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2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor 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 infinity integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology newpage nlab noncommutative noncommutative-geometry number 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 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

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- Discussion Type
- discussion topicJiří Matoušek
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 hour ago

- Discussion Type
- discussion topicmoment map
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by Urs
- Last Active 3 hours ago

I have expanded and edited

*moment map*.

- Discussion Type
- discussion topictransport
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by Urs
- Last Active 4 hours ago

- Discussion Type
- discussion topicidentification
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active 4 hours ago

- Discussion Type
- discussion topicdependent identity type
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by Mike Shulman
- Last Active 4 hours ago

- Discussion Type
- discussion topictype telescope
- Category Latest Changes
- Started by nLab edit announcer
- Comments 6
- Last comment by Mike Shulman
- Last Active 4 hours ago

- Discussion Type
- discussion topicKan complex
- Category Latest Changes
- Started by Urs
- Comments 19
- Last comment by Dmitri Pavlov
- Last Active 5 hours ago

added a sentence to the Idea-section at Kan complex

- Discussion Type
- discussion topicidentity type
- Category Latest Changes
- Started by Urs
- Comments 35
- Last comment by Urs
- Last Active 6 hours ago

added to identity type a mentioning of the alternative definition in terms of inductive types (paths).

- Discussion Type
- discussion topicdependent type
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 6 hours ago

the entries

*dependent type*and*indexed set*did not know of each other.I have now cross-linked them minimally in their “Related entries”-sections. But this would deserve to be expanded on for exposition…

- Discussion Type
- discussion topicfunction field analogy
- Category Latest Changes
- Started by Urs
- Comments 66
- Last comment by Guest
- Last Active 8 hours ago

started

*function field analogy – table*, but didn”t get very far before being interrupted nowwas aiming for the table in section 2.6 of

- Bjorn Poonen, of
*Lectures on rational points on curves*, 2006 (pdf)

- Bjorn Poonen, of

- Discussion Type
- discussion topicdifferential ring groupoid
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 11 hours ago

- Discussion Type
- discussion topicdifferential ring
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 11 hours ago

- Discussion Type
- discussion topicright cancellative category > history
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by nLab edit announcer
- Last Active 11 hours ago

- Discussion Type
- discussion topicleft cancellative category > history
- Category Latest Changes
- Started by Peter Heinig
- Comments 14
- Last comment by nLab edit announcer
- Last Active 11 hours ago

Created left cancellative category. This is a useful technical term.

One natural (and non-posetal) example is the category of fields with ring homomorphisms as the morphisms—provided that the zero ring is removed from it.

Incidentally: is there a usual technical term for, when considering any category $\mathsf{C}$, the full subcategory obtained by

*removing*all terminal objects of $\mathsf{C}$?

- Discussion Type
- discussion topicprincipal ideal
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 12 hours ago

- Discussion Type
- discussion topicunique factorization domain
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Guest
- Last Active 16 hours ago

- Discussion Type
- discussion topicBézout ring
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 16 hours ago

- Discussion Type
- discussion topicconcept with an attitude
- Category Latest Changes
- Started by Urs
- Comments 31
- Last comment by Guest
- Last Active 17 hours ago

This is maybe mainly for entertainment. But don’t forget that for newcomers there is a real issue here which may well be worth explaining:

In mathematics it happens at times that one and the same concept is given two different names to indicate a specific perspective, a certain attitude as to what to do whith such objects.

Here are examples:

A

*quiver*is just a directed graph (pseudograph, to be explicit). But one says*quiver*instead of*directed graph*when one is interested in studying*quiver representations*: functors from the free category on that graph to the category of finite-dimensional vector spaces.A

*presheaf*is just a contravariant functor. But one says*presheaf*instead of*contravariant functor*when one is interested in studying its sheafification, or even if one is just intersted in regarding the category of functors with its structure of a topos: the presheaf topos.

