New page small cardinality selection axiom.
]]>Is the category Hom of bicategories with homomorphisms as the morphisms, in the sense of
Ross Street, Fibrations in bicategories, Cahiers de topologie et géométrie différentielle catégoriques, tome 21, no 2 (1980), p. 111-160
already (recognizably) documented on the nLab ? (I did a sem-cursory search in this respect, but did not find it documented (in its own right, I mean, the article of Street appears.)
Should it be?
Should it have an article of its own?
To me it seems it should (my motivation is that I am using and documenting bicategories currently, and are studying Street’s 1980 paper as a sort of background reading to Garner–Shulman, Adv. Math. 289), but its traditional name Hom seems unfortunate, creating yet another meaning of Hom.
My suggestion would be to call it (and its article)
]]>It’s well known that the category of points of the presheaf topos over , the dual of the category of finitely presented rings, is the category of all rings (without a size or presentation restriction). In fact this holds for any algebraic theory, not only for the theory of commutative rings. One can learn about this in our entries on Gabriel-Ulmer duality, flat functors, and Moerdijk/Mac Lane.
But what if we don’t restrict the site to consist only of the compact objects? What are the points of the presheaf topos over the large category , to the extent that the question is meaningful because of size-related issues? What are the points of the presheaf topos over , the dual of the category of rings admitting a presentation by many generators and relations, where is a regular cardinal? (The category is essentially small, so the question is definitely meaningful.)
The question can be rephrased in the following way: What is an explicit description of the category of finite limit preserving functors ? Any such functor gives rise to a ring by considering , but unlike in the case such a functor is not determined by this ring.
This feels like an extremely basic question to me; it has surely been studied in the literature. I appreciate any pointers! Of course I’ll record any relevant thoughts in the lab.
]]>Partitions of unity for an ordinary cover can be used to write down explicit coboundaries and cocycles for geometric objects specified locally on an open cover.
Suppose now we have a geometric object specified using a hypercover, e.g., a bundle gerbe. Is there an analog of the usual notion of partition of unity that allows us to write down explicit formulas in a similar fashion, e.g., as in the article partitions of unity? For example, can one construct a connection on a bundle gerbe in a similar fashion as in the article connection on a bundle?
]]>Is anything known about the existence of partitions of unity and good open covers for PL-manifolds?
Here a good open cover of a PL-manifold is a locally finite open cover {U_i} such that every finite intersection of U_i is either empty or PL-isomorphic to R^n.
A partition of unity subordinate to an open cover {U_i} of a PL-manifold X is a family of nonnegative PL-functions f_i: X→R such that supp f_i is a subset of U_i, supp f_i form a locally finite family, and the sum of f_i is 1.
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