Not signed in (Sign In)

A discussion forum about contributions to the nLab wiki and related areas of mathematics, physics, and philosophy.

Want to take part in these discussions? Sign in if you have an account, or apply for one below

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry beauty bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-theory cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis 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 integration-theory k-theory lie lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics 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 sheaves simplicial space spin-geometry stable-homotopy-theory string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory type type-theory universal variational-calculus

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

If you want to take part in these discussions either sign in now (if you have an account), apply for one now (if you don't).

- Discussion Type
- discussion topicDaniel Kan
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 7
- Last comment by Dmitri Pavlov
- Last Active Jun 24th 2022

- Discussion Type
- discussion topicmonoidal model category
- Category Latest Changes
- Started by Urs
- Comments 25
- Last comment by Dmitri Pavlov
- Last Active Jun 24th 2022

I have added to

*monoidal model category*statement and proof (here) of the basic statement:

Let $(\mathcal{C}, \otimes)$ be a monoidal model category. Then 1) the left derived functor of the tensor product exsists and makes the homotopy category into a monoidal category $(Ho(\mathcal{C}), \otimes^L, \gamma(I))$. If in in addition $(\mathcal{C}, \otimes)$ satisfies the monoid axiom, then 2) the localization functor $\gamma\colon \mathcal{C}\to Ho(\mathcal{C})$ carries the structure of a lax monoidal functor

$\gamma \;\colon\; (\mathcal{C}, \otimes, I) \longrightarrow (Ho(\mathcal{C}), \otimes^L , \gamma(I)) \,.$

The first part is immediate and is what all authors mention. But this is useful in practice typically only with the second part.

- Discussion Type
- discussion topicHomotopy Limit Functors on Model Categories and Homotopical Categories
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Dmitri Pavlov
- Last Active Jun 24th 2022

created a reference-entry Homotopy Limit Functors on Model Categories and Homotopical Categories and added pointers to it to a bunch of relevant entries

- Discussion Type
- discussion topicflat connection
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jun 24th 2022

This entry didn’t have any really pertinent references yet.

On the equivalence between flat connections and reps of the fundamental group I have now added pointer to:

- Pierre Deligne, §I.1 of:
*Equations différentielles à points singuliers réguliers*, Lecture Notes Math.**163**, Springer (1970) $[$publications.ias:355$]$

- Pierre Deligne, §I.1 of:

- Discussion Type
- discussion topicunivalence axiom
- Category Latest Changes
- Started by spitters
- Comments 30
- Last comment by Urs
- Last Active Jun 24th 2022

- Discussion Type
- discussion topicuniversal fibration of (infinity,1)-categories
- Category Latest Changes
- Started by Hurkyl
- Comments 9
- Last comment by Urs
- Last Active Jun 24th 2022

- Discussion Type
- discussion topichomotopy type theory
- Category Latest Changes
- Started by Urs
- Comments 62
- Last comment by Urs
- Last Active Jun 24th 2022

stub for homotopy type theory

- Discussion Type
- discussion topicenriched monad
- Category Latest Changes
- Started by Sam Staton
- Comments 9
- Last comment by nLab edit announcer
- Last Active Jun 23rd 2022

- Discussion Type
- discussion topicgroup extension
- Category Latest Changes
- Started by Urs
- Comments 12
- Last comment by nLab edit announcer
- Last Active Jun 22nd 2022

added to group extension a section on how group extensions are torsors and on how they are deloopings of principal 2-bundles, see group extension – torsors

- Discussion Type
- discussion topicformal moduli problem
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Jun 22nd 2022

- Discussion Type
- discussion topicsynthetic (infinity,1)-category theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 10
- Last comment by Urs
- Last Active Jun 22nd 2022

- Discussion Type
- discussion topicenriched type
- Category Latest Changes
- Started by nLab edit announcer
- Comments 15
- Last comment by Madeleine Birchfield
- Last Active Jun 22nd 2022

- Discussion Type
- discussion topicsigma-model
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Dmitri Pavlov
- Last Active Jun 21st 2022

I have considerably expanded the entry sigma-model and will probably continue to do so in small steps in the nearer future (with interruptions). This goes in parallel with a discussion we are having on the $n$Café here.

- Discussion Type
- discussion topicKnizhnik-Zamolodchikov equation
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Jun 21st 2022

finally added the actual definition,

`!include`

-ed from*Knizhnik-Zamolodchikov-Kontsevich construction – definition*(as per the discussion here)

- Discussion Type
- discussion topicparameterized homotopy theory
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Jun 21st 2022

added pointer to

- Vincent Braunack-Mayer,
*Combinatorial parametrised spectra*(arXiv:1907.08496)

- Vincent Braunack-Mayer,

- Discussion Type
- discussion topicFrobenius reciprocity
- Category Latest Changes
- Started by zskoda
- Comments 22
- Last comment by Urs
- Last Active Jun 21st 2022

Stub Frobenius reciprocity.

