Homotopy Type Theory
loop space of a wedge of circles

Let $A$ be a set, and consider the HIT$W_A$ with constructors

$base:W_A$

$loop : A \to (base = base)$.

This is the “wedge of $A$ circles”. It can also be defined as the suspension of $A+1$.

Known facts

If $A$ has decidable equality?, then the loop space$\Omega W_A$ is the free group on $A$.

For general $A$, it is probably still true that the fundamental group?$\pi_1 W_A$ is the free group on $A$.

An open problem

For a general set $A$ without decidable equality, is $W_A$ even a 1-type? What can be said about $\Omega W_A$?

In the classical simplicial set model?, $W_A$ is a 1-type for all $A$, since classically all sets have decidable equality. Moreover, since $W_A$ is definable as a colimit, and being a 1-type is a finite-limit property, this fact is inherited by all Grothendieck (infinity,1)-topoi. Thus, no counterexample to $W_A$ being a 1-type can be found in such models, but it is also not clear how to prove that $W_A$ is a 1-type.