Localization in Homotopy Type Theory
Abstract
We study localization at a prime in homotopy type theory, using self maps of the circle. Our main result is that for a pointed, simply connected type $X$, the natural map $X \to X_{(p)}$ induces algebraic localizations on all homotopy groups. In order to prove this, we further develop the theory of reflective subuniverses. In particular, we show that for any reflective subuniverse $L$, the subuniverse of $L$separated types is again a reflective subuniverse, which we call $L'$. Furthermore, we prove results establishing that $L'$ is almost left exact. We next focus on localization with respect to a map, giving results on preservation of coproducts and connectivity. We also study how such localizations interact with other reflective subuniverses and orthogonal factorization systems. As key steps towards proving the main theorem, we show that localization at a prime commutes with taking loop spaces for a pointed, simply connected type, and explicitly describe the localization of an EilenbergMac Lane space $K(G,n)$ with $G$ abelian. We also include a partial converse to the main theorem.
 Publication:

arXiv eprints
 Pub Date:
 July 2018
 arXiv:
 arXiv:1807.04155
 Bibcode:
 2018arXiv180704155C
 Keywords:

 Mathematics  Algebraic Topology;
 Mathematics  Category Theory;
 55P60 (Primary);
 18E35;
 03B15 (Secondary)
 EPrint:
 32 pages