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$\def\Spec{\mathop{\rm Spec}} \def\R{{\bf R}} \def\Ep{{\rm E}^+} \def\L{{\rm L}} \def\EpL{\Ep\L}$ One can argue that an object of the right category of spaces in measure theory is not a set equipped w …

To clarify Chris Heunen's answer, let me point out that most notions of measure theory have analogs in the category of smooth manifolds. For example, the analog of a measure space (X,M,μ), where X is …

An approach to studying that works with many branches of mathematics is to learn the prerequisites first, then study a good textbook on the subject. For instance, one could study subjects like algebra …

The opposite category of commutative von Neumann algebras is not a topos because categorical products with a fixed object do not always preserve small colimits. See Theorem 6.4 in Andre Kornell's Quan …

For a beginner, the more accessible textbooks seem to be the following two. Francis Borceux, Handbook of Categorical Algebra, Volume 3. Saunders Mac Lane, Ieke Moerdijk: Sheaves in Geometry and Logi …

Daniel M. Kan defined adjoint functors in his paper Adjoint functors (written in 1956). In Chapter II he defines limits and colimits of arbitrary small diagrams and proves that the limit and colimit f …

Another application of stacks is in synthetic differential geometry. Start with the opposite category of germ-determined finitely generated C^∞-rings and equip it with the appropriately defined Groth …

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 …

Spaces (in the sense of homotopy theory). These have many Quillen equivalent models: the Serre–Quillen model structure on topological spaces, the Kan–Quillen model structure on simplicial sets, and t …

The category of internal locales in the Grothendieck topos of sheaves on a locale X is equivalent to the slice category over X. In other words, internal locales over X are precisely morphisms of local …

Consider the symmetric monoidal category TVS of complete Hausdorff topological vector spaces equipped with the completed projective, injective, or inductive tensor product. Every finite-dimensional ve …

A good answer to both questions is provided by the following variant of the Gelfand duality for commutative von Neumann algebras, which shows that the following categories are equivalent: The categor …

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 …

Yes, both Theorem A and Theorem B are special cases of a more general construction. Denote by $R$ the category of commutative unital C*-algebras or the category of commutative rings. Denote by $R'$ th …