Topologists love assigning algebraic invariants to spaces - for instance, homotopy groups, homology groups, or cohomology rings. These invariants give fragments of information about the space. Grothendieck's homotopy hypothesis is the claim that one can construct a complete algebraic invariant for spaces, called the ``fundamental $\infty$-groupoid,'' of which all other algebraic invariants are shadows. In this talk, we will introduce $\infty$-groupoids and their relation to the more well-known algebraic invariants. We will then discuss their ``strictifications'' and show that when equipped with a coalgebra structure, these allow one to encode all information about a space (up to homotopy equivalence) using homological information.

No particular background will be assumed, though knowing the definition of the fundamental group will be helpful.