Coming from logic background, the connection of topos theory and geometry motivates my study in algebraic geometry and algebraic topology in my first year graduate study. I'm interested in the intersection of mathematical logic and algebraic topology, especially about homotopy theory and model category. I found so far about this direction is this research Bringing set theory and algebraic topology together which may be promising to solve some open problems in algebraic topology. And I also heard about that results from topos theory may be used in proving some isomorphisms in homotopy theory.
My question is: what are some research topics about the intersection of mathematical logic and algebraic topology, especially by bringing tools from logic to solve problems in algebraic topology?