Archive for the ‘higher category theory’ Category

Often in mathematics we define constructions outputting objects which a priori have a certain amount of structure but which end up having more structure than is immediately obvious. For example:

  • Given a Lie group G, its tangent space T_e(G) at the identity is a priori a vector space, but it ends up having the structure of a Lie algebra.
  • Given a space X, its cohomology H^{\bullet}(X, \mathbb{Z}) is a priori a graded abelian group, but it ends up having the structure of a graded ring.
  • Given a space X, its cohomology H^{\bullet}(X, \mathbb{F}_p) over \mathbb{F}_p is a priori a graded abelian group (or a graded ring, once you make the above discovery), but it ends up having the structure of a module over the mod-p Steenrod algebra.

The following question suggests itself: given a construction which we believe to output objects having a certain amount of structure, can we show that in some sense there is no extra structure to be found? For example, can we rule out the possibility that the tangent space to the identity of a Lie group has some mysterious natural trilinear operation that cannot be built out of the Lie bracket?

In this post we will answer this question for the homotopy groups \pi_n(X) of a space: that is, we will show that, in a suitable sense, each individual homotopy group \pi_n(X) is “only a group” and does not carry any additional structure. (This is not true about the collection of homotopy groups considered together: there are additional operations here like the Whitehead product.)


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In the previous post we learned that it is possible to recover the center Z(R) of a ring R from its category R\text{-Mod} of left modules (as an \text{Ab}-enriched category). For commutative rings, this justifies the idea that it is sensible to study a ring by studying its modules (since the modules know everything about the ring).

For noncommutative rings, the situation is more interesting. Two rings R, S are said to be Morita equivalent if the categories R\text{-Mod}, S\text{-Mod} are equivalent as \text{Ab}-enriched categories. As it turns out, there exist examples of rings which are non-isomorphic but which are Morita equivalent, so Morita equivalence is a strictly coarser equivalence relation on rings than isomorphism. However, many important properties of a ring are invariant under Morita equivalence, and studying Morita equivalence offers an interesting perspective on rings on general.

Moreover, Morita equivalence can be thought of in the context of a fascinating larger structure, the bicategory of bimodules, which we briefly describe.


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The center Z(G) of a group is an interesting construction: it associates to every group G an abelian group Z(G) in what is certainly a canonical way, but not a functorial way: that is, it doesn’t extend (at least in any obvious way) to a functor \text{Grp} \to \text{Ab} (unlike the abelianization G/[G, G]). We might wonder, then, exactly what kind of construction the center is.

Of course, it is actually not hard to come up with a rather general example of a canonical but not functorial construction: in any category C we may associate to an object c \in C its automorphism group \text{Aut}(c) or endomorphism monoid \text{End}(c)), and this is a canonical construction which again doesn’t extend in an obvious way to a functor C \to \text{Grp} or C \to \text{Mon}. (It merely reflects some special part of the bifunctor \text{Hom}(-, -).)

It turns out that the center can actually be thought of in terms of automorphisms (or endomorphisms), not of a group G, but of the identity functor G \to G, where G is regarded as a category with one object. This definition generalizes, and the resulting general definition has some interesting specializations. Moreover, an important general property is that the center is always abelian, and this has a very elegant proof.


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