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Such a timeline is necessarily highly subjective. With this disclaimer in mind, we can identify some important turns in the development of foundations of homotopy theory. The list below concentrates o …

Here is my own favorite construction of the (Lebesgue) integral. Suppose M is an arbitrary smooth manifold. Denote by Or(M) the orientation line bundle of M. This bundle is equipped with a canonical …

The Quillen S⁻¹S construction (not to be confused with the Quillen Q-construction or the Quillen plus-construction), as defined by Grayson in Higher algebraic K-theory: II (page 219), takes as an inpu …

Here is what André Joyal wrote in an email to me: No, I have not discovered the model structure for quasi-categories in the 1980's. I became interested in quasi-categories (without the name) around 1 …

Model categories provide a powerful framework for commuting (homotopy) limits and colimits, and, more generally, for commuting left adjoint functors and (homotopy) limits, as well as right adjoint fun …

Both relative categories and marked simplicial sets (over Δ^0) present the ∞-category of ∞-categories. Naturally, one could ask whether there is a reasonably direct way to pass between these two mode …

There are numerous expositions of simplicial homology in the literature. Munkres in “Elements of Algebraic Topology” develops the homology theory of simplicial complexes. Hatcher in “Algebraic Topol …

A concrete model for Ω^∞ applied to Thom spectra (which is all what we need because Thom spectra are connective) was given by Quinn in his thesis. Very roughly, Ω^∞MG is a simplicial set whose n-simp …

The bicategory of Grothendieck toposes is equivalent to the bicategory of localic groupoids, with appropriately defined 1-morphisms and 2-morphisms. "Ordinary spaces" are topological spaces, or, perh …

The fiberwise Stokes theorem says that given a differential form on a smooth fiber bundle whose fibers have boundary, the difference between the fiberwise integral of the differential and the differen …

The Thom spectrum MO is a module over the ring spectrum π≤0S=HZ, where S is the sphere spectrum. In particular, MO is equivalent to the Eilenberg-MacLane spectrum Hπ*(MO). On the other hand, MU and MS …

Boardman's thesis was (re)published a year later as three separate booklets, and a PDF scan of all three booklets is available on my scans page: J. M. Boardman. Stable homotopy theory. University of W …

Some of the earliest writings on the Joyal model structure on simplicial sets include Jacob Lurie's account in Higher Topos Theory from 2006, as well as Joyal's own account in The Theory of Quasi-Cate …

For sufficiently nice topological spaces $X$ (e.g., locally connected for the last two categories to be equivalent, and semilocally simply connected and locally path-connected for all three to be equi …

Consider a diagram D: I→ChR of real connective chain complexes. In the example I have in mind all chain complexes are concentrated in some fixed degree n. There is a canonical map lim D → holim D fro …