Given two maps between homotopy types with a common codomain, $f: A \to X$ and $g: B \to X$, we denote by $A \ast_X B$ the pushout of their pullback, $A \times_X B$. Then the join of $f$ and $g$, written $f \ast g$, is the induced map from $A \ast_X B$ to $X$.

When $X \equiv \mathbf{1}$, the unit type, we write $A \ast_{\mathbf{1}} B$ as $A \ast B$, the join of the two types.

The join of two maps is the fiberwise join of their respective fibres.

Examples

The sequential colimit$f \ast^{\infty}$ of the join powers $f \ast^{n}$ is equivalent to the image inclusion of $f$.

When $X \equiv \mathbf{1}$, the infinite join power $A \ast^{\infty}$ is the propositional truncation of $A$.

When $pt: \mathbf{1} \to \mathbf{B} G$, the point in a delooped group, $pt \ast^{\infty}$ is the Milnor construction.

The join of two embeddings is an embedding, and an embedding is idempotent for the join construction.

The unique map of type $\mathbf{0} \to X$ is a unit for the join operation.