simplex category



The simplex category Δ\Delta encodes one of the main geometric shapes for higher structures. Its objects are the standard cellular nn-simplices. It is also called the simplicial category, but that term is ambiguous.



The augmented simplex category Δ a\Delta_a is the full subcategory of Cat on the free categories of finite linear directed graphs

{c 0c 1c n}. \{c_0 \to c_1 \to \cdots \to c_n\} \,.

Equivalently this is the category whose objects are finite totally ordered sets, or finite ordinals, and whose morphisms are order-preserving functions between them.


The simplex category Δ\Delta is the full subcategory of Δ a\Delta_a (and hence of CatCat) consisting of the free categories on finite and inhabited directed graphs, hence of non-empty linear orders or non-zero ordinals.


It is common, convenient and without risk to use a skeleton of Δ\Delta or Δ a\Delta_a, where we pick a fixed representative in each isomorphism class of objects. Since isomorphisms of totally ordered sets are unique this step is so trivial that it is often not even mentioned explicitly.

With this the objects of Δ\Delta are in bijection with natural numbers nn \in \mathbb{N} and one usually writes

[n]={01n} [n] = \{0 \to 1 \to \cdots \to n\}

for the object of Δ\Delta given by the category with (n+1)(n+1) objects. Geometrically one may think of this as the spine of the standard cellular nn-simplex, see the discussion of simplicial sets below. In this context one also writes Δ[n]\Delta[n] or Δ n\Delta^n for the simplicial set represented by the object [n][n]: the simplicial nn-simplex. By the Yoneda lemma one may identify the subcategory of simplicial sets on the Δ[n]\Delta[n] with Δ\Delta.

With this convention the first few objects of Δ\Delta are

[0]={0} [0] = \{0\}
[1]={01} [1] = \{0 \to 1\}
[2]={012} [2] = \{0 \to 1 \to 2\}


The category Δ a\Delta_a contains one more object, corresponding to the empty category \emptyset. When sticking to the above standard notation for the objects of Δ\Delta, that extra object is naturally often denoted

[1]=. [-1] = \emptyset \,.

However, in contexts where only Δ a\Delta_a and not Δ\Delta plays a role, some authors prefer to start counting with 0 instead of with 1-1. Then for instance the notation

0= \mathbf{0} = \emptyset
1=[0]={0} \mathbf{1} = [0] = \{0\}
2=[1]={01} \mathbf{2} = [1] = \{0 \to 1\}

and generally

n=[n1] \mathbf{n} = [n-1]

may be used.


The skeletal version of the augmented simplex category Δ a\Delta_a can be presented as follows:

  • objects are the finite totally ordered sets n:={0<1<<n1}\mathbf{n} := \{0 \lt 1 \lt \cdots \lt n-1\} for all nn \in \mathbb{N};

  • morphisms generated by (are all expressible as finite compositions of) the following two elementary kinds of maps

    1. face maps: δ i:=δ i n:n1n\delta_i := \delta_i^n : \mathbf{n-1} \hookrightarrow \mathbf{n} is the injection whose image leaves out i[n]i \in [n];

    2. degeneracy maps: σ i:=σ i n:n+1n\sigma_i := \sigma_i^n : \mathbf{n+1} \to \mathbf{n} is the surjection such that σ i(i)=σ i(i+1)=i\sigma_i(i) = \sigma_i(i+1) = i;

subject to the following relations, called the simplicial relations or simplicial identities:

δ j n+1δ i n=δ i n+1δ j1 n i<j σ j nσ i n+1=σ i n1σ j+1 n ij \array{ \delta_j^{n+1} \circ \delta_i^n = \delta_i^{n+1}\circ \delta_{j-1}^n & \; i \lt j \\ \sigma_j^n \circ \sigma_i^{n+1} = \sigma_i^{n-1} \circ \sigma_{j+1}^n & \; i \leq j }
σ j nδ i n+1={δ i nσ j1 n1 i<j Id n i=jori=j+1 δ i1 nσ j n1 i>j+1 \sigma_j^n \circ \delta_i^{n+1} = \left\lbrace \array{ \delta_i^n \circ \sigma_{j-1}^{n-1} & i \lt j \\ Id_n & \; i = j \;or\; i = j+1 \\ \delta^n_{i-1} \circ \sigma_{j}^{n-1} & i \gt j +1 } \right.


