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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics comma complex complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration finite foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity group-theory higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration-theory k-theory kan lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monad monoidal monoidal-category-theory morphism motives motivic-cohomology newpage nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pasting philosophy physics planar pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal

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- Discussion Type
- discussion topicstar-autonomous category
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by Mike Shulman
- Last Active 14 minutes ago

I added to star-autonomous category a mention of “$\ast$-autonomous functors”.

- Discussion Type
- discussion topiclinear bicategory
- Category Latest Changes
- Started by Mike Shulman
- Comments 9
- Last comment by Mike Shulman
- Last Active 1 hour ago

Created linear bicategory.

- Discussion Type
- discussion topiclinearly distributive category
- Category Latest Changes
- Started by Mike Shulman
- Comments 25
- Last comment by Mike Shulman
- Last Active 4 hours ago

I reorganized linearly distributive category by moving the long block of history down to the bottom, adding an “Idea” section and a description of how $*$-autonomous categories give rise to linearly distributive ones and linearly distributive ones give rise to polycategories. I also cross-linked the page better with polycategory and star-autonomous category.

- Discussion Type
- discussion topiccoordination
- Category Latest Changes
- Started by David_Corfield
- Comments 29
- Last comment by Urs
- Last Active 7 hours ago

Made a start at coordination. I’m unsure whether it’s worth spending too long on the intricate accounts of Schlick and Reichenbach, and then of whose makes best sense of Einstein’s proposals. Then there’s plenty of recent literature on the subject.

For me, it would probably only be worth expanding if we could thrash out an account of what the nPOV has to say on the subject. Urs has suggested we look at Bohrification. That sounds like the best lead. Reading through the Bohr topos entry, however, it seemed to me that little is said there about how to integrate that with other parts of the synthetic QFT story. There’s the idea of the ’fifth axiom’, but shouldn’t we expect these quantum phase spaces to have appeared earlier as part of the quantization process. Or do we see it merely as way to interpret our way back from the weird quantum world to something as classical as possible so as to be able to relate theory to the recordings of our classical instruments?

- Discussion Type
- discussion topicSerre-Swan theorem
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active 10 hours ago

I have expanded a bit at

*Serre-Swan theorem*: gave it an actual Idea-section, mentioned more variants (over general ringed spaces, in higher geometry) and added more references.

- Discussion Type
- discussion topicdualizable object
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Mike Shulman
- Last Active 10 hours ago

edited dualizable object a little, added a brief paragraph on dualizable objects in symmetric monoidal $(\infty,n)$-categories

- Discussion Type
- discussion topiclocal colimit
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active 11 hours ago

Created local colimit.

- Discussion Type
- discussion topicShuffles
- Category Latest Changes
- Started by Tim_Porter
- Comments 43
- Last comment by Tim_Porter
- Last Active 14 hours ago

I have started an entry on shuffles. It is meant to be an ’elementary introduction’ so there will be room for deeper exploration of them in follow-on entries.

- Discussion Type
- discussion topicSimplex category
- Category Latest Changes
- Started by Tim_Porter
- Comments 29
- Last comment by Tim_Porter
- Last Active 15 hours ago

Jonas Frey has raised the question of the notation $[n]$ in the entry for simplex category. I would go along with his choice of notation as it is the one I use myself. (I was surprised to see another convention being used.)

- Discussion Type
- discussion topic[[distribution]]
- Category Latest Changes
- Started by Todd_Trimble
- Comments 10
- Last comment by Urs
- Last Active 2 days ago

I wrote the article distribution. I'm by no means an expert though. I left open a section "Applications" in case someone would like to add some, or if not I'll try to fill this in soon.

- Discussion Type
- discussion topicdistributions are the smooth linear functionals
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

For ease of linking to from various entries, and in order to have all the relevant material in one place, I am creating an entry

Presently this contains

an Idea-section,

some preliminaries to set the scene,

the statement and proof for the case of compactly supported distributions, taken from what I had just writted into the entry

*compactly supported distribution*,the informal statement for general distributions, so far just with a pointer to Kock-Reyes 04,

a section “Applications”, so far with

some comments on the relevance in pQFT;

some vague pointer to Lawvere-Kock’s generalization to a more general theory of “extensive quantity”

both of which deserve to be expanded.

