Category theory

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




The category SimpSetSimpSet, or sSetsSet for short, is the category of simplicial sets.

This is the functor category from the opposite category Δ op\Delta^{op} of the simplex category Δ\Delta to the category Set of sets:

SimpSet:=[Δ op,Set]. Simp Set := [\Delta^{op}, Set] \,.

Its objects are simplicial sets.


Like all categories of presheaves on a small category, the category SimpSet of simplicial sets is complete and cocomplete (with limits and colimits constructed levelwise) and cartesian closed. In fact, like all presheaf categories, it is a topos.

Monoidal structure

As described at closed monoidal structure on presheaves the cartesian tensor product ST=S×TS \otimes T = S \times T of simplicial sets SS and TT is the simplicial set

(ST):[n]S n×T n, (S \otimes T) : [n] \mapsto S_n \times T_n \,,

where the product on the right is the cartesian product in Set.

One central reason why simplicial sets are useful and important is that this simple monoidal structure (“disturbingly simple minded” in the words of Friedman08, p. 24) actually does fully capture the standard monoidal structure on topological spaces under geometric realization ||:SSetTop|\cdot| : SSet \to Top


For SS and TT simplicial sets, we have

|S×T||S|×|T|, |S \times T| \simeq |S| \times |T| \,,

where on the right the cartesian product is in the nice category of compactly generated Hausdorff spaces.

See also products of simplices.

Closed structure

As described at closed monoidal structure on presheaves the internal hom [S,T][S,T] of simplicial sets is the simplicial set

[S,T]:[n]Hom SSet(S×Δ[n],T), [S,T] : [n] \mapsto Hom_{SSet}(S \times \Delta[n], T) \,,

where Δ[n]=Hom Δ(,[n])\Delta[n] = Hom_{\mathbf{\Delta}}(-,[n]) is the standard simplicial nn-simplex, the image of [n]Δ[n] \in \mathbf{\Delta} under the Yoneda embedding. This formula is clearly representing a Kan extension.


The maps N:CatSimpSetN: \Cat \rightarrow \Simp\Set and S:TopSimpSetS: \Top \rightarrow \Simp\Set described in the examples are actually functors, both of which have left adjoints. These adjoint pairs are examples of a very general sort of adjunction involving simplicial sets, of which there are many examples.

Let EE be any cocomplete category and let F:ΔEF: \Delta \rightarrow E be a functor. We define the right adjoint R:ESimpSetR : E \rightarrow \Simp\Set as follows. Given an object eEe \in E the nn-simplices of ReRe are defined to be the set E(F[n],e)E(F[n],e) of morphisms in EE from F[n]F[n] to ee. Face and degeneracy maps are given by precomposition by the appropriate (dual) maps in the image of FF. RR is defined on morphisms by postcomposition.

The left adjoint LL is defined to be the left Kan extension of FF along the Yoneda embedding y:ΔSimpSety: \Delta \rightarrow \Simp\Set. Because the yy is full and faithful, we will have Ly=FLy = F, i.e., L(Δ[n])=F[n]L (\Delta[n]) = F[n]. By specifying FF, we have already defined a functor to EE on the represented simplicial sets; LL is the unique cocontinuous extension of this functor to SimpSet\Simp\Set. It can be described explicitly on objects as a coend, or as a weighted colimit.

(Easy) abstract nonsense shows that LL and RR form an adjoint pair LRL \dashv R.

Here are some examples:

  • Let E=CatE = \Cat and FF be the functor [n][n][n] \mapsto [n] (the inclusion of posets into categories). The right adjoint is the nerve functor NN described above. The left adjoint τ 1{\tau}_1 takes a simplicial set to its fundamental category.

  • Let E=TopE = \Top and FF be the functor [n]Δ n[n] \mapsto {\Delta}_n. The right adjoint is the total singular complex functor SS described above. The left adjoint |||-| is called geometric realization. As a consequence of the Kan extension construction, the geometric realization of the represented simplicial set Δ[n]\Delta[n] is the standard nn-simplex Δ n{\Delta}^n.

  • (Barycentric) subdivision and extension sd:SimpSetSimpSet:ex\sd: \Simp\Set \leftrightarrow \Simp\Set :\ex.

  • The homotopy coherent nerve functor and its left adjoint SimpSetSimpCat\Simp\Set \leftrightarrow \Simp\Cat where SimpCat? denotes the category of simplicially enriched categories, i.e., categories enriched in SimpSet\Simp\Set.

  • The adjunction ×X:SimpSetSimpSet:() X- \times X: \SimpSet \leftrightarrow \SimpSet :(-)^X between the product with a simplicial set XX and the internal-hom, which makes SimpSet\Simp\Set into a cartesian closed category.

  • Let EE be a Grothendieck topos equipped with an “interval” II, i.e. a totally ordered object in the internal logic equipped with distinct top and bottom elements. Then we have the functor ΔE\Delta \to E sending [n][n] to the subobject {(x 1,x 2,,x n)|x 1x 2x n}I n \{ (x_1,x_2,\dots,x_n) \;|\; x_1 \le x_2 \le \dots \le x_n \} \hookrightarrow I^n which gives rise to a geometric morphism ESimpSetE\to \SimpSet. Therefore, SimpSet\SimpSet is the classifying topos of such “intervals”.

Model category structures

There are important model category structures on sSetsSet.

Internal logic

Like any elementary topos, SimpSet\SimpSet has an internal logic. Here we list some properties of this logic.

  • It is a two-valued topos, i.e. the only subobjects of 1=Δ 01 = \Delta^0 are 00 and 11. (This is not really a property of the internal logic, but we include it to contrast with the next point.)

  • It is not Boolean. In general, the complement of a simplicial subset ABA\subseteq B is the full simplicial subset on the vertices of BB not contained in AA (“full” meaning it contains a simplex of BB as soon as it contains all its vertices). Thus, A¬A=BA\cup \neg A = B only if AA is a connected component of BB, i.e. any simplex with at least one vertex in AA lies entirely in AA.

  • By Diaconescu's theorem, SimpSet\SimpSet therefore does not satisfy the axiom of choice.

  • Like any presheaf topos, it satisfies the dependent choice (assuming it holds in the metatheory); see Fourman and Scedrov. Moreover, natural numbers object is simply the discrete simplicial set of ordinary natural numbers.

  • Similarly, it satisfies Markov's principle.

  • Less obviously, it satisfies the Kreisel-Putnam axiom? that (¬p(qr))=((¬pq)(¬pr))(\neg p \to (q\vee r)) = ((\neg p \to q) \vee (\neg p \to r)); see this MO question and answers.


  • Mike Fourman and Ščedrov, The “world’s simplest axiom of choice” fails, manuscripta mathematica 1982, Volume 38, Issue 3, pp 325-332 PDF

category: category

Revised on March 15, 2014 10:04:10 by Bas Spitters (