The rabbit hole.

Research

1. 01
   Operational Semantics $\lambda :\mathrm{\Sigma }B\Rightarrow B\mathrm{\Sigma }$

   Turi–Plotkin Homology

   Rules in the GSOS format are natural transformations, and their models are bialgebras. The free bialgebra gives a comonad, and its bar resolution turns every model into a chain complex. The homology sees the algebra of a model under its rules, not only its behaviour.

   NotesTalk
2. 02
   Quantitative Algebra $T_{U}A$

   Magnitude Meets Persistence

   A quantitative equational theory $U$ gives every metric space $A$ a free algebra $T_{U}A$. Filtering its length nerve by length yields persistent homology, and the associated graded is magnitude homology. A long exact sequence ties the two together, and the barcode in degree $n$ moves by at most $(n+1)\delta$ under a perturbation of size $\delta$.

   PreprintSlides
3. 03
   Categories and Learning ${\mathrm{Lan}}_{J}F$

   Kan Extensions for Transfer

   A change of task is a functor $J:\mathcal{A}\to \mathcal{B}$, and a representation is a functor $F$ into invariants. The left Kan extension ${\mathrm{Lan}}_{J}F$ is the structure the new task has to carry. Its distance to what is observed scores the transfer, and for one-parameter persistence modules that distance is the bottleneck distance.

   PreprintSources
4. 04
   Topological Groupoids $H_{\bullet }(\mathcal{G};A)$

   Homology of Ample Groupoids

   Ample groupoids carry a homology built from compactly supported functions on their nerve. With discrete coefficients a universal coefficient theorem holds, and without them it fails. On the Cantor set this homology differs from that of the classifying space already in degree zero.

   PapersTalk

• free algebras
• adjunctions
• homology
• derived colimits