The String Coffee Table

Monday at the Streetfest I

Posted by Guest

While others report from conferences on Strings or Loops, in Sydney we are enjoying the balmy winter and the calls of the cockatoos and lorakeets. The Streetfest, in honour of Ross Street’s 60th birthday, got under way yesterday. Day one was very maths oriented.

The first talk was by Joyal, a well known collaborator of Ross Street. He spoke about quasicategories; an amazingly clear speaker.

The theory of quasicategories is developed as an extension of both ordinary category theory and homotopical algebra (a la Quillen). The hope is that it will yield insights into the development of both higher category theory and homotopical algebra.

Let $S$ be the category of simplicial sets, that is functors from $\Delta^{0}$ into $\mathbf{Set}$. A simplicial set is called a quasicategory if every horn $$\Lambda^{k}(n) \to X \qquad 0 \lt k \lt n$$ can be filled.

An example of a quasicategory is then a Kan complex.

The left adjoint $\tau$ of the nerve functor $N: \mathbf{Cat} \to S$ preserves products, so can be used to give a 2-category structure on $S$. The Hom category is given by $\tau (B^{A})$ and the product gives the composition law. This 2-categorical structure is used to define equivalences of quasicategories. The category $\mathbf{QCat}$ of quasicategories is Cartesian closed.

As an application of this, whereas in $\mathbf{Cat}$ one looks at discrete fibrations where $X^{\mathbf{2}} \to Y^{\mathbf{2}} \times_Y X$ is an isomorphism, in the extension to $\mathbf{QCat}$ one demands that this is a trivial fibration. Think of fibres of fibrations being Kan complexes.

Next, Borceux spoke about semi-direct products and the representability (in a categorical sense) of actions. Later in the day Johnstone followed this with a talk on his work on bi-Heyting toposes. He had been trying to prove a conjecture: that $E$ bi-Heyting implies “there exists an essential surjection $B \to E$ for $B$ Boolean. Instead he found a counterexample: the sheaves on $[0,1]$ as a complete Heyting algebra.

I’d like to talk about Borceux’s talk - but no time now. If there are any real category theorists reading this who would like to correct any misunderstandings - please comment! Lectures about to start on Tuesday…so I’m off.

Marni Sheppeard

Addendum from David: the talk has started, Marni’s run off, and I’m just trying to tidy this up - failed, I know, so maybe it can be fixed later.

D