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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 combinatorics complex-geometry computable-mathematics computer-science constructive constructive-mathematics cosmology deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry differential-topology digraphs duality education elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph graphs gravity grothendieck group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory history homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory internal-categories k-theory lie-theory limit limits linear linear-algebra locale localization logic 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 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-theory subobject superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory variational-calculus

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- Discussion Type
- discussion topicfinite product
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 5 days ago

added the statement that categories with finite products are cosifted (here)

- Discussion Type
- discussion topiccommutativity of limits and colimits
- Category Latest Changes
- Started by Marc Hoyois
- Comments 31
- Last comment by Urs
- Last Active 5 days ago

Added to commutativity of limits and colimits the case of coproducts commuting with connected limits in a topos, and the generalization to higher topoi. This particular instance of commutativity is not mentioned very often, probably because it’s not very impressive in Set, but its generalization to higher topoi (for which I couldn’t find a reference) is more interesting. For instance, cofiltered limits commute with taking quotients by an ∞-group in an ∞-topos.

- Discussion Type
- discussion topicsifted colimit
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 5 days ago

added the statement that categories with finite products are cosifted (here). Since this is referenced or used in a few other entries, I will give the statement and its proof a stand-alone entry now…

- Discussion Type
- discussion topicchromatic homotopy theory
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 5 days ago

I gave

*chromatic homotopy theory*an Idea-section.To be expanded eventually…

- Discussion Type
- discussion topicinternal category
- Category Latest Changes
- Started by Urs
- Comments 37
- Last comment by Urs
- Last Active 6 days ago

I edited the formatting of internal category a bit and added a link to internal infinity-groupoid

it looks like the first query box discussion there has been resolved. Maybe we can remove that box now?

- Discussion Type
- discussion topicgeometry of physics -- categories and toposes
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active 6 days ago

I’ll be preparing here notes for my lectures

*Categories and Toposes (schreiber)*, later this month.

- Discussion Type
- discussion topichom-functor preserves limits
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 6 days ago

copied over to here (from internal hom) statement and proof that also an internal hom-bifunctor preserves ordinary limits in both arguments (now this prop)

- Discussion Type
- discussion topicpresheaf of groupoids
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active 6 days ago

for convenience of hyperlinking and disambiguation, I need an entry of this title, in between presheaf and (2,1)-presheaf. Started with a brief Idea-section that just scans the different possible meanings and their relation. Might copy over more material from

*geometry of physics – categories and toposes*once that has stabilized more.

- Discussion Type
- discussion topicmatching family
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 days ago

added remark about and pointer to

*Cech groupoid*as co-representing sets of matching families (here)

- Discussion Type
- discussion topicČech groupoid
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 days ago

I have added the actual general definition of the Cech groupoid as presheaf of groupoids, and headlined the definition previously offered here as “Idea”. Then I added detailed statement and proof, that the Cech-groupoid co-represents sets of matching families for set-valued presheaves (now this prop.)

- Discussion Type
- discussion topiclimits commute with limits
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 days ago

for ease of linking, I gave this statement its own entry, to go along with the companion entries

*adjoints preserve (co-)limits*and*hom-functor preserves limits*(and maybe more of this kind, to come)

- Discussion Type
- discussion topiclimit
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 6 days ago

I toiuched the formatting and the hyperlinking of the paragraphs on compatibility of limits with other universal constructions.

Merged the previous tiny subsections on this to a single one, now

*Compatibility with universal constructions*.added the hyperlink to the stand-alone entry

*adjoints preserve (co-)limits*.Will create an analogous stand-alone entry for

*limits commute with limits*.

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

This entry is currently undecided as to whether “full subcategory” inclusion requires the functor to be an injection on objects. It begins by pointing to

*subcategory*which does require this, but before long it speaks about fully faitful functors being full subcategory inclusions.This will be confusing to newcomers. There should be at least some comments about invariance under equivalence of categories.

