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… 
View PDF on arXiv
25 Citations
Background Citations
12
Methods Citations
1

25 Citations

Elementary remarks on units in monoidal categories
  • Joachim Kock
  • Mathematics
    Mathematical 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)
TLDR
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
TLDR
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
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
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
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
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.
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
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
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…
...
...

References

SHOWING 1-10 OF 32 REFERENCES
Elementary remarks on units in monoidal categories
  • Joachim Kock
  • Mathematics
    Mathematical 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…
Homotopical algebraic geometry. I. Topos theory.
The equivalence of $\infty$-groupoids and crossed complexes
Multiple categories, and in particular n-fold categories, have been considered by various authors [8,11]. The object of this paper is to define ∞-categories (which are n-categories for all n) and…
Weak units and homotopy 3-types
We show that every braided monoidal category arises as $\End(I)$ for a weak unit $I$ in an otherwise completely strict monoidal 2-category. This implies a version of Simpson's weak-unit conjecture in…
Higher Operads, Higher Categories
Part I. Background: 1. Classical categorical structures 2. Classical operads and multicategories 3. Notions of monoidal category Part II. Operads. 4. Generalized operads and multicategories: basics…
Categories for the Working Mathematician
I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large…
Descente pour les n-champs (Descent for n-stacks)
We develop the theory of n-stacks (or more generally Segal n-stacks which are $\infty$-stacks such that the morphisms are invertible above degree n). This is done by systematically using the theory…
Operads, algebras, modules and motives
With motivation from algebraic topology, algebraic geometry, and string theory, we study various topics in differential homological algebra. The work is divided into five largely independent parts: I…
Simplicial localization of monoidal structures, and a non-linear version of Deligne's conjecture
We show that if $(M,\otimes,I)$ is a monoidal model category then $\mathbb{R}\underline{\rm End}_{M}(I)$ is a (weak) 2-monoid in sSet. This applies in particular when M is the category of A-bimodules…
Higher dimensional algebra and topological quantum field theory
The study of topological quantum field theories increasingly relies upon concepts from higher‐dimensional algebra such as n‐categories and n‐vector spaces. We review progress towards a definition of…
...
...