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2-category 2-category-theory abelian-categories accessible adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry anomalies arithmetic arithmetic-geometry beauty bundles calculus categories category category-theory chern-simons-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 deformation-theory descent diagrams differential differential-cohomology differential-geometry duality education elliptic-cohomology enriched fibration finite foundations functional-analysis functor galois-theory gauge-theory gebra geometric-quantization geometry goodwillie-calculus graph 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 infinity integration-theory internal-categories k-theory kan lie-theory limit limits linear linear-algebra locale localization logic manifolds mathematics measure-theory modal-logic model model-category-theory monoidal monoidal-category-theory morphism motives motivic-cohomology newpage nonassociative noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory 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 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 topological topology topos topos-theory tqft type type-theory universal web

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- Discussion Type
- discussion topicdisplayed category
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by maxsnew
- Last Active 11 hours ago

New page displayed category.

- Discussion Type
- discussion topicCotopology and cocompact space
- Category Latest Changes
- Started by Daniel Luckhardt
- Comments 3
- Last comment by Daniel Luckhardt
- Last Active 13 hours ago

I created article cotopology including a redirect from cocompact space.

- Discussion Type
- discussion topicp-derivation
- Category Latest Changes
- Started by David_Corfield
- Comments 3
- Last comment by David_Corfield
- Last Active 15 hours ago

I separated p-derivation from Fermat quotient.

- Discussion Type
- discussion topicTopological Concrete Category
- Category Latest Changes
- Started by Tim Campion
- Comments 10
- Last comment by Mike Shulman
- Last Active 19 hours ago

I’ve just edited topological concrete category to correct the claim that topological functors create limits, which is not quite true: for instance, the forgetful $U: \mathrm{Top} \to \mathrm{Set}$ fails to reflect limits because choosing a finer topology on the limit vertex yields a non-limiting cone with the same image in $\mathrm{Set}$. This is correctly reported on wikipedia and in Joy of Cats, p. 227.

It is true that topological functors allow you to calculate limits using the image of the diagram under the functor, which is quite powerful. In Joy of Cats, a topological functor is said to “uniquely lift limits” (definition p. 227, proven p. 363). There doesn’t seem to be an nlab page for this property – I suppose it’s not much used by most category theorists.

- Discussion Type
- discussion topicover-topos
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by IngoBlechschmidt
- Last Active 23 hours ago

created over-topos

- Discussion Type
- discussion topicKleisli category
- Category Latest Changes
- Started by Urs
- Comments 28
- Last comment by Matt Earnshaw
- Last Active 1 day ago

I have tried to brush-up

*Kleisli category*; also made*Kleisli composition*redirect to it and cross-linked with*monad (in computer science)*

- Discussion Type
- discussion topicmonoidal adjunction
- Category Latest Changes
- Started by Urs
- Comments 17
- Last comment by Mike Shulman
- Last Active 1 day ago

at

*monoidal adjunction*the second item sayswhile the left adjoint is necessarily strong

but should it not say

while the left adjoint is necessarily oplax

?

- Discussion Type
- discussion topicuniform space
- Category Latest Changes
- Started by Mike Shulman
- Comments 20
- Last comment by David_Corfield
- Last Active 1 day ago

I added the definition of uniform space in terms of covering families. But I don’t know the covering version of the constructive “axiom (0)”.

- Discussion Type
- discussion topicarithmetic jet space
- Category Latest Changes
- Started by Urs
- Comments 10
- Last comment by David_Corfield
- Last Active 1 day ago

created

*arithmetic jet space*, so far only highlighting the statement that at prime $p$ these are $X \underset{Spec(\mathbb{Z})}{\times}Spec(\mathbb{Z}_p)$ (regarded so in Borger’s absolute geometry by applying the Witt ring construction $(W_n)_\ast$ to it).This is what I had hoped that the definition/characterization would be, so I am relieved. Because this is of course just the definition of synthetic differential geometry with $Spec(\mathbb{Z}_p)$ regarded as the $p$th abstract formal disk.

