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Isomorphism classes of finitely generated right Hilbert modules over a II_1 factor are in a bijective correspondence with the nonnegative reals. The correspondence sends every module to its dimension. …

Yes. In fact, Grothendieck topologies on any small category constitute a locale. See Proposition 3.2.13 in Borceux's Handbook of Categorical Algebra 3.

I think both uhom(Y × X, Z) and uhom(Y, uhom(X, Z)) represent the same functor: hom(W, uhom(Y × X, Z)) = hom(W × Y × X, Z) hom(W, uhom(Y, uhom(X, Z))) = hom(W × Y, uhom(X, Z)) = hom(W × Y × X, Z), hen …

It is certainly possible to define the class of functions as something else than ordered pairs: for example, we can define a function to be a set $Z$, whose individual elements must be ordered pairs $ …

Say, we take G to be U(1) and then form BU(1). Is it a 1-groupoid or 2-groupoid? The delooping of the Lie group $\def\B{{\sf B}} \def\U{{\sf U}} \U(1)$ is the Lie 1-groupoid $\B\U(1)$. The shape $\d …

As demonstrated by Andre Kornell in http://arxiv.org/abs/1202.2994, the category of measurable spaces is closed with respect to the monoidal structure given by the spatial tensor product (Theorem 9.5) …

$T$ can be formalized as a natural transformation $\def\Vect{{\rm Vect}} \def\Vectc{\Vect_\nabla} \Vectc→Ω^n$ of functors $\def\Man{{\sf Man}} \def\op{{\sf op}} \def\Grpd{{\sf Grpd}} \Man^\op → \Grpd$ …

Yes, there is such a category, namely the category of measurable spaces and morphisms of measurable spaces equipped with fiberwise measures (also known as operator valued weights). See this answer for …

If we write down the lifting square for an arbitrary cofibration $f\colon A→B$ and the unit map $η\colon X→RLX$ (with the bottom map being $b\colon B→RLX$), and then use the adjunction to pass to the …

This answer serves to record two explicit proofs of this fact in the literature: Corollary 5.1 in Raptis and Rosický, “The accessibility rank of weak equivalences”, arXiv:1403.3042v2. Theory and Appl …

The analogous adjunction for the enriched category of functors can be deduced from the adjunction for the ordinary category of functors using the Yoneda lemma. Indeed, to establish a natural isomorphi …

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 …

There are two theses by Jean Celeyrette indexed by various libraries: Theoreme de Kan dans un topos (Lille, 1974). Catégories internes et fibrations ; Cohomologie de Gel'fand-Fuks (Paris-Nord, 1975). …

One can do even better than a Quillen adjunction: Theorem 3.8 in A model structure for Grothendieck fibrations establishes a Quillen equivalence between the projective model structure on presheaves of …

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 …