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Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

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• Stub. For the moment just for providing a place to record this reference:

• Jean Thierry-Mieg, Connections between physics, mathematics and deep learning, Letters in High Energy Physics, vol 2 no 3 (2019) (doi:10.31526/lhep.3.2019.110)
• stub article on alternative algorithm to backpropagation in neural networks

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• stub article for now

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• Link to article on taboos

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• adding proof of the inverse function theorem

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• 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!)

• Auke Booij said in his thesis that one could replace locally nonconstant with locally non-zero in his proof of the intermediate value theorem.

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• starting something, but just a stub so far

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• Francis Borceux, Section 6.1 of: Handbook of Categorical Algebra Vol. 2: Categories and Structures $[$doi:10.1017/CBO9780511525865$]$, Encyclopedia of Mathematics and its Applications 50, Cambridge University Press (1994)
• 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!

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• brief category: people-entry for hyperlinking references

• 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.

• 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 $\dagger$-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 $\dagger$-categories.

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The term “h-cofibration” can refer to two closely related, but different notions:

## Definition

A map $f\colon X\to Y$ in a relative category $C$ is an h-cofibration if the cobase change functor $X/C\to Y/C$ is a relative functor, i.e., preserves weak equivalences.

## Properties

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.

## References

!redirects h-cofibrations

• creating disambiguation page

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• brief category:people-entry for hyperlinking references

• 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

• Angelos Anastopoulos, Marco Benini, Sec. 2.4 of Homotopy theory of net representations $[$arXiv:2201.06464$]$

is genuinely new, or a special case of an existing theorem. The category $Rep(\mathcal{A})$ 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.

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• brief category:people-entry, since the link was requested at uniform locale

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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 $x \lt y$ implies $f(x) \lt f(y)$, 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 $T$ is a partial order; in general you want to say $x\leq y$ whenever $f(x) \cong f(y)$. The version with $=$ is still not invariant under equivalence of $T$.

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.” =–

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• I added more info on pseudo double categories and double bicategories to double category. I also simplified the picture of a square, which had been bristling with scary unnecessary detail. There's a slight blemish in the left vertical arrow, which I can't see how to fix.
• Just for completeness, such as to compile links to related entries we have:

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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.

• Added a theorem with a reference.

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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.

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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.

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.

## Selected works

• J. v. Neumann, Eine Axiomatisierung der Mengenlehre, Journal für die reine und angewandte Mathematik 154 (1925), 219–240. doi.

PhD thesis (journal version):

• J. v. Neumann, Die Axiomatisierung der Mengenlehre, Mathematische Zeitschrift 27 (1928), 669–752. doi.

category: people

• PhD-thesis on categorical aspects of vN-algebras. We’ll probably want to include more results from it.

• 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).

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• brief category:people-entry for hyperlinking references

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