Is there a meaningful difference between biased and unbiased composition?

Question

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 review of Leinster's book Higher Operads, Higher Categories (arXiv:math/0305049, doi:10.1017/CBO9780511525896) (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?

Answer 1

As you probably know, I definitely agree that the difference is technical rather than foundational. That said, the technicalities can be formidable. There is an initial hurdle to get over in starting out with unbiased things, in that you have to have infinitely many operations and axioms, but once you're past that, I agree that the operations and axioms are usually much more natural and proofs happen much more easily.

My favorite example is that while it's true that biased and unbiased monoidal categories are equivalent, the proof of that requires essentially the entire strength of Mac Lane's coherence theorem! The coherence theorem for unbiased monoidal categories can be shown via 2-categorical abstract nonsense about 2-monads; basically all the difficulty comes from insisting on a biased definition. I also think that unbiased definitions often lead to more natural categorifications. For instance, Leinster's unbiased notion of "contractible globular operad" fits nicely into a language of weak factorization systems, while I'm not sure if the same is true of Batanin's original biased version.

There is certainly an unbiased definition of Lie algebra, since Lie algebras are the algebras for an operad. Whether that unbiased definition admits a nice presentation in terms of n-ary operations, I don't know.

Answer 2

Certainly unbiased definitions are the norm in modern homotopy theory. I guess an example of a biased definition is the (original?) definition by Stasheff of an $A_\infty$ space—the homotopy theorist's monoid. (For simplicity, I'll ignore matters related to the unit.) Informally, it consists of a space $X$, a multiplication $\mu : X \times X \to X$, a homotopy $\alpha$ between the maps $\mu(\mu(x,y),z)$ and $\mu(x,\mu(y,z))$ from $X \times X \times X$ to $X$, a "pentagonator" $\pi$ which extends a map $S^1 \times X^4 \to X$ built from $\mu$ and $\alpha$ to $D^2 \times X^4$, and so on. This seems to be the appropriate generalization of your biased definition of monoids to the situation where we need infinitely many higher coherence homotopies. More precisely, an $A_\infty$ space under this definition consists of a space $X$ together with maps $\mu_n : A_n \times X^n \to X$, where $A_n$ is the $n$th associahedron—a certain convex polyhedron in $\mathbb{R}^{n-2}$—and on the boundary of $A_n$, $\mu_n$ is required to be equal to a certain expression built out of the $\mu_k$ for $k$ smaller than $n$. Thus, we may think of it as a null-homotopy of a prescribed map $\partial A_n \times X^n \to X$.

This definition is quite complicated combinatorially, since it depends on the polyhedra $A_n$. Peter May realized that certain features of the $A_n$ could be abstracted into the notion of a (non-symmetric) operad. I don't want to define exactly what an operad is, but instead just observe the change in perspective: rather than thinking of $A_n$ as a space which encodes a null-homotopy of a map defined on its boundary, we think of the points of $A_n$ as describing various $n$-ary multiplications on $X$ via the map $\mu_n : A_n \to \mathrm{Map}(X^n, X)$. The contractibility of $A_n$ guarantees that there is essentially just one way to multiply $n$-tuples. So we see that the same definition can be viewed as either biased or unbiased. However, in the world of operads, we are free to consider algebras over any operad whose $n$th space is contractible for all $n$, and any such space is an $A_\infty$ algebra. This definition no longer has the combinatorial complexity of the associahedra, although operads require a certain amount of machinery. I would call it an unbiased definition. From this point of view, the biased definition consists of giving a certain kind of homotopical presentation of the terminal operad, which is a useful thing to have, but quite unnecessary from the point of view of giving a definition!

Since homotopy theory is "just the easy part of ∞-category theory", I expect that as higher-categorical structures become more commonplace, we will see a similar phenomenon: the easy definitions to write down and work with abstractly are the unbiased ones, and biased definitions are certain presentations of the unbiased notions, which might become quite complicated as the categorical level increases.