational. I imagine any biased gadget should be equivalent to an unbiased one and vice versa, so I'm not envisaging the creation of new types of objects by taking an unbiased viewpoint. Rather, I'm asking if the unbiased viewpoint can provide fundamental insights that the biased viewpoint cannot (or vice versa).

An example of such a foundational reformulation would be the use of the functor of points in algebraic geometry. That you can view schemes in terms of their functors of points is a triviality, but this change of reference frame is more than a technical convenience; in fact, it buys us a great deal. For instance, it leads to the theory of algebraic stacks. It might have been possible for algebraic stacks to have been developed in the absence of the functor of points, but I imagine that it would have taken much longer and would be less elegant and more inaccessible.

So my overarching question is, might viewing composition in one way or the other be more than just a technical convenience?