**
higher category theory**
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category theory
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homotopy theory
## Basic concepts
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k-morphism,
coherence
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looping and delooping
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looping and suspension
## Basic theorems
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homotopy hypothesis-theorem
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delooping hypothesis-theorem
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periodic table
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stabilization hypothesis-theorem
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exactness hypothesis
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holographic principle
## Applications
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applications of (higher) category theory
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higher category theory and physics
## Models
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(n,r)-category
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Theta-space
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∞-category/
∞-category
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(∞,n)-category
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n-fold complete Segal space
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(∞,2)-category
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(∞,1)-category
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quasi-category
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algebraic quasi-category
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simplicially enriched category
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complete Segal space
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model category
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(∞,0)-category/
∞-groupoid
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Kan complex
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algebraic Kan complex
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simplicial T-complex
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(∞,Z)-category
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n-category = (n,n)-category
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2-category,
(2,1)-category
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1-category
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0-category
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(-1)-category
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(-2)-category
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n-poset =
(n-1,n)-category
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poset =
(0,1)-category
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2-poset =
(1,2)-category
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n-groupoid = (n,0)-category
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2-groupoid,
3-groupoid
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categorification/
decategorification
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geometric definition of higher category
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Kan complex
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quasi-category
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simplicial model for weak ∞-categories
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complicial set
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weak complicial set
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algebraic definition of higher category
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bicategory
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bigroupoid
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tricategory
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tetracategory
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strict ∞-category
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Batanin ∞-category
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Trimble ∞-category
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Grothendieck-Maltsiniotis ∞-categories
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stable homotopy theory
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symmetric monoidal category
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symmetric monoidal (∞,1)-category
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stable (∞,1)-category
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dg-category
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A-∞ category
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triangulated category
## Morphisms
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k-morphism
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2-morphism
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transfor
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natural transformation
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modification
## Functors
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functor
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2-functor
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pseudofunctor
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lax functor
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(∞,1)-functor
## Universal constructions
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2-limit
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(∞,1)-adjunction
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(∞,1)-Kan extension
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(∞,1)-limit
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(∞,1)-Grothendieck construction
## Extra properties and structure
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cosmic cube
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k-tuply monoidal n-category
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strict ∞-category,
strict ∞-groupoid
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stable (∞,1)-category
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(∞,1)-topos
## 1-categorical presentations
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homotopical category
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model category theory
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enriched category theory
Contents
Idea
A simplicial T-complex is essentially a Kan complex equipped with a choice of horn fillers – an algebraic Kan complex.
There is quite a difference between the Kan complex structure
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of the nerve of a groupoid, ,
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and that of, say, a singular complex of some topological space .
In the first, if we are given a -horn, then there is exactly one -simplex in , since the -horn has a chain of -composable arrows of in it (at least unless or , which cases are slightly different) and that chain gives the required -simplex. In other words, there is a ‘canonical’ filler for any horn. In , there will usually be many fillers.
Abstracting from this idea, Brown and Higgins developed the idea of a cubical T-complex. This was a cubical set with in each dimension , a subset of the -cubes being declared ‘thin’. The term was adopted to indicate that they, somehow, were of lower dimension than they looked to be. The theory was initially developed in the thesis of Keith Dakin in a simplicial context, and used by Brown and Higgins who showed that cubical -complexes were equivalent to crossed complexes. The corresponding simplicial -complex theory was developed in another thesis, this time by Nick Ashley, (see below).
Definition
A simplicial -complex is a pair , where is a simplicial set and is a graded subset of with . Elements of are called thin. The thin structure satisfies the following axioms:
- Every degenerate element is thin.
- Every horn in has a unique thin filler.
- A thin filler of a thin box also has its last face thin.
This is closely related to the definition of algebraic Kan complex.
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Together with very similar ideas of John Roberts, adapted by Ross Street, the notion of -complex is one of the precursors of Verity’s notion of complicial set.
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A closely related idea is that of group T-complex. Group -complexes form a category equivalent tor reduced crossed complexes
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The nerve of a crossed complex has a natural T-complex structure.
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If is a group T-complex, then its classifying space has a natural simplicial T-complex structure.
References
Relevant references for simplicial T-complexes include:
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N. Ashley, Simplicial T-Complexes: a non abelian version of a theorem of Dold-Kan, Disser- tations Math., 165, (1989), 11 – 58.
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G. Nan Tie, A Dold-Kan theorem for crossed complexes, J. Pure Appl. Alg., 56, (1989.), 177 – 194. 29, 37, 146, 254
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G. Nan Tie, Iterated W and T-groupoids, J. Pure Appl. Alg., 56, (1989), 195 – 209.