created a simple entry ring object, just for completeness

]]>starting page on ordered local rings to reflect the analogy

local ring : Heyting field :: ordered local ring : ordered field

Anonymous

]]>Added a link to the PDF file with full text:

]]>separating stable equality out from decidable equality

Anonymous

]]>included pointer to the new preprint:

Will add this also to *G-structure*, *Cartan geometry* and maybe other related entries.

In this entry I mean to write out a full proof for the transgression formula for (discrete) group cocycles, using just basic homotopy theory and the Eilenberg-Zilber theorem.

Currently there is an Idea-section and the raw ingredients of the proof. Still need to write connecting text. But have to interrupt for the moment.

]]>some minimum, for the moment just for the convenience that the link works

(in creating this entry I noticed that we have an ancient stub entry *mapping complex* that deserves some attention)

Added:

Suppose $C$ is a category that admits small coproducts.

Given simplicial objects $A,B\in C^{\Delta^{op}}$,
their **function complex** is a simplicial set

whose set of $n$-simplices is the set of maps

$\Delta^n\otimes A\to B,$where $\otimes$ denotes the copowering of simplicial objects over simplicial sets given by

$(C\otimes D)_n=\coprod_{i\in C_n}D_n.$The original definition of a function complex in the generality stated above is due to Daniel M. Kan:

- Daniel M. Kan,
*On c.s.s. categories*, Boletín de la Sociedad Matemática Mexicana 2 (1957), 82–94. PDF.

moved text from higher inductive type and added references

Anonymous

]]>added definition

Anonymous

]]>started bracket type, just for completeness, but don’t really have time for it

]]>starting page on axiom C0 in cohesive homotopy type theory

Anonymous

]]>added pointer to this discussion of possible realization of the SYK-model in condensed matter physics:

- D. I. Pikulin, M. Franz,
*Black hole on a chip: proposal for a physical realization of the SYK model in a solid-state system*, Phys. Rev. X 7, 031006 (2017) (arXiv:1702.04426)

brief `category:people`

-entry for hyperlinking references

stub for *braid group statistics* (again, for the moment mainly in order to record a reference)

This is a bare list of references, to be `!include`

-ed into the References-lists of relevant entries (such as at *anyon*, *topological order*, *fusion category*, *unitary fusion category*, *modular tensor category*).

There is a **question** which I am after here:

This seems to be CMT folklore, as all authors state it without argument or reference.

*Who is really the originator of the claim that anyonic topological order is characterized by certain unitary braided fusions categories/MTCs?*

Is it Kitaev 06 (which argues via a concrete model, in Section 8 and appendix E)?

]]>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.

]]>Page created, but author did not leave any comments.

Anonymous

]]>creating the counterpart to preconvergence space but for sequences only instead of for all nets.

Anonymous

]]>One can use Cartan geometry to provide the so-called first-order formulation of gravity. The discussion in https://math.ucr.edu/home/baez/octonions/node19.html makes clear that one can realize E8 as the symmetry group of some space involving the octonions. Is there a meaningful application of these two facts to a formulation of the heterotic string in terms of octonions? John Baez considered this idea (as seen in https://golem.ph.utexas.edu/category/2014/11/integral_octonions_part_8.html) but I was not able to find a concretization of this.

]]>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)”.

]]>I added some discussion to Hausdorff space of how the localic and spatial versions compare in classical and constructive mathematics, including in particular the fact that I just learned (in discussion with Martin Escardo and Andrej Bauer) that a discrete locale is Hausdorff iff it has decidable equality.

]]>Added wikipedia link to initial value problem

Anonymous

]]>A simple characterization I just came across in some synthetic domain theory.

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