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
Introductions
Definitions
Paths and cylinders
Homotopy groups
Basic facts
Theorems
Given two maps between homotopy types with a common codomain, and , we denote by the pushout of their pullback, . Then the join of and , written , is the induced map from to .
When , the unit type, we write as , the join of the two types.
The join of two maps is the fiberwise join of their respective fibres.
The sequential colimit of the join powers is equivalent to the image inclusion of .
When , the infinite join power is the propositional truncation of .
When , the point in a delooped group, 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 is a unit for the join operation.
The join of two mere propositions, , is equal to the propositional truncation of their sum, .
Last revised on June 11, 2022 at 16:35:39. See the history of this page for a list of all contributions to it.