Lorenzo Riva

There used to be a little blurb here about what my research is about, but lately I've been thinking about a lot of different things that are somewhat interconnected but that, together, do not really form a cohesive research plan. I spend a lot of time thinking about higher categories, both pluri-dimensional (like n-categories) and homotopical (like (∞,1)-categories): how to build them, their model-independent properties, what they tell us about low-dimensional phenomena, and so on. I am also motivated by manifold topology, logic (categorical and not), and combinatorics. Here are some questions I am pondering:

• Is there a simplicial space model for (∞,n)-categories that resembles n-complicial sets? The combinatorics of orientals, which dictates the lifting properties of complicial sets, is pretty complicated; can it be simplified so as to have one "Segal style" condition in every dimension?

• How do we freely add adjoints to k-morphisms in an n-category? The cobordism & tangle hypotheses provide an answer for very simple n-categories as long as we add all adjoints at once. What if we only want to add some adjoints? Do we still get cobordisms?

• Given a (complete) Segal space, its colimit computes the completion of the associated (∞,1)-category, i.e. the space obtained by inverting all of its morphisms. This space is initial with respect to maps out of the (∞,1)-category that land into the invertible part of the target. Since adjoints can be thought of as lax inverses, is there a context in which the process of freely adding adjoints to an (∞,1)-category can be expressed in terms of a (possibly lax) colimit?