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

• Add some discussion of the equivalence between the two definitions, and how in practice we usually use the family-of-collections-of-morphisms one.

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Anonymous

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Anonymous

Shane Kelly

• for completeness

• finally a stub for Segal condition. Just for completeness (and to have a sensible place to put the references about Segal conditions in terms of sheaf conditions).

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Anonymous

• Partially ordered abelian groups whose partial order is a pseudolattice

Anonymous

• Todd,

you added to Yoneda lemma the sentence

In brief, the principle is that the identity morphism $id_x: x \to x$ is the universal generalized element of $x$. This simple principle is surprisingly pervasive throughout category theory.

Maybe it would be good to expand on that. One might think that the universal property of a genralized element is that every other one factors through it uniquely. That this is true for the generalized element $id_x$ is a tautological statement that does not need or imply the Yoneda lemma, it seems.

• for the moment just to satisfy links

• I have added the characterization of Quillen equivalences in the case that the right adjoint creates weak equivalences, here.

• Created with the following content:

### Definition

A locale is paracompact if it is regular and every open cover has a locally finite refinement.

### Properties

Paracompact locales are very closely related to fully normal locales. In fact, for regular locales these two properties are equivalent.

Any metrizable locale is paracompact.

Any Lindelöf locale is paracompact.

A locale is paracompact if and only if it admits a complete uniformity.

The full subcategory of paracompact locales is a reflective subcategory of the category of completely regular locales as well as the category of all [[locales].

In particular, the inclusion functor from paracompact locales to locales preserves small limits, so in particular, products of paracompact locales are paracompact.

This last property clearly distinguishes paracompact locales from paracompact spaces, since products of paracompact spaces need not be paracompact.

### Related concets

• Added the property that final functors and discrete fibrations form an orthogonal factorisation system.

• created cohesive topos.

• wrote an Idea-section that is meant to explain why the concept is very natural, trying to provide some of the chat that one cannot find in the terse (but beautiful in its own way) article by Lawvere

• spelled out the definition in some detail, here, too, trying to fill in things that Lawvere is glossing over, making it all very explicit;

• started an Examples-section:

• copied over the discussion that $Sh(CartSp)$ is a connected topos. checking the remaining axioms for cohesive topos are easy, but i have not typed that yet

• included a little discussion of how diffeological spaces fit in, following our conversation in another thread

• started an analogous section for $\infty Sh(CartSp)$, but just a stub so far

• but added in a section that goes rgrough the various items in Lawvere’s definition and discusses their meaning in a cohesive oo-topos

• added pointer to the second original article by Souriau on diffeological groups:

• Jean-Marie Souriau, Groupes différentiels et physique mathématique, In: Denardo G., Ghirardi G., Weber T. (eds.) Group Theoretical Methods in Physics. Lecture Notes in Physics, vol 201. Springer 1984 (doi:10.1007/BFb0016198)

from James Borger’s comment

But I am sure that there’s a rich, rich overlap between the higher-categorical world and the biring/plethory world. Just to mention one data point, in Drinfeld’s recent paper “Prismatization”, he explains how ring stacks give rise to cohomology theories. He’s particularly interested in prismatic cohomology, but crystalline, de Rham, Dolbeault are super fun baby cases. A ring stack is, sort of by definition, just Spec of a derived biring which is concentrated in two degrees. So “slightly categorified birings” = de-Rham-like cohomology theories.

• I looked at real number and thought I could maybe try to improve the way the Idea section flows. Now it reads as follows:

A real number is something that may be approximated by rational numbers. Equipped with the operations of addition and multiplication induced from the rational numbers, real numbers form a number field, denoted $\mathbb{R}$. The underlying set is the completion of the ordered field $\mathbb{Q}$ of rational numbers: the result of adjoining to $\mathbb{Q}$ suprema for every bounded subset with respect to the natural ordering of rational numbers.

The set of real numbers also carries naturally the structure of a topological space and as such $\mathbb{R}$ is called the real line also known as the continuum. Equipped with both the topology and the field structure, $\mathbb{R}$ is a topological field and as such is the uniform completion of $\mathbb{Q}$ equipped with the absolute value metric.

Together with its cartesian products – the Cartesian spaces $\mathbb{R}^n$ for natural numbers $n \in \mathbb{N}$ – the real line $\mathbb{R}$ is a standard formalization of the idea of continuous space. The more general concept of (smooth) manifold is modeled on these Cartesian spaces. These, in turnm are standard models for the notion of space in particular in physics (see spacetime), or at least in classical physics. See at geometry of physics for more on this.

