Roff - Iterated Magnitude homology

Abstract

Abstract : In 2017, extending Hepworth and Willerton’s construction for graphs, Leinster and Shulman introduced the magnitude homology of an enriched category. Their construction generalizes ordinary categorical homology: the magnitude homology of an ordinary small category is the homology of its classifying space. But categorical homology can also be generalized in a different direction. Duskin and Street established back in the 1980s that the classifying space extends naturally from categories to bicategories, lifting the theory to a second categorical dimension. This talk will explore a hybrid of the two ideas: an iterated magnitude homology theory for categories with a second-order enrichment. Such objects occur in nature more often than one might imagine---in particular, various commonly-encountered structures on groups can be described as second-order enrichments in familiar base categories. To keep things down-to-earth we will focus on such examples, investigating the content and behaviour of iterated magnitude homology when interpreted for groups equipped with extra structure such as a congruence, a partial ordering, or a bi-invariant metric. This talk will be based on the preprint https://arxiv.org/abs/2309.00577.

Magnitude Homology

We remind ourselves that categories are also enriched categories (as enrichements over sets).

A Magnitude is given by

• a monoidal category
• a ring
• A size homomorphism from the cat to the ring

A magnitude homology

• needs a specialized semicartesian monoidal category
• a monoidal abelian category (can multiply)
• a strong symmetric monoidal size functor $\Sigma :(V,\otimes )\to (\mathbb{A},\otimes )$

Construction:

• Construct the nerve of the category
• Take chains defined by Folge of objects, connected by morphisms.
• Make some homology and category magic

For a metric space there is a specialization of the homology which has already been discussed before (Magnitudenhomologie (Metrischer Raum)).

Call $(x,y)$ adjacent, if they are distinct and if there is no point between them (in a geodesic sense).

Enriched Groups

Often a group has extra structure. e.g.

• partially ordered group, coxeter groups with bruhat order
• normed groups (prop. $|gh|\le |g|+|h|$) Easy norms are got by word-lengths (think of generators in mapping class groups!)

E.g. Asa used normed fundamental groups to classify metric fibrations.

Enriched Group: A category, whose underlying ordinary category is a group. i.e. an enrichment of a group in the usual sense. Going back to the earlier examples:

• partially ordered groups are groups enriched in Poset
• normed groups are enriched in Met if the norm is conjugation invariant.

Definition Strict-2-Group: A group object enriched in $(Cat,\times )$.

Mac lane and whitehead showed that strict 2-groups classifiy homotopy 2-types of the classifying space.

Such a group canbe constructed rom a normal subgroup $N\subseteq G$.

• the objects are the elemnets of G
• Arros are the elemente of $N\times G$ with $(k,g):g\to kg$

Roff then goes on about the classifying space of a $2$-category There are apparently two constrcutions by “Duskin or Street Approach” and the “Segal approach”. But it turs out that both constrctions reslt in isomorphic spaces.

Iterated Magnitude Homology

Using the classifying space defined above we can define a so colled iterated magnitude homology.

Iterated magnitude homology for enriched groups

Summary

Various valuable structures on groups are instances of secnd-order enrichment. Iterated magnitude homology captures …