- Discussion Type
- discussion topicsimplicial type theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active Jun 21st 2022

- Discussion Type
- discussion topicinternal category in homotopy type theory
- Category Latest Changes
- Started by Urs
- Comments 19
- Last comment by Urs
- Last Active Jun 21st 2022

I finally created an entry

*internal category in homotopy type theory*.There is old discussion of this topic which I had once written at

*category object in an (infinity,1)-category*in the sub-section*HoTT formulation*, but it’s probably good to give this a stand-alone entry, for ease of linking (such as from*equivalence of categories*now).

- Discussion Type
- discussion topicJonathan Weinberger
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 21st 2022

- Discussion Type
- discussion topicone-parameter semigroup
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 2
- Last comment by Urs
- Last Active Jun 20th 2022

Created:

A

$\mathbf{R} \to U(H),$**one-parameter group**(of unitary operators in a Hilbert space) is a homomorphism of groupswhere $H$ is a Hilbert spaces and $U(H)$ denotes its group of unitary operators.

More generally, one can define

$\mathbf{R} \to B(X),$**one-parameter semigroups**of operators in a Banach space $X$ as homomomorphisms of groupswhere $B(X)$ denotes the semigroup of bounded operators $X\to X$.

Typically, we also require a continuity condition such as continuity in the strong topology.

## Stone theorem

Strongly continuous one-parameter unitary groups $(U_t)_{t\ge0}$ of operators in a Hilbert space $H$ are in bijection with self-adjoint unbounded operators $A$ on $H$

The bijection sends

$A\mapsto (t\mapsto \exp(itA)).$The operator $A$ is bounded if and only if $U$ is norm-continuous.

## Hille–Yosida theorem

Strongly continuous one-parameter semigroups $T$ of bounded operators on a Banach space $X$ (alias

$\|(\lambda I-A)^{-n}\|\le M (\lambda-\omega)^{-n}.$**$C_0$-semigroups**) satisfying $\|T(t)\|\le M\exp(\omega t)$ are in bijection with closed operators $A\colon X\to X$ with dense domain such that any $\lambda\gt \omega$ belongs to the resolvent set of $A$ and for any $\lambda\gt\omega$ we have## References

[…]

- Discussion Type
- discussion topicHugo Steinhaus
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Jun 20th 2022

Created:

Hugo Dyonizy Steinhaus was a mathematician at the University of Lwów and the University of Wrocław.

He get his PhD in 1911 from the University of Göttingen, advised by David Hilbert.

His PhD students include Mark Kac.

- Discussion Type
- discussion topicJames R. Norris
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Jun 20th 2022

James Ritchie Norris is a mathematician at the University of Cambridge.

He got his PhD in 1985 from the University of Oxford, advised by David Edwards.

## Selected writings

On the Feynman–Kac formula on smooth manifolds:

- James R. Norris,
*A complete differential formalism for stochastic calculus in manifolds*, Séminaire de Probabilités XXVI, Lecture Notes in Mathematics (1992), 189–209. doi.

- James R. Norris,

- Discussion Type
- discussion topicMark Kac
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Jun 20th 2022

Created:

Mark Kac was a mathematician at Cornell University and Rockefeller University.

He got his PhD from the University of Lwów in 1937, advised by Hugo Steinhaus.

## Selected writings

On the Feynman–Kac formula:

- Discussion Type
- discussion topicFeynman-Kac formula
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 1
- Last comment by Dmitri Pavlov
- Last Active Jun 20th 2022

Created:

## Idea

The Feynman–Kac formula expresses the integral kernel of the one-parameter semigroup generated by a Laplacian on a smooth manifold as the path integral of the parallel transport map associated to the given connection with respect to all paths of a given length connecting the two given points.

## References

The original reference is

- Mark Kac,
*On distributions of certain Wiener functionals*, Transactions of the American Mathematical Society 65:1 (1949), 1–13. doi.

The case of smooth manifolds is treated in

- James R. Norris,
*A complete differential formalism for stochastic calculus in manifolds*, Séminaire de Probabilités XXVI, Lecture Notes in Mathematics (1992), 189–209. doi.

