# Decomposition spaces, incidence algebras and Möbius inversion I: Basic theory

@article{GalvezCarrillo2018DecompositionSI, title={Decomposition spaces, incidence algebras and M{\"o}bius inversion I: Basic theory}, author={Imma G'alvez-Carrillo and Joachim Kock and Andrew Tonks}, journal={Advances in Mathematics}, year={2018} }

## 66 Citations

### Decomposition spaces, incidence algebras and Möbius inversion II: Completeness, length filtration, and finiteness

- MathematicsAdvances in Mathematics
- 2018

### Decomposition spaces, incidence algebras and Möbius inversion III: The decomposition space of Möbius intervals

- MathematicsAdvances in Mathematics
- 2018

### Decomposition spaces in combinatorics

- Mathematics
- 2016

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…

### Free decomposition spaces

- Mathematics
- 2022

We introduce the notion of free decomposition spaces: they are simplicial spaces freely generated by their inert maps. We show that left Kan extension along the inclusion j : ∆ inert → ∆ takes…

### C T ] 2 S ep 2 01 9 Decomposition-space slices are toposes

- Mathematics
- 2019

Decomposition spaces were introduced for purposes in combinatorics by Gálvez, Kock, and Tonks [7, 8, 9], and purposes in homological algebra, representation theory, and geometry by Dyckerhoff and…

### Categorical bialgebras arising from 2-Segal spaces

- Mathematics
- 2017

Topological field theory (TFT) is the study of representations of the cobordism category of manifolds. Such representations are often constructed from purely algebraic inputs. For example, 2d TFTs…

### Decomposition Spaces and Restriction Species

- MathematicsInternational Mathematics Research Notices
- 2018

We show that Schmitt’s restriction species (such as graphs, matroids, posets, etc.) naturally induce decomposition spaces (a.k.a. unital $2$-Segal spaces), and that their associated coalgebras are…

### THE GÁLVEZ–KOCK–TONKS CONJECTURE FOR DISCRETE DECOMPOSITION SPACES

- Mathematics
- 2021

Gálvez-Carrillo, Kock, and Tonks [17] constructed a decomposition space U of all Möbius intervals, as a recipient of Lawvere’s interval construction for Möbius categories, and conjectured that U…

### The G\'alvez-Kock-Tonks conjecture for locally discrete decomposition spaces

- Mathematics
- 2021

G´alvez-Carrillo, Kock, and Tonks [15] constructed a decomposition space U of all M¨obius intervals, as a recipient of Lawvere’s interval construction for M¨obius categories, and conjectured that U…

### The universal Hall bialgebra of a double 2-Segal space

- Mathematics
- 2017

Hall algebras and related constructions have had diverse applications in mathematics and physics, ranging from representation theory and quantum groups to Donaldson-Thomas theory and the algebra of…

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### Decomposition spaces, incidence algebras and Möbius inversion II: Completeness, length filtration, and finiteness

- MathematicsAdvances in Mathematics
- 2018

### Decomposition spaces, incidence algebras and Möbius inversion III: The decomposition space of Möbius intervals

- MathematicsAdvances in Mathematics
- 2018

### Decomposition spaces in combinatorics

- Mathematics
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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…

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We show that Schmitt’s restriction species (such as graphs, matroids, posets, etc.) naturally induce decomposition spaces (a.k.a. unital $2$-Segal spaces), and that their associated coalgebras are…

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