Eventually I want to have more details on the page, but I’ll leave it at that for the time being. Please feel invited to join in.

I’ll go now and add pointers to this page from “distribution” and from other pages that mention the fact.

- Discussion Type
- discussion topiccompactly supported distribution
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

added statement and proof that compactly supported distributions are equivalently the

*smooth*linear functionals: here(in the sense of either diffeological spaces, or smooth sets, or formal smooth sets/Cahiers topos).

- Discussion Type
- discussion topicFréchet space
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 2 days ago

At

*Fréchet space*I have added to the Idea-section a paragraph motivating the definition via families of seminorms from the example of $\mathbb{R}^\infty = \underset{\longleftarrow}{\lim}_n \mathbb{R}^n$. And I touched the description of this example in the main text, now here.

- Discussion Type
- discussion topicA First Idea of Quantum Fields
- Category Latest Changes
- Started by Urs
- Comments 32
- Last comment by Urs
- Last Active 2 days ago

At

*field (physics)*I am beginning to write an actual introduction to the topic, now in a new section titled “A first idea of quantum fields”.This means to introduce the concept with precise detail, but in a simple context (trivial and bosonic field bundles over Minkowski spacetime, perturbatively quantized) that allows to get a quick idea of the idea of the concept of (quantum) fields as such, without being distracted by other details.

So far I made it up to the derivation of the EOMs. Discussion of (deformation) quantization is to follow (maybe by tonight, depending on how much trouble I have with the trains) and I plan to sprinkle in the detailed example from

*scalar field*in parallel with the abstract discussion.

- Discussion Type
- discussion topicconvenient vector space
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 2 days ago

added to convenient vector space a Properties-section mentioning their embedding into the Cahiers topos, and added the reference by Kock where this is proven.

- Discussion Type
- discussion topicmeaning explanation
- Category Latest Changes
- Started by Mike Shulman
- Comments 29
- Last comment by DavidRoberts
- Last Active 3 days ago

I wrote something at meaning explanation, but I didn’t add any links to it yet because I’m hoping to get some feedback from type theorists as to its correctness (or lack thereof).

- Discussion Type
- discussion topictangent bundle categories
- Category Latest Changes
- Started by Mike Shulman
- Comments 4
- Last comment by David_Corfield
- Last Active 3 days ago

Created a stub tangent bundle categories as a link target to be disambiguated from tangent categories (with a hatnote at the latter). What I’m calling “tangent bundle categories” here are usually called just “tangent categories”, but that clashes with our page tangent category, so I invented a variation. Better suggestions are welcome.

- Discussion Type
- discussion topicenriched category
- Category Latest Changes
- Started by Mike Shulman
- Comments 5
- Last comment by Mike Shulman
- Last Active 3 days ago

I added two recent examples of enriched categories: tangent bundle categories and Lawvere theories.

- Discussion Type
- discussion topicpremonoidal category
- Category Latest Changes
- Started by Mike Shulman
- Comments 14
- Last comment by maxsnew
- Last Active 3 days ago

Somebody named Adam left a comment box a while ago at premonoidal category saying that naturality of the associator requires three naturality squares. I believe that this is true when phrased explicitly in terms of one-variable functors, but the slick approach using the “funny tensor product” allows us to rephrase it as a single natural transformation between functors $C\otimes C\otimes C\to C$. I’ve edited the page accordingly. I also added the motivating example (the Kleisli category of a strong monad) and a link to sesquicategory.

There is a comment on the page that “It may be possible to weaken the above make $(Cat,\otimes)$ a symmetric monoidal 2-category, in which a monoid object is precisely a premonoidal category”. However, the Power-Robinson paper says that “We remark that $(C \otimes -) : Cat \to Cat$ is not a 2-functor,” which seems to throw some cold water on the obvious approach to that idea. Was the thought to define a different 2-categorical structure on $Cat$ than the usual one, e.g. using unnatural transformations? It seems that at least one would still have to explicitly require centrality of the coherence isomorphisms.

- Discussion Type
- discussion topicorder category
- Category Latest Changes
- Started by zskoda
- Comments 20
- Last comment by zskoda
- Last Active 3 days ago

New stub order category (redirecting also the more general case of preorder category).