- Mark Kac,

- Discussion Type
- discussion topicAtsushi Matsuo
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 20th 2022

- Discussion Type
- discussion topichypergeometric KZ-solutions -- references
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active Jun 20th 2022

- Discussion Type
- discussion topiccompanion pair
- Category Latest Changes
- Started by mattecapu
- Comments 1
- Last comment by mattecapu
- Last Active Jun 20th 2022

- Discussion Type
- discussion topicPursuing Stacks
- Category Latest Changes
- Started by Tim_Porter
- Comments 28
- Last comment by DavidRoberts
- Last Active Jun 20th 2022

@Todd. Thanks for correcting my atrocious English!

Does anyone have any ideas as to how we could provide a bit more for this entry?

- Discussion Type
- discussion topicepimorphism
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by nLab edit announcer
- Last Active Jun 20th 2022

felt like adding a handful of basic properties to epimorphism

- Discussion Type
- discussion topicexponential law for parameterized topological spaces -- references
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jun 20th 2022

this is a bare list of references, to be

`!included`

into the lists of references of relevant entries (such as at*compactly generated topological space*,*parameterized homotopy theory*,*exponential law for spaces*)

- Discussion Type
- discussion topicReinhold Heckmann
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 20th 2022

- Discussion Type
- discussion topicexponential law for spaces
- Category Latest Changes
- Started by nLab edit announcer
- Comments 6
- Last comment by Urs
- Last Active Jun 20th 2022

The induced map most likely isn’t a homeomorphism when $X, Y$ are locally compact Hausdorff.

The original statement was in monograph by Postnikov without proof.

Not only that, in the current form it couldn’t possibly be true, since the map could lack to be bijective.

For more details see here: https://math.stackexchange.com/questions/3934265/adjunction-of-pointed-maps-is-a-homeomorphism .

I’ve added a reference in the case when $X, Y$ are compact Hausdorff though.

Adam

- Discussion Type
- discussion topicStone Spaces
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 20th 2022

reformatted the bib-item, added link to the publisher page, and cross-link with

*Topos Theory*

- Discussion Type
- discussion topicC. R. F. Maunder
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 20th 2022

- Discussion Type
- discussion topichomotopy theory and algebraic topology -- references
- Category Latest Changes
- Started by Urs
- Comments 23
- Last comment by Urs
- Last Active Jun 20th 2022

following discussion here I am starting an entry with a bare list of references (sub-sectioned), to be

`!include`

-ed into the References sections of relevant entries (mainly at*homotopy theory*and at*algebraic topology*) for ease of updating and syncing these lists.The organization of the subsections and their items here needs work, this is just a start. Let’s work on it.

I’ll just check now that I have all items copied, and then I will

`!include`

this entry here into*homotopy theory*and*algebraic topology*. It may best be*viewed*withing these entries, because there – but not here – will there be a table of contents showing the subsections here.

- Discussion Type
- discussion topicPeter I. Booth
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Tim_Porter
- Last Active Jun 20th 2022

- Discussion Type
- discussion topicMichael C. Crabb
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 20th 2022

- Discussion Type
- discussion topicsurgery theory
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jun 20th 2022

- Discussion Type
- discussion topiccohesive homotopy type theory with two kinds of types
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jun 19th 2022

- Discussion Type
- discussion topicdifferential cohesive homotopy type theory
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Guest
- Last Active Jun 19th 2022

While working at

*geometry of physics*on the next chapter*Differentiation*I am naturally led back to think again about how to best expose/introduce infinitesimal cohesion. To the reader but also, eventually, to Coq.First the trivial bit, concerning terminology: I am now tending to want to call it

*differential cohesion*, and*differential cohesive homotopy type theory*. What do you think?Secondly, I have come to think that the extra right adjoint in an infinitesimally cohesive neighbourhood need not be part of the axioms (although it happens to be there for $Sh_\infty(CartSp) \hookrightarrow Sh_\infty(CartSp_{th})$ ).

So I am now tending to say

**Definition.**A*differential structure*on a cohesive topos is an ∞-connected and locally ∞-connected geometric embedding into another cohesive topos.And that’s it. This induces a homotopy cofiber sequence

$\array{ CohesiveType &\hookrightarrow& InfThickenedCohesiveType &\to& InfinitesimalType \\ & \searrow & \downarrow & \swarrow \\ && DiscreteType }$Certainly that alone is enough axioms to say in the model of smooth cohesion all of the following:

- reduced type, infinitesimal path ∞-groupoid, de Rham space, jet bundle, D-geometry, ∞-Lie algebra (synthetically), Lie differentiation, hence “Formal Moduli Problems and DG-Lie Algebras” , formally etale morphism, formally smooth morphism, formally unramified morphism, smooth etale ∞-groupoid, hence ∞-orbifold etc.

