# Weak identity arrows in higher categories

@inproceedings{Kock2005WeakIA, title={Weak identity arrows in higher categories}, author={Joachim Kock}, year={2005} }

There are a dozen definitions of weak higher categories, all of which loosen the notion of composition of arrows. A new approach is presented here, where instead the notion of identity arrow is weakened — these are tentatively called fair categories. The approach is simplicial in spirit, but the usual simplicial category D is replaced by a certain ‘fat’ delta of ‘coloured ordinals’, where the degeneracy maps are only up to homotopy. The first part of this exposition is aimed at a broad…

## 25 Citations

Elementary remarks on units in monoidal categories

- MathematicsMathematical Proceedings of the Cambridge Philosophical Society
- 2008

Abstract We explore an alternative definition of unit in a monoidal category originally due to Saavedra: a Saavedra unit is a cancellable idempotent (in a 1-categorical sense). This notion is more…

Space-Valued Diagrams, Type-Theoretically (Extended Abstract)

- MathematicsArXiv
- 2017

It is shown how to define homotopy coherent diagrams which come with all higher coherence laws explicitly, with two variants that come with assumption on the index category or on the type theory.

Linear Logic without Units

- MathematicsArXiv
- 2013

A higher-dimensional analogue of Cayley's theorem is proved, and used to deduce a novel characterisation of the unit of a promonoidal category, and two characterisations of the categories that model the unitless fragment of intuitionistic multiplicative linear logic are given.

Coherence for Weak Units

- Mathematics
- 2009

We define weak units in a semi-monoidal 2-category C as cancellable pseudo-idempotents: they are pairs (I,α) where I is an object such that tensoring with I from either side constitutes a…

The periodic table of n-categories for low dimensions I: degenerate categories and degenerate bicategories

- Mathematics
- 2007

We examine the periodic table of weak n-categories for the low-dimensional cases. It is widely understood that degenerate categories give rise to monoids, doubly degenerate bicategories to…

NOTE ON COMMUTATIVITY IN DOUBLE SEMIGROUPS AND TWO-FOLD MONOIDAL CATEGORIES

- Mathematics
- 2007

A concrete computation — twelve slidings with sixteen tiles — reveals that certain commutativity phenomena occur in every double semigroup. This can be seen as a sort of EckmannHilton argument, but…

Non-unital polygraphs form a presheaf categories

- Mathematics
- 2017

We prove, as claimed by A.Carboni and P.T.Johnstone, that the category of non-unital polygraphs, i.e. polygraphs where the source and target of each generator are not identity arrows, is a presheaf…

Regular polygraphs and the Simpson conjecture.

- Mathematics
- 2018

We prove Carlos Simpson's "semi-strictification" (or "weak unit") conjecture in the case of infinity-groupoids. More precisely, we introduce two precise versions of the conjecture, the "general" and…

Non-unital polygraphs form a presheaf category

- Mathematics
- 2019

We prove, as claimed by A.Carboni and P.T.Johnstone, that the category of non-unital polygraphs, i.e. polygraphs where the source and target of each generator are not identity arrows, is a presheaf…

The periodic table of $n$-categories for low dimensions II: degenerate tricategories

- Mathematics
- 2007

We continue the project begun in ``The periodic table of $n$-categories for low dimensions I'' by examining degenerate tricategories and comparing them with the structures predicted by the Periodic…

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