Musings

Consider a topological space, $X$, and its fundamental groupoid, ${\pi }_{\le 1}(X)$, whose objects are points in $X$ and whose morphisms are paths in $X$ up to homotopy. This efficiently captures ${\pi }_0(X)$ and ${\pi }_1$ of each connected component of $X$.

An obvious generalization is to try to define a fundamental $n$-groupoid (which would capture the homotopy groups up to ${\pi }_n$) as an $n$-category with all $k$-morphisms invertible ($k\le n$).

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

• object = points in $X$
• 1-morphism = paths in $X$
• 2-morphisms = homotopies of paths in $X$ 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 $n=2$, and you can show that the above weak 2-category is equivalent to a certain strict 2-category. But for $n\ge 3$, you’re basically stuck with the weak case, where associativity of the composition of $k$-morphisms hold only up to coherent isomorphisms (and precisely stating those coherence relations, for weak $n$-categories, becomes very cumbersome as $n$ increases).

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

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

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

Conventionally, a $d$-dimensional TQFT is a functor, $Z$, from the 1-category, $\mathrm{Cob}(d)$, to the 1-category $\mathrm{Vect}$. For $M$ a closed $(d-1)$-manifold, $Z(M)$ is a finite dimensional complex vector space. Instead, we would like to consider a generalization where $Z(M)$ is a chain complex of vector spaces.

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

That is, $Z$ is a functor between $(\mathrm{\infty },1)$-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 “$Q$”). 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 $(\mathrm{\infty },1)$ Bordism category presumably involves manifolds with Riemannian metrics, because that’s what we typically need, in physics, to define a cohomological TQFT.