(…)

- Discussion Type
- discussion topicBézout domain
- Category Latest Changes
- Started by nLab edit announcer
- Comments 4
- Last comment by nLab edit announcer
- Last Active 18 hours ago

added section about Bézout domains in constructive mathematics, most of the text copied from principal ideal domain

Anonymous

- Discussion Type
- discussion topicstability of persistence diagrams
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by Urs
- Last Active 20 hours ago

- Discussion Type
- discussion topicpersistent homology
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 20 hours ago

added pointer to today’s

- Gunnar Carlsson,
*Persistent Homology and Applied Homotopy Theory*, in: Handbook of Homotopy Theory, CRC Press, 2019 (arXiv:2004.00738)

- Gunnar Carlsson,

- Discussion Type
- discussion topicisomorphism class
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by nLab edit announcer
- Last Active 21 hours ago

I noticed that “isomorphism class” was just re-directing to

*decategorification*. Just for completeness, I am giving it its own little entry hereby.

- Discussion Type
- discussion topicequivalence class
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 21 hours ago

- Discussion Type
- discussion topicUlrich Bauer
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicMichael Lesnick
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicMagnus Bakke Botnan
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicLeonidas J. Guibas
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicPaul Bendich
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicAmit Patel
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicdiamond principle
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicMarc Glisse
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicFrédéric Chazal
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicpersistence diagram
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicDavid Cohen-Steiner
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicDmitriy Morozov
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicSara Kališnik
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topiczigzag persistence module
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicrational zero theorem
- Category Latest Changes
- Started by nLab edit announcer
- Comments 1
- Last comment by nLab edit announcer
- Last Active 1 day ago

- Discussion Type
- discussion topiccancellative binary function
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicpartition of unity
- Category Latest Changes
- Started by DavidRoberts
- Comments 23
- Last comment by Urs
- Last Active 1 day ago

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?’.

- Discussion Type
- discussion topiclogical relation
- Category Latest Changes
- Started by Urs
- Comments 14
- Last comment by maxsnew
- Last Active 1 day ago

I gave the entry

*logical relation*an Idea-section, blindly stolen from a pdf by Ghani that I found on the web. Please improve, I still don’t know what a “logical relation” in this sense actually is.Also, I cross-linked with

*polymorphism*. I hope its right that “parametricity” may redirect there?

- Discussion Type
- discussion topicVin de Silva
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicGabriel's theorem
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicsynthetic mathematics
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by jonsterling
- Last Active 1 day ago

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.

- Discussion Type
- discussion topicRalf Schiffler
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicSteve Y. Oudot
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicquiver representation
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 1 day ago

I just discovered that, all along, the term “quiver representation” was just redirecting to

*representation*. Have started this dedicated page now, with the bare minimum

- Discussion Type
- discussion topicsynthetic Tait computability
- Category Latest Changes
- Started by David_Corfield
- Comments 3
- Last comment by maxsnew
- Last Active 1 day ago

- Discussion Type
- discussion topictorsion subgroup
- Category Latest Changes
- Started by Richard Williamson
- Comments 4
- Last comment by nLab edit announcer
- Last Active 1 day ago

Linking to the new page torsion points of an elliptic curve.

- Discussion Type
- discussion topichigher category theory and physics
- Category Latest Changes
- Started by Urs
- Comments 15
- Last comment by nLab edit announcer
- Last Active 1 day ago

I have been further working on the entry higher category theory and physics. There is still a huge gap between the current state of the entry and the situation that I am hoping to eventually reach, but at least now I have a version that I no longer feel ashamed of.

Here is what i did:

Partitioned the entry in two pieces: 1. “Survey”, and 2. “More details”.

The survey bit is supposed to give a quick idea of what the set of the scene of fundamental physics is. It starts with a kind of creation story of physics from $\infty$-topos theory, which – I think – serves to provide a solid route from just the general abstract concept of space and process to the existence and nature of all $\sigma$-model quantum field theories of “$\infty$-Chern-Simons theory”-type (which includes quite a few) and moreover – by invoking the “holographic principle of higher category theory” – all their boundary theories, which includes all classical phase space physics.

The Survey-bit continues with indicating the formalization of the result of quantizing all these to full extended quantum field theories. It ends with a section meant to indicate what is and what is not yet known about the quantization step itself. This is currently the largest gap in the mathematical (and necessarily higher categorical) formalization of physics: we have a fairly good idea of the mathematics that describes geometric background structure for physics and a fairly good idea of the axioms satisfied by the quantum theories obtained from these, but the step which takes the former to the latter is not yet well understood.