Well, or at least this is what Buium defines. Borger instead calls $(W_n)_\ast$ itself already the arithmetic jet space functor. I am not sure yet if I follow that.

I am hoping to realize the following: in ordinary differential geometry then synthetic differential infinity-groupoids is cohesive over “formal moduli problems” and here the flat modality $\flat$ is exactly the analog of the above “jet space” construction, in that it evaluates everything on formal disks. Moreover, $\flat$ canonically sits in a fracture suare together with the “cohesive rationalization” operation $[\Pi_{dR}(-),-]$ and hence plays exactly the role of the arithmetic fracture square, but in smooth geometry. I am hoping that Borger’s absolute geometry may be massaged into a cohesive structure over the base $Et(Spec(\mathbb{F}_1))$ that makes the cohesive fracture square reproduce the arithmetic one.

If Borger’s absolute direct image were base change to $Spec(\mathbb{Z}_p)$ followed by the Witt vector construction, then this would come really close to being true. Not sure what to make of it being just that Witt vector construction. Presently I have no real idea of what good that actually is (apart from giving any base topos for $Et(Spec(Z))$, fine, but why this one? Need to further think about it.)

- Discussion Type
- discussion topiccontinuous metric space valued function on compact metric space is uniformly continuous
- Category Latest Changes
- Started by Urs
- Comments 5
- Last comment by Urs
- Last Active 2 days ago

while writing out the proof of the fundamental product theorem in K-theory I had occasion to record that

- Discussion Type
- discussion topicdoctrinal adjunction
- Category Latest Changes
- Started by FinnLawler
- Comments 5
- Last comment by Urs
- Last Active 2 days ago

Created doctrinal adjunction. The page could probably use some examples and/or fleshing out.

- Discussion Type
- discussion topicdablock at [[inverse image]]
- Category Latest Changes
- Started by TobyBartels
- Comments 3
- Last comment by Mike Shulman
- Last Active 2 days ago

I put a disambiguation block at the top of inverse image to point to preimage.

- Discussion Type
- discussion topicinduced metric
- Category Latest Changes
- Started by Urs
- Comments 1
- Last comment by Urs
- Last Active 2 days ago

created

*induced metric*, just for completeness

- Discussion Type
- discussion topicparametric right adjoints
- Category Latest Changes
- Started by Mike Shulman
- Comments 2
- Last comment by Mike Shulman
- Last Active 3 days ago

Created parametric right adjoint.

- Discussion Type
- discussion topicdiagram chasing lemmas - contents
- Category Latest Changes
- Started by Urs
- Comments 37
- Last comment by Peter Heinig
- Last Active 3 days ago

created

*diagram chasing lemmas - contents*(with some information on what implies what) and included it as a floating TOC into the relevant entries.

- Discussion Type
- discussion topicexponential object
- Category Latest Changes
- Started by Urs
- Comments 9
- Last comment by Mike Shulman
- Last Active 4 days ago

I had started an entry “exponentiation” but then thought better of it and instead expanded the existing exponential object: added an examples-section specifically for $Set$ and made some remarks on exponentiation of numbers.

- Discussion Type
- discussion topicYoneda embedding
- Category Latest Changes
- Started by Urs
- Comments 8
- Last comment by Todd_Trimble
- Last Active 4 days ago

Added Eric’s illustrations to the Idea-section at Yoneda embedding.

- Discussion Type
- discussion topicPROP
- Category Latest Changes
- Started by fpaugam
- Comments 23
- Last comment by Mike Shulman
- Last Active 4 days ago

- I would like to include something on wheeled properads (or wheeled PROPs) in the nlab. It seems to me that a wheeled prop is something like a symmetric monoidal category with duals for every object generated by one object. Is this right? Is there a place in the litterature where i can find the relation between wheeled properads used by Merkulov and some kinds of symmetric monoidal categories with duality?

