Cellular Cohomology in Homotopy Type Theory

Wepresent a development of cellular cohomology in homotopy type theory. Cohomology associates to each space a sequence of abelian groups capturing part of its structure, and has the advantage over homotopy groups in that these abelian groups of many common spaces are easier to compute in many cases. Cellular cohomology is a special kind of cohomology designed for cell complexes: these are built in stages by aŠaching spheres of progressively higher dimension, and cellular cohomology de€nes the groups out of the combinatorial description of how spheres are aŠached. Our main result is that for €nite cell complexes, a wide class of cohomology theories (including the ones de€ned through Eilenberg-MacLane spaces) can be calculated via cellular cohomology. Œis result was formalized in the Agda proof assistant. ACKNOWLEDGMENTS Œis research was sponsored by the National Science Foundation under grant number DMS-1638352 and the Air Force Oce of Scienti€c Research under grant number FA9550-15-1-0053. Œe authors would also like to thank the Isaac Newton Institute for Mathematical Sciences for its support and hospitality during the program “Big Proof” when part of work on this paper was undertaken; the program was supported by Engineering and Physical Sciences Research Council under grant number EP/K032208/1. Œe views and conclusions contained in this document are those of the authors and should not be interpreted as representing the ocial policies, either expressed or implied, of any sponsoring institution, government or any other entity.