nLab loop space of a wedge of circles

Idea

This page considers the homotopy type of the loop space of a wedge sum of circles. For the moment, the following is entirely geared towards the language of homotopy type theory.

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.

category: homotopy theory