Lawvere's universal Hopf algebra

Lawvere was one of the early proponents of objective combinatorics,
where one seeks to work directly with finite sets rather than with
their numbers. In this context he discovered that there is a kind of
universal Hopf algebra H, whose Möbius function induces the Möbius
function of every incidence coalgebra by pullback along a certain
interval construction. He made the discovery in the 1980s, but it
remained unpublished until [Lawvere-Menni, TAC 2010], where they also
establish the objective Möbius inversion principle. The slightly
disturbing fact that H is not itself the incidence coalgebra of any
poset or category was overcome by Gálvez-Kock-Tonks in 2014 with the
discovery that simplicial spaces more general than posets and
categories admit the incidence coalgebra construction and a general
Möbius inversion principle at the objective level. These simplicial
spaces are called decomposition spaces. Lawvere's Hopf algebra H is in
fact the incidence coalgebra of a decomposition space U, and it was
conjectured that U is universal among decomposition spaces and culf
maps (another notion introduced by Lawvere, in his work on dynamical
systems, and exploited in Lawvere-Menni). The conjecture was recently
proved by Forero, pin-pointing finally the precise universal property
of Lawvere's Hopf algebra. The talk will be an introduction to all
these ideas, including background on Möbius inversion, without going
too much into technical details and proofs.