Objective combinatorics and decomposition spaces

Coalgebras and Hopf algebras in combinatorics arise from the
ability to decompose combinatorial structures. The theory of
decomposition spaces (also called 2-Segal spaces) serves among
other things to give an objective approach to combinatorial Hopf
algebras. 'Objective' here means working with the combinatorial
objects themselves, instead of with vector spaces derived from
them. Compared to the classical theory of incidence coalgebras of
posets, there are three generalisations involved: the passage
from vector spaces to slice categories; the passage from sets to
groupoids (and infinity-groupoids); the simplicial viewpoint
which prompts the passage from categories to decomposition
spaces.

The first lecture, after some introductory stuff, will outline
the main points of objective linear algebra, where slice
categories replace vector spaces, and spans and linear functors
replace linear maps. One advantage of this abstraction is the
ability to deal with infinite quantities. (While one should
generally avoid infinite numbers, there is no reason to be afraid
of an infinite set.) This leads to an neat interpretation of the
duality between vector spaces and pro-finite-dimensional vector
spaces, and a combinatorial interpretation of continuity.

The second lecture will start with incidence coalgebras of
posets, then generalise to Möbius categories, and see how these
constructions look objectively. Then we shall look at some
combinatorial Hopf algebras that cannot arise directly in this
way. Analysing this from a simplicial viewpoint leads to the
notion of decomposition spaces, which are simplicial objects
subject to some conditions tailor-made to make the incidence
coalgebra construction work. For the theory to work smoothly,
it is necessary to upgrade from sets to groupoids and use 
homotopy methods. This prompts the further upgrade from
groupoids to infinity-groupoids, and leads to surprising 
connections with other areas of mathematics (the theory of
higher Segal spaces).

References: for the viewpoint of combinatorics taken here, see
various papers of Gálvez-Kock-Tonks, starting perhaps with
"Decomposition spaces in combinatorics". For an ampler
perspective, see Dyckerhoff-Kapranov, "Higher Segal Spaces"
(Springer Lecture Notes).