Maths - Quasi-Category

Infinity Categories

In addition to objects and morphisms infinity categories have unlimited higher level morphisms (morphisms between morphisms and morphisms between morphisms between morphisms and so on).

Such categories can be characterised by two numbers (n,k) where:

  • n: is the maximum dimension, in this case ∞.
  • k: the dimension above which all morphisms are invertible.

I am looking at this from a topological point of view. Where a morphisms is invertible this looks like an isomorphism or equivilance but here it may only be upto a homotopy equivalence. That is, there may be many arrows between a given source and target, but they at all related.

Here the most commonly used infinity categories are ( ∞ , 0) and ( ∞ , 1) :

  ( ∞ , 0) category
or
∞-groupoid
( ∞ , 1) category
Models for the category: Kan complexes

quasi-category

or
weak Kan complex

More about infinity-categories on this page

see Wiki

Quasi-Category

A quasi-category is a generalisation of the notion of a category. In a category the composition of two morphisms is uniquely defined but in a quasi-category all the morphisms that can serve as composition of two given morphisms are related to each other by higher order invertible morphisms.

Definition of Quasi-Category as a Simplicial Set

A quasi-category is a simplicial set A in which every inner horn can be extended (not necessarily uniquely) to a simplex. That is, for every map /\k[n] -> A (0<k<n), there exists an extension Δ[n] -> A along the inclusion /\k[n] ->Δ[n].

diagram

Where:

  • A is a simplicial set
  • Δ[n] is an n-dimensional simplex.
  • /\k[n] is the k-th horn of an n-dimensional simplex.
diagram

Example of a 2-dimensional simplex.

diagram

Example of a 2-dimensional horn with face opposite 0 missing.

More about:

diagram

This blue area is intended to illustrate some general topological space (which may have holes in it).

There could be many paths from the composition of paths A and B

Higher Category Theory

In higher category theory diagrams may commute up to homotopy, but not on the nose, so higher homotopical information is needed to characterise the morphisms below it. (homotopy coherence)

A ( ∞ , 1) category is a category with

Quasi-categories and Simplicial Sets

Quasi-categories can be represented by simplicial sets where:

diagram The triangle is intended to represent all the possible compositions of A and B that can be continuously deformed into each other.
We can combine these together so A is composed with B and also with C. diagram
diagram Or here B•A=D•C
If we compose a sequence of three morphisms we get homotopies between homotopies. diagram
diagram This is best shown in 3 dimensions where each morphism is shown in a different dimension.

Nerve and Geometric Realisation

Nerve and geometric realisation are a pair of adjoint functors

The nerve is the right adjoint.

Weak Kan Complexes

Weak Kan complexes are a model for (infinity,1) categories.


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see also:

http://stackoverflow.com/questions/13352205/what-are-free-monads

Correspondence about this page

Book Shop - Further reading.

Where I can, I have put links to Amazon for books that are relevant to the subject, click on the appropriate country flag to get more details of the book or to buy it from them.

flag flag flag flag flag flag The Princeton Companion to Mathematics - This is a big book that attempts to give a wide overview of the whole of mathematics, inevitably there are many things missing, but it gives a good insight into the history, concepts, branches, theorems and wider perspective of mathematics. It is well written and, if you are interested in maths, this is the type of book where you can open a page at random and find something interesting to read. To some extent it can be used as a reference book, although it doesn't have tables of formula for trig functions and so on, but where it is most useful is when you want to read about various topics to find out which topics are interesting and relevant to you.

 

Terminology and Notation

Specific to this page here:

 

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