Topological spaces can be studied by breaking them into building blocks, called n-types, using a classical construction in homotopy theory, the Postnikov decomposition. The desire to model algebraically the building blocks of spaces was one of the motivations for the development of higher groupoids, generalizing the fundamental groupoid of a space. In this talk I will first illustrate how this naturally leads to the need to encode weakly associative and weakly unital compositions of higher morphisms in a higher groupoid and I will discuss the challenges that this poses.

More generally, structures arising in mathematical physics, namely topological quantum field theories, call for the need to define a notion of higher category, in which higher morphisms are not necessarily invertible.

The precise formalization of the notions of higher groupoids and higher categories can be achieved through several combinatorial machineries. I will introduce one of the approaches arising from homotopy theory, based on the notion of multisimplicial sets. I will finally briefly discuss why this approach is promising in terms of proving a long-standing open conjecture in higher category theory.