Corrigendum to “Decomposition spaces, incidence algebras and Möbius inversion II: Completeness, length filtration, and finiteness” [Adv. Math. 333 (2018) 1242–1292] (Advances in Mathematics (2018) 333 (1242–1292), (S000187081830104X), (10.1016/j.aim.2018.03.017))

Alex Cebrian has pointed out a mistake in Corollary 7.12 of our paper [4], which in turn affects Corollary 7.13 and Examples 7.15 and 7.17. He has also provided an explicit counter-example and indicated a fix of the statements to make the proofs valid. The mistake does not affect the main results of any of the three papers in the trilogy [3–5]. The erroneous result is an explicit formula for the section coefficients in the incidence coalgebra of a Segal groupoid. Since this formula makes sense independently of the theory it illustrates, and is easy to apply in many situations, there is a certain risk it could be used by other mathematicians; to minimise the potential damage we feel obliged to emit the present corrigendum, which contains corrected statements and proofs. The corollaries concern the section coefficients of the incidence coalgebra of a Segal space X with 1-truncated [Formula presented], such as notably the fat nerve of a small category. For simplicity we explain the problem in this case. The section coefficients are the numbers [Formula presented] such that [Formula presented] in the numerical incidence coalgebra of X, where [Formula presented] is the homotopy cardinality of the map [Formula presented]. The argument amounts to calculating the homotopy cardinality of the space [Formula presented], where the letter subscripts indicate fibre of the face maps [Formula presented], [Formula presented] and [Formula presented], respectively. Lemma 7.11 of [4] shows that [Formula presented] is discrete, and is naturally equivalent to the discrete groupoid [Formula presented]. Assuming that a and b are composable up to isomorphism, so that we have isomorphisms [Formula presented] this discrete groupoid can also be described as the loop space [Formula presented]. We had claimed in Corollary 7.12 that [Formula presented], but the factor [Formula presented] is not correct even for fat nerves of small categories: it does map to [Formula presented], by [Formula presented] (with left-to-right composition), but not every [Formula presented] is mapped to something isomorphic to [Formula presented]. The explicit counter-example provided by Cebrian is in the fat nerve of the category of finite sets and surjections, whose incidence bialgebra is the Faà di Bruno bialgebra. Consider [Formula presented], [Formula presented], and [Formula presented] given as [Formula presented] in the picture, only [Formula presented], [Formula presented], [Formula presented], [Formula presented] give a composite isomorphic to f, whereas [Formula presented], [Formula presented] do not. Our incorrect formula for the section coefficient in Corollary 7.13 gave [Formula presented] (rather than passing to the loop space [Formula presented]). To compute the pullback [Formula presented] is discrete (which in turn is a consequence of the fact that [Formula presented] is assumed 1-truncated). Note that if X is the fat nerve of a category, then [Formula presented] defined as the fibre [Formula presented]. Proof We compute [Formula presented] as the sum of the fibres for the upper square in the diagram [Formula presented] Since D is discrete, we get [Formula presented] □ Applying this to the situation of (1) now yields the correct version of Corollary 7.12: [Formula presented] is a 1-groupoid. Given [Formula presented] we have [Formula presented] The mapping space depends only on whether f and [Formula presented] are equivalent or not, in the space [Formula presented]. Precisely: [Formula presented] So we need as many copies of the loop space as there are φ giving f. Taking cardinality, we therefore arrive at the following corrected formula for the section coefficients. Recall that in the case of the fat nerve of a category, [Formula presented]. Corrected Corollary 7.13 Suppose X is the fat nerve of a category. Then the section coefficients of the incidence coalgebra are given by [Formula presented] Here [Formula presented] is the set consisting of those [Formula presented] for which there exists an isomorphism [Formula presented]. Remark A version of the argument was used already by Cebrian [1] for calculation of section coefficients in the plethystic bialgebra. To complete the Corrigendum, we amend the statements of the discussion following Corollary 7.13 and the examples given there. Coassociativity can be interpreted as stating that the section coefficients constitute a 2-cocycle. In the paper we moved on from Corollary 7.13 to show that this 2-cocycle [Formula presented] is trivial, by exhibiting it as the coboundary of the 1-cocycle [Formula presented] This argument breaks down, as it relied on the wrong formula [Formula presented]. It is not true that the 2-cocycle [Formula presented] is trivial (except for posets and preorders). (This is actually good, because one should not expect 2-dimensional data (such as composition in a category) to be determined by 1-dimensional data.) In Examples 7.15 and 7.17, we used the corollaries to describe the multiplication formula for the convolution algebra of the fat nerve of a category (the ‘fat category algebra’). The given formula [Formula presented] is false. (Here [Formula presented] denotes the representable (covariant) presheaf [Formula presented], [Formula presented].) As the counter-example shows, some elements [Formula presented] may yield different composites, not isomorphic to ab. The correct formula is [Formula presented] as follows readily by dualising the corrected Corollary 7.12. This formula has a clear intuitive content (more so than the original wrong formula): to multiply two arrows in the ‘fat category algebra’, sum over all connecting isos φ to make a and b composable, and return the triple composite. It should be stressed here that the cardinality of [Formula presented] is not the basis element [Formula presented] dual to [Formula presented]. Rather [Formula presented]. With this, at the numerical level the convolution algebra has [Formula presented], so that the section coefficients are the same as for the coalgebra, as expected. The subtleties regarding this duality are carefully treated in [2].