Ah, now I see that at

*subcategory*there is such a discussion (here). Hm, there is some room for cleaning-up here.

- Discussion Type
- discussion topicsoft linear logic
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active 7 days ago

- Discussion Type
- discussion topicspace and quantity
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by David_Corfield
- Last Active 7 days ago

worked on space and quantity a bit

tried to polish the introduction and the Examples-section a bit

added a section on the adjunction with a detailed end/coend computation of the fact that it is an adjunction.

- Discussion Type
- discussion topicsmall site
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

- Discussion Type
- discussion topicpoints-to-pieces transform
- Category Latest Changes
- Started by Urs
- Comments 4
- Last comment by Urs
- Last Active 7 days ago

I have split off the section on

*points-to-pieces transform*from*cohesive topos*and expanded slightly, pointing also to*comparison map between algebraic and topological K-theory*

- Discussion Type
- discussion topicsifted category
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active 7 days ago

I have expanded a little at

*sifted category*: added the example of the reflexive-coequalizer diagram, added the counter-examples of the non-reflexive coequalizer diagram, added a references.

- Discussion Type
- discussion topicterminal category
- Category Latest Changes
- Started by Richard Williamson
- Comments 3
- Last comment by Urs
- Last Active 7 days ago

Added a redirect for final category, and made a couple of tiny additions.

- Discussion Type
- discussion topicempty category
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

discovered this old entry. Touched the formatting and added cross-links with

*terminal category*.

- Discussion Type
- discussion topicgeneralized quantifier
- Category Latest Changes
- Started by Thomas Holder
- Comments 2
- Last comment by Urs
- Last Active 7 days ago

I created a new page generalized quantifier mostly to drop some references.

- Discussion Type
- discussion topicgeometry of physics -- coordinate systems
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 7 days ago

did a little bit of reorganization. Removed one layer of $sub^n$-sections, moved the lead-in paragraphs to before the table of contents, added cross-links to

*geometry of physics – categories and toposes*at the point where the concept of categories appears.

- Discussion Type
- discussion topicIsbell duality
- Category Latest Changes
- Started by spitters
- Comments 14
- Last comment by Urs
- Last Active 7 days ago

http://ncatlab.org/nlab/show/Isbell+duality

Suggests that Stone, Gelfand, … duality are special cases of the adjunction between CoPresheaves and Presheaves. A similar question is raised here. http://mathoverflow.net/questions/84641/theme-of-isbell-duality

However, this paper http://www.emis.ams.org/journals/TAC/volumes/20/15/20-15.pdf

seems to use another definition. Could someone please clarify?

- Discussion Type
- discussion topicspectral symmetric algebra
- Category Latest Changes
- Started by Urs
- Comments 22
- Last comment by David_Corfield
- Last Active 7 days ago

I have created an entry

*spectral symmetric algebra*with some basics, and with pointers to Strickland-Turner’s Hopf ring spectra and Charles Rezk’s power operations.In particular I have added amplification that even the case that comes out fairly trivial in ordinary algebra, namely $Sym_R R$ is interesting here in stable homotopy theory, and similarly $Sym_R (\Sigma^n R)$.

I am wondering about the following:

In view of the discussion at spectral super scheme, then for $R$ an even periodic ring spectrum, the superpoint over $R$ has to be

$R^{0 \vert 1} \;=\; Spec(Sym_R \Sigma R) \simeq Spec\left( R \wedge \left( \underset{n \in \mathbb{N}}{\coprod} B\Sigma(n)^{\mathbb{R}^n} \right)_+ \right) \,.$This of course is just the base change/extension of scalars under Spec of the “absolute superpoint”

$\mathbb{S}^{0\vert 1} \simeq Spec(Sym_{\mathbb{S}} (\Sigma \mathbb{S}))$(which might deserve this notation even though the sphere spectrum is of course not even periodic).

