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January 23, 2009

Lurie II

Jacob Lurie continued his series of lectures with a romp through higher category theory.

Consider a topological space, XX, and its fundamental groupoid, π 1(X)\pi_{\leq 1}(X), whose objects are points in XX and whose morphisms are paths in XX up to homotopy. This efficiently captures π 0(X)\pi_0(X) and π 1\pi_1 of each connected component of XX.

An obvious generalization is to try to define a fundamental nn-groupoid (which would capture the homotopy groups up to π n\pi_n) as an nn-category with all kk-morphisms invertible (knk\leq n).

If you try to make it a strict nn-category, you run into trouble. For instance, the “fundamental 2-groupoid” would have

  • object = points in XX
  • 1-morphism = paths in XX
  • 2-morphisms = homotopies of paths in XX up to homotopy

Unfortunately, thus-defined, the composition of 1-morphisms isn’t associative1. Instead, we should define a weak 2-category, where associativity holds only up to homotopy.

This can actually be fixed for n=2n=2, and you can show that the above weak 2-category is equivalent to a certain strict 2-category. But for n3n\geq 3, you’re basically stuck with the weak case, where associativity of the composition of kk-morphisms hold only up to coherent isomorphisms (and precisely stating those coherence relations, for weak nn-categories, becomes very cumbersome as nn increases).

In the case of nn-groupoids, however, the theory is relatively nice, and we can contemplate the \infty-groupoid, π (X)\pi_{\leq\infty}(X), which captures all information about the homotopy type of XX. In fact (this seems to be not so much a theorem, as a definition), every \infty-groupoid is realized as π (X)\pi_{\leq\infty}(X) for some topological space XX. In fact, we might go so far as to say that an \infty-groupoid “is” a topological space, XX.

In any case, the \infty-groupoids can be rechristened (,0)(\infty,0)-categories, and form the first in a series of (,n)(\infty,n)-categories, which are \infty-categories in which all the kk-morphisms, for k>nk\gt n are isomorphisms.

For present purposes, it’s (,1)(\infty,1)-categories that we’re interested in.

Conventionally, a dd-dimensional TQFT is a functor, ZZ, from the 1-category, Cob(d)Cob(d), to the 1-category VectVect. For MM a closed (d1)(d-1)-manifold, Z(M)Z(M) is a finite dimensional complex vector space. Instead, we would like to consider a generalization where Z(M)Z(M) is a chain complex of vector spaces.

If BB is dd-manifold which is a bordism from MM to NN, then Z(B)Z(B) is a chain map from Z(M)Z(N)Z(M)\to Z(N). If BB and BB' are diffeomorphic, then Z(B)Z(B) and Z(B)Z(B') are chain-homotopic. For every isotopy of diffeomorphism, we should get a chain homotopy between chain homotopies. Etc.

That is, ZZ is a functor between (,1)(\infty,1)-categories.

Note that these are “ordinary” TQFTs, not the “enriched” TQFTs discussed in the previous lecture.

Why, you might ask, are we interested in TQFTs with values in chain complexes, rather than finite-dimensional vector spaces? Well, one reason is that, in physics, that’s usually the way they come to us. With a few notable exceptions, what we typically have is an (infinite-dimensional) \mathbb{Z}-graded vector space, with a differential (usually called “QQ”). That is to say, we have a chain complex. The finite-dimensional vector space that we usually call the TQFT Hilbert space is the cohomology of this complex. Jacob’s setup is intended to capture that situation. One model of his (,1)(\infty,1) Bordism category presumably involves manifolds with Riemannian metrics, because that’s what we typically need, in physics, to define a cohomological TQFT.

1 Let x,yXx,y\in X. A path, pp, from xx to yy is a map p:[0,1]X,such thatp(0)=x,p(1)=y p: [0,1]\to X,\, \text{such that}\, p(0)=x,\, p(1)=y To compose a path pp from xx to yy with a path qq from yy to zz, we need to define something like (qp):[0,1]X,(qp)(t)={p(2t) 0t1/2 q(2t1) 1/2t1 (q\circ p): [0,1]\to X,\, (q\circ p)(t) = \begin{cases}p(2t)& 0\leq t\leq 1/2\\ q(2t-1)& 1/2\leq t \leq 1\end{cases} In the case of the fundamental groupoid, where we considered paths up to homotopy, this was no problem, but for the fundamental 2-groupoid, where the 1-morphisms are paths (not paths up to homotopy), this clearly fails to be associative. In this case, we can fix the problem by letting the interval be of arbitrary length. But that’s only a temporary respite. In the fundamental 3-groupoid of S 2S^2, we’ll encounter the Hopf map.

Posted by distler at January 23, 2009 1:16 AM

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Re: Lurie II

Hello Jacques,

what do you think about Lovelace’s paper?

It would be very interesting to read a post about that one on your blog…

Posted by: Michael on February 3, 2009 9:18 AM | Permalink | Reply to this

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