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

• 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.

• at monoidal adjunction the second item says

while the left adjoint is necessarily strong

but should it not say

while the left adjoint is necessarily oplax

?

• 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)”.

• 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.)

• 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.

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

• 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).

• 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.

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

• 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.)

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