This looks like a plausible answer to the quest that David C. and myself were on in another thread, to find a plausible candidate in spectral geometry of the ordinary superpoint $\mathbb{R}^{0 \vert 1}$, regarded as the base of the brane bouquet.

- Discussion Type
- discussion topicnormal form
- Category Latest Changes
- Started by David_Corfield
- Comments 3
- Last comment by David_Corfield
- Last Active Jun 11th 2018

Started this page normal form, but I see there might be a difference between the no-further-rewrites idea and the designated set of normal terms idea (as in disjunctive normal form).

- Discussion Type
- discussion topicSpectral Algebraic Geometry
- Category Latest Changes
- Started by Urs
- Comments 2
- Last comment by Urs
- Last Active Jun 11th 2018

I have cross-linked with

*Structured Spaces*, indicating that*Structured Spaces*has morphed into Part I of*Spectral Algebraic Geometry*, I suppose(?) (both of these are category:reference entries for recording the documents of these titles by Jacob Lurie)

- Discussion Type
- discussion topicgeometry of physics -- perturbative quantum field theory
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active Jun 11th 2018

I am changing the page title – this used to be “A first idea of quantum field theory”, which of course still redirects. The “A first idea…” seemed a good title for when this was an ongoing lecture that was being posted to PhysicsForums. I enjoyed the double meaning one could read into it, but it’s a bad idea to carve such jokes into stone. And now that the material takes its place among the other chapters of

*geometry of physics*, with the web of cross-links becoming thicker, the canonical page name clearly is “perturbative quantum field theory”.

- Discussion Type
- discussion topicfunctorial geometry
- Category Latest Changes
- Started by Urs
- Comments 16
- Last comment by Urs
- Last Active Jun 11th 2018

The scan of the writeup of Grothendieck’s 73 Buffalo lecture that we point to at

*functorial geometry*is really badly done. Is there a better scan or any other re-typing available?

- Discussion Type
- discussion topicGrp
- Category Latest Changes
- Started by Todd_Trimble
- Comments 1
- Last comment by Todd_Trimble
- Last Active Jun 11th 2018

- Discussion Type
- discussion topicadjoint quadruple
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Urs
- Last Active Jun 11th 2018

I have fleshed out (and corrected) and then spelled out the proof of the statement (here) that Kan extension of an adjoint pair is an adjoint quadruple:

For $\mathcal{V}$ a symmetric closed monoidal category with all limits and colimits, let $\mathcal{C}$, $\mathcal{D}$ be two small $\mathcal{V}$-enriched categoriesand let

$\mathcal{C} \underoverset {\underset{p}{\longrightarrow}} {\overset{q}{\longleftarrow}} {\bot} \mathcal{D}$be a $\mathcal{V}$-enriched adjunction. Then there are $\mathcal{V}$-enriched natural isomorphisms

$(q^{op})^\ast \;\simeq\; Lan_{p^{op}} \;\colon\; [\mathcal{C}^{op},\mathcal{V}] \longrightarrow [\mathcal{D}^{op},\mathcal{V}]$ $(p^{op})^\ast \;\simeq\; Ran_{q^{op}} \;\colon\; [\mathcal{D}^{op},\mathcal{V}] \longrightarrow [\mathcal{C}^{op},\mathcal{V}]$between the precomposition on enriched presheaves with one functor and the left/right Kan extension of the other.

By essential uniqueness of adjoint functors, this means that the two Kan extension adjoint triples of $q$ and $p$

$\array{ Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& Ran_{q^{op}} \\ && Lan_{p^{op}} &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}} }$merge into an adjoint quadruple

$\array{ Lan_{q^{op}} &\dashv& (q^{op})^\ast &\dashv& (p^{op})^\ast &\dashv& Ran_{p^{op}} } \;\colon\; [\mathcal{C}^{op},\mathcal{V}] \leftrightarrow [\mathcal{D}^{op}, \mathcal{V}]$