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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).
Todd,
you added to Yoneda lemma the sentence
In brief, the principle is that the identity morphism is the universal generalized element of . 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 is a tautological statement that does not need or imply the Yoneda lemma, it seems.
started something stubby at Liouville theory, for the moment just so as to record some references and provide for a minimum of cross-links (e.g. with Chern-Simons gravity).
(also created a stub for quantum Teichmüller theory in the course of this, but nothing there yet except a pointer to reviews)
for hyperlinking references at conformal bootstrap
for hyperlinking references at Liouville theory
for hyperlinking references at Liouville theory
for hyperlinking references at Liouville theory
for hyperlinking references at Liouville theory
made more explicit (here) the back-link to topological vector bundle for classical concordance for topological vector bundles
I have added the characterization of Quillen equivalences in the case that the right adjoint creates weak equivalences, here.
Created with the following content:
A locale is paracompact if it is regular and every open cover has a locally finite refinement.
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.
Cleaned up partition of unity and fine sheaf a bit, so I could link to them from this MO answer to the question ’Why are there so many smooth functions?’.
I have spelled out the proof at paracompact Hausdorff spaces equivalently admit subordinate partitions of unity.
This uses Urysohn’s lemma and the shrinking lemma, whose proofs are not yet spelled out on the Lab.
I have added to principal bundle
a remark on their definition As quotients;
statements about (classes of) (counter-)examples of quotients
Thanks for pointers to the literature from this MO thread!
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 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 , 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:
Added the writing
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 . The underlying set is the completion of the ordered field of rational numbers: the result of adjoining to 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 is called the real line also known as the continuum. Equipped with both the topology and the field structure, is a topological field and as such is the uniform completion of equipped with the absolute value metric.
Together with its cartesian products – the Cartesian spaces for natural numbers – the real line 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.
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.
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 . Well, nice but not that standard among representation theorists themselves. Over there one takes such a thing – representation by permutations of a finite group on a set , and looks what happens in the vector space of functions into a field . As we know, for a group element the definition is, , for is the way to induce a representation on the function space . The latter representation is called the permutation representation in the standard representation theory books like in
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 definition to beta-gamma system
have splitt off an entry Stone-Cech compactification from the entry compactum, (where it was a bit hidden) such as to make the content here a bit easier to find and to link to.
at species it says that this is a presheaf of sets on . At structure type this then makes me expect the words “is a presheaf of groupoids” on . Is there a deeper reason why it does not say that?
It seems clear that the Gepner-Kock homotopical species are precisely the -presheaves on , i.e. the -groupoid valued ones.
I’d think it would be good to emphasize this presheaf-point of view in our entries.
fianlly added the details of Dugger’s description of cofibrant objects in the projective model structure on simplicial presheaves in the section Cofibrant objects.
Someone deleted the contents of the entry simplicial localization on th 4 April, then another reinstated it on the 5th. Curious!
started adding some genuine substance to model structure on sSet-categories (which used to be just a template).
added pointer to
This would reduce the question of Quillen right adjoints representing right -adjoints between presentable -categories to coreflections and monadic functors…
brief category:people-entry for hyperlinking references at equivariant stable homotopy theory and enriched (infinity,1)-category theory
added references (also to Burnside ring):
Erkki Laitinen, On the Burnside ring and stable cohomotopy of a finite group, Mathematica Scandinavica Vol. 44, No. 1 (August 30, 1979), pp. 37-72 (jstor:24491306, Laitinen79.pdf:file)
Wolfgang Lück, The Burnside Ring and Equivariant Stable Cohomotopy for Infinite Groups (arXiv:math/0504051)
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 -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 -category and of -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 -categories.
[end old forwarded discussion]
Created cartesian multicategory.
added pointer to:
added publication data for:
added pointer to:
felt like splitting off simplicial Quillen adjunction, so that one can better link to it
added pointer to Belopolsky97b etc., regarding “picture number” induced by a choice of integral top-forms.
added disambiguation with the notion of “integral form” in the sense of integration over supermanifolds