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Online seminář z algebry - 17.6.2021 PDF Tisk

Další seminář z algebry se koná 17.6.2021 od 13.00 online na platformě ZOOM. Informace pro připojení a další program semináře je zde.

Leonardo Larizza

Lax factorisation systems and categories of partial maps

Abstrakt:
Lax factorisation systems and categories of partial maps
Factorisation systems describe morphisms in a category by factorising them into pairs of composable morphisms. Their definition depends on a  kind of orthogonality relation between morphisms, which entails the existence of some diagonal morphisms for certain squares. In this seminar we present the new notion of lax weak orthogonality between morphisms, which involves lax squares and the factorisation systems it generates. Then we will introduce lax versions of functorial and algebraic weak factorisation systems and some of their properties. These lax factorisation systems are discussed, keeping the theory of ordinary factorisation systems as a blueprint and providing useful properties.
An overview of the examples of such lax factorisation systems is presented in the context of partial maps. We conclude with a discussion of general constructions of these examples and their description in the particular case of sets with partial maps.

Aktualizováno Úterý, 15 Červen 2021 13:52
 
Online seminář z algebry - 10.6.2021 PDF Tisk

Další seminář z algebry se koná 10.6.2021 od 13.00 online na platformě ZOOM. Informace pro připojení a další program semináře je zde.

Noam Zeilberger

Skew monoidal categories and the proof-theoretic anatomy of associativity (and unitality)

Abstrakt:
Based on joint work with Tarmo Uustalu and Niccolò Veltri.

The talk will survey a recent line of work, which takes a proof-theoretic approach to solving the coherence problem(s) for skew monoidal categories and related structures. I will begin by discussing the so-called Tamari order on fully-bracketed words induced by a semi-associative law (AB)C <= A(BC), and explain how a simple sequent calculus may account for some of its fascinating properties, such as the fact that the set of fully-bracketed words on n+1 letters forms a lattice Y_n under this order,as well as a remarkable formula counting the number of intervals in Y_n.
Then I will recall the definition of skew monoidal categories, and explain how a more refined sequent calculus may be used to solve two related coherence problems: deciding equality of maps and enumerating homsets in free skew monoidal categories. Closely related to recent work by Bourke and Lack, this sequent calculus may be considered as a canonical construction of the free left representable skew multicategory over aset of atoms.
Finally, I will briefly discuss variations of the sequent calculus capturing "partially skew" monoidal categories with different normality conditions.

References:
[1] https://arxiv.org/abs/1803.10080
[2] https://arxiv.org/abs/2003.05213
[3] https://arxiv.org/abs/2101.10487




Aktualizováno Středa, 09 Červen 2021 07:49
 
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