So that seems to be plenty of justification for these axioms.

We should, I think, decide which name is best (“differential cohesion”?, “infinitesimal cohesion”?) and then get serious about the “differential cohesive homotopy type theory” or “infinitesimal cohesive homotopy type theory” or maybe just “differential homotopy type theory” respectively.

- Discussion Type
- discussion topicAndré Joyal
- Category Latest Changes
- Started by Tim_Porter
- Comments 6
- Last comment by Dmitri Pavlov
- Last Active Jun 19th 2022

- Discussion Type
- discussion topiclinear order
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Guest
- Last Active Jun 19th 2022

- Discussion Type
- discussion topichigher observational type theory
- Category Latest Changes
- Started by nLab edit announcer
- Comments 6
- Last comment by nLab edit announcer
- Last Active Jun 19th 2022

- Discussion Type
- discussion topiccompactly generated topological space
- Category Latest Changes
- Started by Todd_Trimble
- Comments 81
- Last comment by Urs
- Last Active Jun 19th 2022

I left a counter-query underneath Zoran’s query at compactly generated space. It may be time for a clean-up of this article; the query boxes have been left dangling and unanswered for quite some time. Either proofs or references to detailed proofs would be welcome.

- Discussion Type
- discussion topicabelian infinity-group
- Category Latest Changes
- Started by Hurkyl
- Comments 2
- Last comment by Urs
- Last Active Jun 19th 2022

Should this topic be renamed to something like “$E_\infty$ group” or some similar thing? I haven’t seen “abelian” used elsewhere to describe this notion.

IMO that choice of name is potentially misleading. For example, it could also refer to a model of the usual finite product theory of abelian groups: i.e. an object of the $\infty$-category of by connective chain complexes of abelian groups modulo quasi-isomorphism. This is actually specifically what I would have expected from the term.

This example is, in some sense, also “more commutative” than being a grouplike $E_\infty$ monoid, which makes the description of being “maximally abelian” misleading as well.

Link to topic: abelian infinity-group

- Discussion Type
- discussion topictype checking
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 18th 2022

This content used to be sitting inside

*decidability*, and “type checking” was redirecting to there. But clearly type checking deserves its own entry (though currently it remains a stub.)

- Discussion Type
- discussion topicdecidability
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active Jun 18th 2022

created a stub for

*decidability*, mainly only so that the mainy pointers to it do point somewhere

- Discussion Type
- discussion topicconnected space
- Category Latest Changes
- Started by Todd_Trimble
- Comments 21
- Last comment by Urs
- Last Active Jun 18th 2022

I added a bunch of things to connected space: stuff on the path components functor, an example of a countable connected Hausdorff space, and the observation that the quasi-components functor is left adjoint to the discrete space functor $Set \to Top$ (Wikipedia reports that the connected components functor is left adjoint to the discrete space functor, but that’s wrong).

This bit about quasi-components functor had never occurred to me before, although it seems to be true. I’m having difficulty getting much information on this functor. For example, does it preserve finite products? I don’t know, but I doubt it. Does anyone reading this know?

- Discussion Type
- discussion topicMartti Karvonen
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active Jun 18th 2022

- Discussion Type
- discussion topicbiproduct
- Category Latest Changes
- Started by Colin Tan
- Comments 16
- Last comment by nLab edit announcer
- Last Active Jun 18th 2022

- Discussion Type
- discussion topiccontractible space
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Urs
- Last Active Jun 18th 2022

I have briefly fixed the clause for topological spaces at

*contractible space*, making manifest the distinction between contractible and weakly contractible.

- Discussion Type
- discussion topicGünther Trautmann
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active Jun 18th 2022

- Discussion Type
- discussion topiccompact connected space
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by Urs
- Last Active Jun 18th 2022

- Discussion Type
- discussion topicHoTT in Bonn2018
- Category Latest Changes
- Started by nLab edit announcer
- Comments 6
- Last comment by Urs
- Last Active Jun 18th 2022

- Discussion Type
- discussion topicHoTT at DMV2015
- Category Latest Changes
- Started by nLab edit announcer
- Comments 3
- Last comment by Urs
- Last Active Jun 18th 2022

- Discussion Type
- discussion topicHoTT Mini-Course 2016
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active Jun 18th 2022

- Discussion Type
- discussion topicencode-decode method
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jun 17th 2022

- Discussion Type
- discussion topichomotopy type theory events
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jun 17th 2022

- Discussion Type
- discussion topicCMU HoTT Research Group's local activities
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jun 17th 2022

- Discussion Type
- discussion topichalving group
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active Jun 17th 2022