Homotopy theory

homotopy theory, (∞,1)-category theory, homotopy type theory

flavors: stable, equivariant, rational, p-adic, proper, geometric, cohesive, directed

models: topological, simplicial, localic, …

see also algebraic topology



Paths and cylinders

Homotopy groups

Basic facts




Subdivision is an (often) functorial process which takes as input some combinatorial notion of space (for example, a simplicial complex or simplicial set) and produces as output a more finely meshed space. It is related to the notion of classical subdivision.

There is a more general use of the term in which one simplicial set or complex is a subdivision of another, although the relation may not be functorially generated. We will briefly look at one such example later.

There are various standard forms of functorial subdivision, the most common of which is barycentric subdivision.


For simplicial complexes

Barycentric subdivision is easiest to define for simplicial complexes. We have a pair of functors

SimpComplexFlagUPosSimpComplex \stackrel{\overset{U}{\to}}{\underset{Flag}{\leftarrow}} Pos


  • The functor UU sends a simplicial complex (V,Σ)(V, \Sigma) to Σ\Sigma, regarded as a poset ordered by inclusion, and

  • The functor FlagFlag sends a poset PP to the simplicial complex whose vertex set is PP and whose simplices are the underlying sets {x 0,,x n}\{x_0, \ldots, x_n\} of flags x 0<<x nx_0 \lt \ldots \lt x_n.

The composite FlagUFlag \circ U is called the subdivision SdSd; it is an endofunctor of SimpComplexSimpComplex. A vertex of Sd(X)Sd(X) is a simplex of XX.

Let α X:XSd(X)\alpha_X : X \to Sd(X) be a morphism of simplicial complexes that sends a vertex vv of XX to the vertex {v}\{v\} of Sd(X)Sd(X). Then |f|:|X||Sd(X)|{|f|}: {|X|}\xrightarrow{\cong} {|Sd(X)|} is an isomorphism, where |||-| is the usual geometric realization of simplicial complexes. In terms of categories, α\alpha is a natural transformation from the identity to the endofunctor SdSd whose geometric realization is a natural isomorphism.

This is what it looks like for XX the 2-simplex.

simplicial subdivision of 2-simplex (better resolution, better size than previous images)

The simplicial complex which is the 2-simplex has as its poset of simplices the partially ordered set of non-empty subsets of {0,1,2}\{0,1,2\}. The subdivided simplex therefore is the flags of that poset. The diagram shows that nerve of the corresponding category, (or rather its dual), as this is the simplicial set that one gets from this. The central vertex is the whole set {0,1,2}\{0,1,2\}, the vertices on the long edges are the 2-element subsets and the three extremal vertices are the singletons.

For simplicial sets

We can now define the barycentric subdivision of a simplicial set as follows. Recall that we can make any simplicial complex into a simplicial set. If we do this for the standard nn-simplex, which as a simplicial complex is the set ([n],P([n])([n],P([n]) for [n]={0,1,,n}[n] = \{0,1,\dots,n\}, then we get the standard nn-simplex simplicial set Δ n\Delta^n. We can then define the subdivision of Δ n\Delta^n, as a simplicial set, to be the simplicial set corresponding to its simplicial-complex subdivision. Finally, we can make this functorial on maps between standard simplices, and left Kan extend to a cocontinuous endofunctor of SSetSSet.



Write SdΔ[n]Sd \Delta[n] for the nerve of category of non-degenerate simplices in the standard nn-simplex.

For XX an arbitrary simplicial set, its barycentric subdivision is

SdXlim Δ[n]XSdΔ[n]. Sd X \simeq \underset{\to}{\lim}_{\Delta[n] \to X} Sd \Delta[n] \,.

A review is for instance around (Fiore-Paoli def. 3.1).


Adjointness with the ExEx-functor

By construction (or the adjoint functor theorem), the subdivision functor SdSd on simplicial sets has a right adjoint, called ExEx. A countably infinite iteration of ExEx, called Ex Ex^\infty, can be used to construct Kan fibrant replacements in the model structure on simplicial sets.

This functorial subdivision corresponds to the classical barycentric subdivision.

Other types of subdivision

Other classical subdivisions that are frequently encountered include the middle edge subdivision. This latter is closely related to the ordinal subdivision of simplicial sets.

Relation to category of simplices


If every non-degenerate simplex in XX is given by a monomorphism Δ nX\Delta^n \to X, then the barycentric subdivision of def. 1 is equivalently given by the nerve of the full subcategory of its category of simplices on the non-degenerate simplices.

(Thomason, p. 311)


  • Rick Jardine, Simplicial approximation, Theory and Applications of Categories, Vol. 12, 2004, No. 2, pp 34-72. (web)

A review is in section 3 of

The relation to the nerve of the category of simplices is disucssed in

  • R. W. Thomason, Cat as a closed model category, Cahiers Topologie Géom. Différentielle, 21(3):305–324, 1980.

Revised on September 9, 2016 12:39:27 by Tim Porter (