The “More details”-bit is stubby. I mainly added one fairly long subsection on the topic of “Gauge theory”, where I roughly follow the historical route that eventually led to the understanding that gauge fields are modeled by cocycles in higher (nonabelian) differential cohomology.

Apart from this I added more references and some cross-links.

I know that the entry is still very imperfect. If you feel like pointing out all the stuff that is still missing, consider adding at least some keywords directly into the entry.

- Discussion Type
- discussion topicHaldane model
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active 1 day ago

- Discussion Type
- discussion topicsynthetic computability theory
- Category Latest Changes
- Started by maxsnew
- Comments 2
- Last comment by Urs
- Last Active 2 days ago

- Discussion Type
- discussion topicenriched poset
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Urs
- Last Active 2 days ago

- Discussion Type
- discussion topicseparated presheaf
- Category Latest Changes
- Started by Urs
- Comments 6
- Last comment by Urs
- Last Active 2 days ago

I worked a little on the entry separated presheaf. Apart from some general editing I

added construction and proof of the separafication functor;

began a section on the full notion of bi-separated presheaf.

More deserves to be done here, but I have to stop for the moment.

- Discussion Type
- discussion topicElements of Mathematics
- Category Latest Changes
- Started by Dmitri Pavlov
- Comments 4
- Last comment by Urs
- Last Active 2 days ago

Created:

Elements of Mathematics (French: Éléments de mathématique) is a series of books by Nicolas Bourbaki.

Website: https://www.bourbaki.fr/Ouvrages.html

Most of these books were quite influential, some of them especially so.

In particular, Bourbaki’s presentation of multilinear algebra in Chapter III of the second book (Algèbre) was the first detailed expository presentation of the Grassmann algebra.

Another influential book was

*Groupes et algèbres de Lie*, its presentation of root systems and related topics in Chapter 4–6 was the first of its kind.Some criticisms raised against this series:

Almost complete absence of categories (a deliberate choice);

*Théorie des ensembles*receives a lot of criticism for its somewhat nonstandard treatment of logic, as well as the rather awkward notion of a*structure*, which occupies a similar niche to categories.The use of Daniell’s approach in

*Intégration*, mostly ignoring abstract measure spaces.Chapter 10 of

*Algèbre*(homological algebra) only covers the very classical results, (intentionally) ignoring derived categories.

## Related entries

- Discussion Type
- discussion topicconcrete category
- Category Latest Changes
- Started by Mike Shulman
- Comments 7
- Last comment by Hurkyl
- Last Active 2 days ago

Added the statement of the Isbell-Freyd characterization of concrete categories, in the special case of finitely complete categories for which it looks more familiar, along with the proof of necessity.

- Discussion Type
- discussion topicnoncommutative geometry
- Category Latest Changes
- Started by Urs
- Comments 7
- Last comment by nLab edit announcer
- Last Active 2 days ago

added textbook pointer:

- José Gracia-Bondia, Joseph Varilly, Héctor Figueroa,
*Elements of noncommutative geometry*, Birkhäuser 2001. xviii+685 pp. (doi:10.1007/978-1-4612-0005-5, gBooks)

- José Gracia-Bondia, Joseph Varilly, Héctor Figueroa,

- Discussion Type
- discussion topiccontinuous map
- Category Latest Changes
- Started by Urs
- Comments 11
- Last comment by nLab edit announcer
- Last Active 2 days ago

I gave

*continuous map*a little bit of substance by giving it an actual Idea-paragraph and by writing out the epsilontic definition for the case of metric spaces, together with its equivalence to the “abstract” definition in terms of opens.

- Discussion Type
- discussion topicreal-cohesive (infinity,1)-topos
- Category Latest Changes
- Started by nLab edit announcer
- Comments 2
- Last comment by Guest
- Last Active 2 days ago

- Discussion Type
- discussion topicstably compact space
- Category Latest Changes
- Started by PaoloPerrone
- Comments 4
- Last comment by Urs
- Last Active 2 days ago