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?

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    $\begingroup$ A significant recent area is homotopy type theory. This provides a bidirectional bridge to applying type theoretic/logical argumenst to homotopy theory and vice versa. At this point it is largely in a theory-building stage where it is a matter of establishing foundational results and working out the language. To my knowledge, it hasn't been much used to try to advance algebraic topology yet, but there are new proof approaches to standard results. $\endgroup$ – Derek Elkins Mar 1 at 20:38

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