The theory of decomposition spaces arose out of a need for a more flexbible framework for incidence (co)algebras and Möbius inversion than the classical setting of locally finite posets. As the name suggests, it is a theory of decomposition rather than composiition (as in categories), and it appears that all combinatorial coalgebras are incidence coalgebras of a decomposition space (whereas many of them cannot be realised directly as the incidence coalgebra of any poset). The lectures will be an introduction to decomposition spaces from the viewpoint of combinatorics, but the tools come from category theory and simplicial homotopy theory.
The first lecture will start with a brief review of the classical Möbius functioin in number theory, and Rota's theory for locally finite posets. Here the notion of reduction is important: often it has the flavour of cooking up an auxiliary poset in order to apply the classical theory, and then impose an equivalence relation so as to arrive at the coalgebra actually needed. With this motivation, we then move on to Möbius categories and decomposition spaces, and go through some examples beyond posets, such as the chromatic Hopf algebra of graphs and the Butcher-Connes-Kreimer Hopf algebra of rooted trees.

The second lecture will be a bit more categorical, including an introduction to objective algebraic combinatorics using objective linear algebra: this is linear algebra with coefficients in sets and groupoids instead of number coefficients. Slice categories play the role of vector spaces, and linear functors (given by spans) play the role of linear maps. By working at this level, the theory of decomposition spaces can be seen as a systematic way of lifting coalgebraic identities to 'bijective proofs'. I'll spend some time explaining the need for groupoids and homotopy theory, exemplified with the Faà di Bruno bialgebra.

The third lecture focuses on functorialities, explaining how certain simplicial maps called CULF and IKEO induce coalgebra homomorphisms. Especially important is the simplicial notion of decalage, which underlies many of the reductions in classical theory. CULF maps also enter in the definition of monoidal decomposition spaces, in turn inducing bialgebras and Hopf algebras (rather than just coalgebras). I hope to be able to finish with some potential applications to process calculi and quantum logic. Sometimes processes cannot be composed, or can be composed in many ways. Often this means there is instead a well-defined decomposition operation, and likely a decomposition space. Similarly, effect algebras and effect algebroids describe situations where composition is only partially defined, with axioms ensuring that they are in fact examples of decomposition spaces.