Before changing the PROP entry to add this variant, i would like to have a nice reference on this.

- Discussion Type
- discussion topicspatial topos
- Category Latest Changes
- Started by David_Corfield
- Comments 1
- Last comment by David_Corfield
- Last Active 4 days ago

Although referred to in a couple of place, it seems we had no entry for spatial topos, so I’ve made a start.

- Discussion Type
- discussion topictwosets_nonames20170618
- Category Latest Changes
- Started by Peter Heinig
- Comments 1
- Last comment by Peter Heinig
- Last Active 5 days ago

- created twosets_nonames20170618

- Discussion Type
- discussion topictwosets_op_with_names20170618
- Category Latest Changes
- Started by Peter Heinig
- Comments 1
- Last comment by Peter Heinig
- Last Active 5 days ago

- created twosets_op_with_names20170618, which is Illustration of opposite of the category that twosets20170617 is an illustration of.

twosets_op_with_names20170618 is identical to twosets_op_nonames20170618 except for the morphisms having names.

- Discussion Type
- discussion topictwosets_op_nonames20170618
- Category Latest Changes
- Started by Peter Heinig
- Comments 1
- Last comment by Peter Heinig
- Last Active 5 days ago

- created twosets_op_nonames20170618

- Discussion Type
- discussion topiccategory of simple graphs
- Category Latest Changes
- Started by Todd_Trimble
- Comments 62
- Last comment by Mike Shulman
- Last Active 5 days ago

A long time ago we had a discussion at graph about notions of morphism. I have written an article category of simple graphs which collects some properties of the category under one of those definitions (corresponding better, I think, to graph-theoretic practice).

- Discussion Type
- discussion topictwosets20170617
- Category Latest Changes
- Started by Peter Heinig
- Comments 1
- Last comment by Peter Heinig
- Last Active 6 days ago

Created twosets20170617. Contains an svg illustration of a full subcategory of $\mathsf{Set}$ consisting of a terminal object and a two-element set. Uses the convention that an identity arrow is labelled by its object. Intended for use in some graph-theoretical considerations from an nPOV. Sufficiently general to be possibly of use in some other nLab articles too.

- Discussion Type
- discussion topicinternal type theory
- Category Latest Changes
- Started by spitters
- Comments 1
- Last comment by spitters
- Last Active 6 days ago

Created a stub for internal type theory to collect some references. At some point we could move more from Mike’s blog post.

- Discussion Type
- discussion topicamazing right adjoint
- Category Latest Changes
- Started by Urs
- Comments 3
- Last comment by Peter Heinig
- Last Active 6 days ago

created amazing right adjoint

- Discussion Type
- discussion topicnonstandard model of arithmetic
- Category Latest Changes
- Started by jesse
- Comments 2
- Last comment by Todd_Trimble
- Last Active 6 days ago

I wrote down some standard facts about nonstandard models of Peano arithmetic at nonstandard model of arithmetic, and spelled out two proofs (really the same proof) that the ordering on the nonstandard part is always dense.

- Discussion Type
- discussion topicetendue
- Category Latest Changes
- Started by Todd_Trimble
- Comments 7
- Last comment by Todd_Trimble
- Last Active 6 days ago

I changed “irreflexive graph” to “directed graph” (aka quiver), as such a presheaf is more commonly known. (Irreflexive to my mind means loops at a vertex are

*forbidden*.)

- Discussion Type
- discussion topicpseudo-distributive law
- Category Latest Changes
- Started by Mike Shulman
- Comments 1
- Last comment by Mike Shulman
- Last Active 7 days ago

Somehow we didn’t yet have a page about pseudo-distributive laws, so I made a stubby one with a few examples and references.

- Discussion Type
- discussion topicregular epimorphisms
- Category Latest Changes
- Started by Mike Shulman
- Comments 33
- Last comment by Todd_Trimble
- Last Active 7 days ago

Comment/question on terminology in the (?,1)-case at regular epimorphism