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Anonymous

• Added alternative terminology “local right adjoint” and “strongly cartesian monad” from Berger-Mellies-Weber. They claim the former “has become the more accepted terminology” than “parametric right adjoint”; does anyone know other references to support this? (I think it’s certainly more logical, in that it fits with the general principle of “local” meaning “on slice categories” — not to be confused with the different general principle of “local” meaning “in hom-objects”.)

• felt the desire to have an entry on the general idea (if any) of synthetic mathematics, cross-linking with the relevant examples-entries.

This has much room for being further expanded, of course.

• Did some editing on this page. There’s a query there about whether the Grothendieck ring of a braided monoidal category is commutative. Seems so from here, so I’ll remove it.

• Unfortunately, I need to discuss with you another terminological problem. I am lightly doing a circle of entries related to combinatorial aspects of representation theory. I stumbled accross permutation representation entry. It says that the permutation representation is the representation in category $Set$. Well, nice but not that standard among representation theorists themselves. Over there one takes such a thing – representation by permutations of a finite group $G$ on a set $X$, and looks what happens in the vector space of functions into a field $K$. As we know, for a group element $g$ the definition is, $(g f)(x) = f(g^{-1} x)$, for $f: X\to K$ is the way to induce a representation on the function space $K^X$. The latter representation is called the permutation representation in the standard representation theory books like in

• Claudio Procesi, Lie groups, an approach through invariants and representations, Universitext, Springer 2006, gBooks

I know what to do approximately, we should probably keep both notions in the entry (and be careful when refering to this page – do we mean representation by permutations, what is current content or permutation representation in the rep. theory on vector spaces sense). But maybe people (Todd?) have some experience with this terminology.

Edit: new (related) entries for Claudio Procesi and Arun Ram.

• there used to be, all along, a section titled “Derived adjunction”, which however fell short of saying anything about the derived adjunction as such.

Have added the statement now, with pointer to a new stand-alone entry derived adjunction.

• I added an explicit definition of cartesian model category to cartesian closed model category to highlight the convention that the terminal object is assumed cofibrant.

• Added a reference to Tall–Wraith. Changed $\circ: P \otimes_k P \to P$ to $\circ: P \odot_k P \to P$. Added redirects.

• This page doesn’t seem to have a discussion page on the nForum.

Anonymous

• Update url for generalized species paper

Vikraman Choudhury

• at species it says that this is a presheaf of sets on $core(FinSet)$. At structure type this then makes me expect the words “is a presheaf of groupoids” on $core(FinSet)$. Is there a deeper reason why it does not say that?

It seems clear that the Gepner-Kock homotopical species are precisely the $(\infty,1)$-presheaves on $core(FinSet)$, i.e. the $\infty$-groupoid valued ones.

I’d think it would be good to emphasize this presheaf-point of view in our entries.

• Added the description of the unit morphism, as it was not present before.

Anonymous

• Someone deleted the contents of the entry simplicial localization on th 4 April, then another reinstated it on the 5th. Curious!

• Lior Yanovski, The Monadic Tower for $\infty$-Categories (arXiv:2104.01816)

This would reduce the question of Quillen right adjoints representing right $\infty$-adjoints between presentable $\infty$-categories to coreflections and monadic functors…

• I thought we needed an entry enriched (infinity,1)-category, so I created one. Added an Idea-section that mentions the evident general abstract definition (which hasn’t been worked out) and mentions the evident model (which has).

I have used links to this now at table - models for (infinity,1)-operads in an attempt to clarify the “general pattern” of the table (now the first part of the table itself).

I notice/rememberd that we have two equally orphaned and equally stubby entries, titled weak enrichment and titled homotopical enrichment. Something should be done about that unfortunate state of affairs, but for the moment I just added more links between these.

There was also this ancient discussion, which we don’t need to keep there:

[begin old forwarded discussion]

Urs: can anyone point me to – or write an entry containing – a discussion of systematical “homotopical enrichment” where we enrich over a homotopical category systematically weakening everything up to coherent homotopy. If/when we have this we should also link it to (infinity,n)-category, as that is built by iteratively doing homotopical enrichement starting with Top.

Mike: If anyone ever does anything like that, I would love to see it. As far as I know there is no general theory. You can define Segal categories in any homotopical category with finite products. You can define complete Segal spaces in any model category, at least, and less may suffice. And you can define $A_\infty$-categories in any monoidal homotopical category. But the problem is finding some way to get a handle on them, like lifting a model structure to them. Of course, people have iterated the existing definitions to get notions of $n$-category and of $(\infty,n)$-category (Simpson-Tamsamani, Trimble, Barwick, Lurie, etc), but I’ve never seen a general theory. Peter May and I have been planning for a while to think about iterating enriched $A_\infty$-categories.

[end old forwarded discussion]

• Only a stub at the moment, but I thought we needed to start a page on this. Looks like it’s going to become important.