### More Papers on Magnitude

#### Posted by Simon Willerton

I’ve been distracted by other things for the last few months, but in that time several interesting-looking papers on magnitude (co)homology have appeared on the arXiv. I will just list them here with some vague comments. If anyone (including the author!) would like to write a guest post on any of them then do email me.

For years a standing question was whether magnitude was connected with persistent homology, as both had a similar feel to them. Here Nina relates magnitude homology with persistent homology.

- Magnitude cohomology by Richard Hepworth

In both mine and Richard’s paper on graphs and Tom Leinster and Mike Shulman’s paper on general enriched categories, it was magnitude *homology* that was considered. Here Richard introduces the dual theory which he shows has the structure of a non-commutative ring.

- Smoothness filtration of the magnitude complex by Kiyonori Gomi

I haven’t looked at this yet as I only discovered it last night. However, when I used to think a lot about gerbes and Deligne cohomology I was a fan of Kiyonori Gomi’s work with Yuji Terashima on higher dimensional parallel transport.

- Graph magnitude homology via algebraic Morse theory by Yuzhou Gu

This is the write-up of some results he announced in a discussion here at the Café. These results answered questions asked by me and Richard in our original magnitude homology for graphs paper, for instance proving the expression for magnitude homology of cyclic graphs that we’d conjectured and giving pairs of graphs with the same magnitude but different magnitude homology.

Posted at November 2, 2018 11:05 AM UTC
## Re: More Papers on Magnitude

Here are the main points of my paper on magnitude cohomology:

Magnitude cohomology is the “dual” to magnitude homology in the same sense that singular cohomology is “dual” to singular homology. It has the structure of a graded associative ring, but not necessarily commutative.

For metric spaces with a certain separation condition (there exists positive $\delta$ such that if $x\neq y$ then $d(x,y)\geq\delta$), which includes all graphs, the magnitude cohomology ring determines the metric space up to isometry!

For enriched categories, the magnitude cohomology ring is graded-commutative so long as the enriching category is cartesian. This applies to posets (where the enriching category is $\{\text{true},\text{false}\}$ and the magnitude cohomology is the ordinary cohomology of the order complex) and to ordinary categories (where the enriching category is Set and the magnitude cohomology is the ordinary cohomology of the classifying space). It does not apply to metric spaces (where the enriching category is $[0,\infty]$).

Magnitude cohomology rings are sometimes computable! For any diagonal graph this is super-easy, and the result is a specific quotient of the quiver algebra. I also work out what happens for odd cyclic graphs, where I need to apply Yuzhou Gu’s computation of the magnitude homology.