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The cotangent complex of a map of commutative rings is a central object in deformation theory. Since the 1990s, it has been generalized to the homotopical setting of E-infinity-ring spectra in various
We prove that every elementary (infinity, 1)-topos has a natural number object. We achieve this by defining the loop space of the circle and showing that we can construct a natural number object out o
We prove that four different ways of defining Cartesian fibrations and the Cartesian model structure are all Quillen equivalent: 1.On marked simplicial sets (due to Lurie [31]), 2.On bisimplicial spac