Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Decomposition spaces in Combinatorics   A decomposition space (also called unital 2-Segal space) is a simplicial
object satisfying an exactness condition weaker than the Segal condition: just
as the Segal condition expresses (up to homotopy) composition, the new
condition expresses decomposition. It is a general framework for incidence
(co)algebras. In the present contribution, after establishing a formula for the
section coefficients, we survey a large supply of examples, emphasising the
notion's firm roots in classical combinatorics. The first batch of examples,
similar to binomial posets, serves to illustrate two key points: (1) the
incidence algebra in question is realised directly from a decomposition space,
without a reduction step, and reductions are often given by CULF functors; (2)
at the objective level, the convolution algebra is a monoidal structure of
species. Specifically, we encounter the usual Cauchy product of species, the
shuffle product of L-species, the Dirichlet product of arithmetic species, the
Joyal-Street external product of q-species and the Morrison `Cauchy' product of
q-species, and in each case a power series representation results from taking
cardinality. The external product of q-species exemplifies the fact that
Waldhausen's S-construction on an abelian category is a decomposition space,
yielding Hall algebras. The next class of examples includes Schmitt's chromatic
Hopf algebra, the Fa\`a di Bruno bialgebra, the Butcher-Connes-Kreimer Hopf
algebra of trees and several variations from operad theory. Similar structures
on posets and directed graphs exemplify a general construction of decomposition
spaces from directed restriction species. We finish by computing the M\"obius
function in a few cases, and commenting on certain cancellations that occur in
the process of taking cardinality, substantiating that these cancellations are
not possible at the objective level.

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The categories of this paper are: solv-int
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The categories of this paper are: comp-gas
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The categories of this paper are: stat
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The categories of this paper are: astro-ph
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The categories of this paper are: bayes-an
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The categories of this paper are: cmp-lg
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The categories of this paper are: chao-dyn
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The categories of this paper are: physics
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The categories of this paper are: q-alg
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The categories of this paper are: q-fin
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The categories of this paper are: quant-ph
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The categories of this paper are: atom-ph
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The categories of this paper are: hep-th
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The categories of this paper are: alg-geom
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The categories of this paper are: cond-mat
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The categories of this paper are: hep-ph