- Discussion Type
- discussion topicGrothendieck construction
- Category Latest Changes
- Started by Urs
- Comments 24
- Last comment by Mike Shulman
- Last Active 3 days ago

added to Grothendieck construction a section Adjoints to the Grothendieck construction

There I talk about the left adjoint to the Grothendieck construction the way it is traditionally written in the literature, and then make a remark on how one can look at this from a slightly different perspective, which then is the perspective that seamlessly leads over to Lurie's realization of the (oo,1)-Grothendieck construction.

There is a CLAIM there which is maybe not entirely obvious, but straightforward to check. I'll provide the proof later.

- Discussion Type
- discussion topicauxiliary field
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 4 days ago

just for completeness, some terminology explanation at

*auxiliary field*.

- Discussion Type
- discussion topicphilosophy
- Category Latest Changes
- Started by zskoda
- Comments 2
- Last comment by zskoda
- Last Active 5 days ago

Several recent updates to literature at philosophy, the latest being

- Mikhail Gromov,
*Ergostructures, Ergologic and the Universal Learning Problem: Chapters 1, 2.*, pdf;*Structures, Learning and Ergosystems: Chapters 1-4, 6*(2011) pdf

which is more into cognition and language problem, but still very relevant, and by a top mathematician. As these 2 are still manuscripts I put them under articles, though I should eventually classify those as books…

- Mikhail Gromov,

- Discussion Type
- discussion topicbornological set
- Category Latest Changes
- Started by Todd_Trimble
- Comments 28
- Last comment by trent
- Last Active 6 days ago

Started bornological set. Some people call it a bornological space, but that conflicts with the terminology in functional analysis which refers to a locally convex TVS with a suitable “bounded = continuous” property. I quickly wrote that uniformly continuous maps between metric spaces induce bounded maps, but I’ll recheck when I have a free moment.

- Discussion Type
- discussion topicNakanishi-Lautrup field
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Oct 12th 2017

Gave

*Nakanishi-Lautrup field*a brief entry, and touched the corresponding entry*antighost field*.Presently the actual technical content that should go into this entry is this example at

*A first idea of quantum field theory*. But before I paste that into the entry, I will expand and polish it a bit more tomorrow, when I am more awake.

- Discussion Type
- discussion topiccontractible chain complex
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Oct 12th 2017

Needed to be able to point to

*contractible chain complex*and discovered that we didn’t have an entry for that, so I quickly created one.

- Discussion Type
- discussion topicFrobenius pseudomonoid
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by Urs
- Last Active Oct 12th 2017

Created Frobenius pseudomonoid (i.e. $\ast$-autonomous pseudomonoid).

- Discussion Type
- discussion topicKoszul complex
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Oct 12th 2017

I gave

*Koszul complex*and Idea-section and stated two key Properties in citable form (but without proof), one of them the statement that a sufficient condition for the Koszul complex to be a resolution of $R/(x_1, \cdots, x_n)$ is that $R$ is Noetherian, the $x_i$ are in the Jacobson radical, and the cohomology in degree -1 vanishes.Finally I stated the special case of this (here) where $R$ is a formal power series algebra over a field and the elements $x_i$ are formal power series with vanishing constant term.

(I have added the relevant facts as citable numbered examples at

*Noetherian ring*and at*Jacobson radical*.)This happens to be the case that one need in BV-formalism in field theory. I am writing this out now at

*A first idea of quantum field theory*(here).

- Discussion Type
- discussion topicMyles Tierney
- Category Latest Changes
- Started by Tim_Porter
- Comments 8
- Last comment by Tim_Porter
- Last Active Oct 11th 2017

I have added a brief paragraph to the entry Myles Tierney. Can others please check (especially Todd, of course) if we want to add more? Myles was a leading category theorist and will be missed.

- Discussion Type
- discussion topicbalancedmonoidal category
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active Oct 11th 2017

balanced monoidal category used to redirect to twist, which started off with a length disambiguation list before moving on to talk about balanced monoidal categories. I thought this was ugly and unhelpful, so I renamed this page to balanced monoidal category and made twist a pure disambiguation page.