join of maps



Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




Given two maps between homotopy types with a common codomain, f:AXf: A \to X and g:BXg: B \to X, we denote by A* XBA \ast_X B the pushout of their pullback, A× XBA \times_X B. Then the join of ff and gg, written f*gf \ast g, is the induced map from A* XBA \ast_X B to XX.

When X1X \equiv \mathbf{1}, the unit type, we write A* 1BA \ast_{\mathbf{1}} B as A*BA \ast B, the join of the two types.

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


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

When X1X \equiv \mathbf{1}, the infinite join power A* A \ast^{\infty} is the propositional truncation of AA.

When pt:1BGpt: \mathbf{1} \to \mathbf{B} G, the point in a delooped group, pt* 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 0X\mathbf{0} \to X is a unit for the join operation.

The join of two mere propositions, P*QP \ast Q, is equal to the propositional truncation of their sum, ||P+Q||\vert \vert P + Q \vert \vert.


Last revised on April 16, 2017 at 06:17:00. See the history of this page for a list of all contributions to it.