Want to take part in these discussions? Sign in if you have an account, or apply for one below
Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.
Stub. For the moment just for providing a place to record this reference:
just for completeness, since we now also have counterexample
I wrote out a proof which uses very little machinery at fundamental theorem of algebra. It is just about at the point where it is not only short and rigorous, but could be understood by an eighteenth-century mathematician. (Nothing important, just fun!)
merging this into real polynomial function; this article is a stub and should belong as a section of real polynomial function along with all the other properties of real polynomial functions
Anonymous
added to symmetric monoidal category a new Properties-section As models for connective spectra with remarks on the theorems by Thomason and Mandell.
added pointer to:
Todd,
when you see this here and have a minute, would you mind having a look at monoidal category to see if you can remove the query-box discussion there and maybe replace it by some crisp statement?
Thanks!
brief category:people-entry for hyperlinking references and edits at Giry monad
brief category:people-entry for satisfying a hyperlink requested at Giry monad
Added a reference of Robert Furber, Bart Jacobs at Giry monad.
Created the stubby atomic site.
I’ve removed this query box from metric space and incorporated its information into the text:
Mike: Perhaps it would be more accurate to say that the symmetry axiom gives us enriched -categories?
Toby: Yeah, that could work. I was thinking of arguing that it makes sense to enrich groupoids in any monoidal poset, cartesian or otherwise, since we can write down the operations and all equations are trivial in a poset. But maybe it makes more sense to call those enriched -categories.
Gave proper reference for (Kieboom 1987).
Created:
The term “h-cofibration” can refer to two closely related, but different notions:
the dual notion to sharp maps.
In this article, we concentrate on the latter.
A map in a relative category is an h-cofibration if the cobase change functor is a relative functor, i.e., preserves weak equivalences.
A model category is left proper if and only if all cofibrations are h-cofibrations.
In a left proper model category, cobase changes along h-cofibrations are homotopy cobase changes.
The notion of h-cofibrations is most useful in the left proper case, and one can argue that in the non-left proper case, the above property should be taken as the definition instead.
starting a stub here, for an entry which we should have had long ago, to complete the pattern of entries listed under “Related entries”.
What I was really after is seeing if the construction in section 2.4 of
is genuinely new, or a special case of an existing theorem. The category considered by these authors should equivalently be that of modules over a monoid in the monoidal category of copresheaves with values in a fixed monoidal model category. Phrased this way, the model structure on this category which these authors present might be expected to be a special case of the general construction of Schewede & Shipley. I haven’t checked yet.
brief category:people-entry, since the link was requested at uniform locale
Created a stubby page uniform locale.
have adjusted the section-outline
am hereby moving the following ancient query-box discussion from the entry to here:
+– {: .query} Mike Shulman: Is that really the right definition? I think of “strictly increasing” as meaning that implies , which is equivalent to the above for linear orders but weaker for partial orders. But I don’t have much experience with strictly increasing functions between non-linear orders, so maybe that is the right definition for partial orders.
However, I don’t think it is the right definition for preorders; among other things, it’s not invariant under equivalence of categories. It seems to me that what you really want to say is that it is pseudomonic as a functor (whereas my weaker definition would become the statement that it is conservative as a functor.)
Toby: This is the definition in HAF (Section 3.17), which defines it for posets (and is a smart enough book that it wouldn't blindly extend a definition from a special case). Although I don't have a reference, I'm pretty sure that this also used in analysis and topology when thinking about convergence and nets, where they may be prosets. However, I think that you have a good point about preordered sets, so I've changed the wording above. (I'll also try to confirm how covergence theorists define ’strictly increasing’ functions between directed prosets.)
It occurs to me that, in the absence of the axiom of choice, one ought to accept even anafunctors between prosets as morphisms, even though these may not be representable as strict functions at all. I'll save that for another day, however.
Mike Shulman: Of course, the definition you gave above isn’t the same as pseudomonic unless is a partial order; in general you want to say whenever . The version with is still not invariant under equivalence of .
I don’t know a whole lot about convergence and nets, but I don’t remember seeing strictly increasing functions used there; I look forward to seeing what you find. Does HAF use the poset version for any application that makes clear why this is a good definition? Of course, monomorphisms of posets may quite naturally something to be interested in, but the question is why they should be called “strictly increasing.” =–
I am giving this bare list of references its own entry, so that it may be !include-ed into related entries (such as topological quantum computation, anyon and Chern-Simons theory but maybe also elsewhere) for ease of updating and synchronizing
Just for completeness, such as to compile links to related entries we have:
Created:
Victor Bouniakowsky (Russian: Ви́ктор Я́ковлевич Буняко́вский) was a Russian mathematician.
He defended his PhD thesis in 1825 at Sorbonne supervised by Augustin Cauchy.
He is known for publishing a proof of the Cauchy inequality in 1859, also known as Cauch–Bouniakowsky–Schwarz inequality.
Created:
Hermann Schwarz was a German mathematician.
He got his PhD degree in 1864 at the University of Berline, advised by Kummer and Weierstrass.
His PhD students include Lipót Fejér and Ernst Zermelo.
His name is often attached to the Cauchy inequality, for which he published a proof in an 1888 paper, although proofs were previously published by Cauchy in 1821 and Bouniakowsky in 1859.
Created:
Lipót Fejér was a Hungarian mathematician. He defended his PhD thesis in 1902 at the University of Budapest, advised by Hermann Schwarz, where he proved his theorem on summability of Fourier series.
His PhD students include John von Neumann.
An AnonymousCoward started NBG last month.
Added new material:
John von Neumann (German: Johann von Neumann; Hungarian: Neumann János Lajos) was a Hungarian mathematician.
He defended his PhD thesis in 1925 advised by Lipót Fejér, with the title Az általános halmazelmélet axiomatikus felépítése (Axiomatic construction of general set theory), which introduced the NBG set theory, as well as classes and von Neumann ordinals.
PhD thesis (journal version):
category: people
Added to commutativity of limits and colimits the case of coproducts commuting with connected limits in a topos, and the generalization to higher topoi. This particular instance of commutativity is not mentioned very often, probably because it’s not very impressive in Set, but its generalization to higher topoi (for which I couldn’t find a reference) is more interesting. For instance, cofiltered limits commute with taking quotients by an ∞-group in an ∞-topos.
Added some stuff about Barthel–Riehl (2013)’s functorial factorization (in four steps to get past the spam filter).
Anonymous