An autohomeomorphism of the Čech--Stone remainder $\beta N\setminus N$ is called trivial if it has a continuous extension to a map from $\beta N$ into itself. Such map is determined by an almost permutation, which is a bijection between cofinite subsets of $N$. By results of W. Rudin and S. Shelah, the question whether nontrivial autohomeomorphisms of $\beta N\setminus N$ exist is independent from ZFC. We will be considering the so-called rotary autohomeomorphisms. An autohomeomorphism is called rotary if it corresponds to a permutation of $N$ all of whose cycles are finite. If all autohomeomorphisms are trivial, then the problem of their conjugacy is also trivial (in the usual sense of the word). However the Continuum Hypothesis makes the conjugacy relation nontrivial. While our results are somewhat incomplete, they suffice to decide whether for example the rotary autohomeomorphisms whose cycles have lengths $2^{2n}$, for $n\in N$, and $2^{2n+1}$, for $n\in N$, are conjugate. This is a joint work with Will Brian.

Two ideas underpin categorical logic: first, that a theory can be identified with its associated syntactic category; secondly, that the set-theoretic models of a theory can be identified with functors from its syntactic category to the category of sets and functions preserving suitable structure. This point of view is useful because it often helps us to establish existence of free models. After reviewing these ideas, I will present joint work with Giacomo Tendas on variant in the context of 2-dimensional category theory, in which we are interested in categories, rather than sets, equipped with additional structure (often subject to universal properties). Extra subtleties emerge, as one needs to deal with a form of quantification in between `there exists’ and `there exists a unique’. As an application, we obtain a variety of free 2-categorical constructions, including Joyal’s free bicompletions.