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The join construction, Egbert Rijke, 26 Jan 2017, arXiv:1701.07538

Abstract

In homotopy type theory we can define the join of maps? as a binary operation on maps? with a common co-domain. This operation is commutative, associative, and the unique map from the empty type into the common codomain is a neutral element. Moreover, we show that the idempotents of the join of maps are precisely the embeddings?, and we prove the ‘join connectivity theorem’, which states that the connectivity of the join of maps equals the join of the connectivities of the individual maps. We define the image of a map $f:A \to X$ in $U$ via the join construction, as the colimit? of the finite join powers of $f$. The join powers therefore provide approximations of the image inclusion, and the join connectivity theorem implies that the approximating maps into the image increase in connectivity. A modified version of the join construction can be used to show that for any map $f:A\to X$ in which $X$ is only assumed to be locally small, the image is a small type. We use the modified join construction to give an alternative construction of set-quotients?, the Rezk completion of a precategory, and we define the $n$-truncation? for any $n: \mathbb{N}$. Thus we see that each of these are definable operations on a univalent universe for Martin-Löf type theory with a natural numbers object, that is moreover closed under homotopy coequalizers?.