What I'm doing for my PhD
[This is the abstract for a talk I'm giving in a couple of weeks at The Burn, which is a wonderful jolly for Scottish maths-PhD students. If you were totally confused by the paper from my last-but-one post, maybe this will help. The symbol ∈ is pronounced "in", and "x ∈ X" means that x is a member of the collection X. And a "category" can be thought of as a collection of interesting mathematical objects and the structure-preserving transformations between them.]
Consider the natural numbers N = {0, 1, 2, ...} and addition. For any p, q and r, we have that (p + q) + r = p + (q + r) (associativity) and p + 0 = 0 + p = p (0 is a unit). Similarly, for n x n matrices A, B and C, we have that (AB)C = A(BC) and IA = AI = A. We summarize these statements by saying that (N, +, 0) and (Mn(ℜ), . , I) are monoids. Any theorem proved about monoids therefore applies in both of these cases (and many more). As mathematicians are by nature Lazy, this kind of abstraction is a Good Thing. Hence, the theory of monoids is one of a large number of such theories: other examples are the theories of groups, rings, vector spaces, Lie algebras, rigs, crossed monoids, semigroups ...
In order to save effort, and to bring some order to this chaos, the field of universal algebra was invented. Universal algebra attempts to work with arbitrary algebraic theories, and prove general results that hold in all such cases.
In the last forty years, it has become increasingly apparent that the traditional "one-dimensional" theories are only the tip of a much larger iceberg, which is called higher-dimensional algebra. Higher-dimensional analogues of monoids, groups, Lie algebras, etc, have all been defined, in which the defining equations only hold "up to isomorphism". For instance, consider three sets, A, B and C. The repeated Cartesian product A x (B x C) = { (a, (b, c)) : a ∈ A, b ∈ B, c ∈ C } is not equal to (A x B ) x C = { ((a, b), c) : a ∈ A, b ∈ B, c ∈ C }, but they are isomorphic (can be transformed into each other without loss or gain of information) in an obvious way, and similarly {*} x A = {(*,a) : a ∈ A} ≅ A. We say that Set is a weak monoidal category, but we could say that it's a higher-dimensional sort of monoid.
Defining these higher-dimensional structures one at a time is far too much effort: what's needed is a higher-dimensional sort of universal algebra, that can take a one-dimensional theory and give us its higher-dimensional analogue. Unfortunately, this is less easy than it sounds. We can't just replace sets with categories, functions with functors and equations with natural isomorphisms: then we'd have too many isomorphisms around, and our theory would be "incoherent", which would be tedious and messy. To get a useful theory, the isomorphisms themselves must satisfy equations. Finding these equations is an example of a "coherence problem", and I am trying to solve the general coherence problem of this type.
Consider the natural numbers N = {0, 1, 2, ...} and addition. For any p, q and r, we have that (p + q) + r = p + (q + r) (associativity) and p + 0 = 0 + p = p (0 is a unit). Similarly, for n x n matrices A, B and C, we have that (AB)C = A(BC) and IA = AI = A. We summarize these statements by saying that (N, +, 0) and (Mn(ℜ), . , I) are monoids. Any theorem proved about monoids therefore applies in both of these cases (and many more). As mathematicians are by nature Lazy, this kind of abstraction is a Good Thing. Hence, the theory of monoids is one of a large number of such theories: other examples are the theories of groups, rings, vector spaces, Lie algebras, rigs, crossed monoids, semigroups ...
In order to save effort, and to bring some order to this chaos, the field of universal algebra was invented. Universal algebra attempts to work with arbitrary algebraic theories, and prove general results that hold in all such cases.
In the last forty years, it has become increasingly apparent that the traditional "one-dimensional" theories are only the tip of a much larger iceberg, which is called higher-dimensional algebra. Higher-dimensional analogues of monoids, groups, Lie algebras, etc, have all been defined, in which the defining equations only hold "up to isomorphism". For instance, consider three sets, A, B and C. The repeated Cartesian product A x (B x C) = { (a, (b, c)) : a ∈ A, b ∈ B, c ∈ C } is not equal to (A x B ) x C = { ((a, b), c) : a ∈ A, b ∈ B, c ∈ C }, but they are isomorphic (can be transformed into each other without loss or gain of information) in an obvious way, and similarly {*} x A = {(*,a) : a ∈ A} ≅ A. We say that Set is a weak monoidal category, but we could say that it's a higher-dimensional sort of monoid.
Defining these higher-dimensional structures one at a time is far too much effort: what's needed is a higher-dimensional sort of universal algebra, that can take a one-dimensional theory and give us its higher-dimensional analogue. Unfortunately, this is less easy than it sounds. We can't just replace sets with categories, functions with functors and equations with natural isomorphisms: then we'd have too many isomorphisms around, and our theory would be "incoherent", which would be tedious and messy. To get a useful theory, the isomorphisms themselves must satisfy equations. Finding these equations is an example of a "coherence problem", and I am trying to solve the general coherence problem of this type.