Monoidal structure

The addition of natural numbers extends to a functor :Δ a×Δ aΔ a\oplus : \Delta_a \times \Delta_a \to \Delta_a and :Δ×ΔΔ\oplus : \Delta \times \Delta \to \Delta, by taking mn\mathbf{m} \oplus \mathbf{n} to be the disjoint union of the underlying sets of m\mathbf{m} and n\mathbf{n}, with the linear order that extends those on m\mathbf{m} and n\mathbf{n} by putting every element of m\mathbf{m} below every element of n\mathbf{n}. This is called the ordinal sum functor. If we visualise n\mathbf{n} as a totally ordered set {0<1<<n1}\{0 \lt 1 \lt \cdots \lt n-1\}, and similarly for m\mathbf{m}, then mn\mathbf{m} \oplus \mathbf{n} looks like

mn={0<1<<m1<0 *<1 *<<(n1) *} \mathbf{m} \oplus \mathbf{n} = \{0 \lt 1 \lt \cdots \lt m-1 \lt 0^*\lt 1^* \lt \cdots \lt (n-1)^*\}

where k *k^* denotes kk considered as an element of n\mathbf{n}.

Clearly :Δ a×Δ aΔ a\oplus : \Delta_a \times \Delta_a \to \Delta_a acts on objects as

nm=n+m, \mathbf{n} \oplus \mathbf{m} = \mathbf{n+m},

On morphisms, given f:mmf : \mathbf{m} \to \mathbf{m}' and g:nng : \mathbf{n} \to \mathbf{n}', we have

(fg)(i)={f(i) if0im1 m+g(im) ifmi(m+n1). (f\oplus g)(i) = \left\lbrace \array{ f(i) & if \; 0 \leq i \leq m - 1 \\ m' + g(i-m) & if \; m \leq i \leq (m+n-1) } \right. \,.

so that fgf \oplus g can be visualised as ff and gg placed side by side.

It is easy to see now that (Δ a,,[0])(\Delta_a,\oplus,[0]) is a strict monoidal category.

It is important to note that this tensor does not give a monoidal structure to Δ\Delta, as that does not have the unit 0=[1]=\mathbf{0} = [-1] = \emptyset.

Also note that this monoidal structure is not symmetric! One does have isomorphisms mnn+mnm\mathbf{m} \oplus \mathbf{n} \simeq \mathbf{n+m} \simeq \mathbf{n} \oplus \mathbf{m} for all m,nm,n, but it is easy to see that they are not bifunctorial.

Under Day convolution this monoidal structure induces the join of simplicial sets.

Δ\Delta and Δ a\Delta_a as 2-categories

Being full subcategories of the 2-category CatCat, Δ\Delta and Δ a\Delta_a are themselves 2-categories: their 2-cells fgf \Rightarrow g are given by the pointwise order on monotone functions. Equivalently, they are generated under (vertical and horizontal) composition by the inequalities

δ i+1 nδ i nσ i nσ i+1 n. \delta^n_{i+1} \leq \delta^n_i \qquad \qquad \sigma^n_i \leq \sigma^n_{i+1} \, .

Of course, the ordinal sum functor \oplus extends to a 2-functor in the obvious way.

For each nn there is a string of adjunctions

δ n1 nσ n2 nδ n2 nδ 1 nσ 0 nδ 0 n \delta^n_{n-1} \dashv \sigma^n_{n-2} \dashv \delta^n_{n-2} \dashv \cdots \dashv \delta^n_1 \dashv \sigma^n_0 \dashv \delta^n_0

where the counit of σ iδ i\sigma_i \dashv \delta_i and the unit of δ i+1σ i\delta_{i+1} \dashv \sigma_i are identities.

For each n2n \geq 2, the object n+1\mathbf{n+1} is given by the pushout

n1 δ 0 n δ n1 δ 0 n δ 0 n+1 \array{ \mathbf{n-1} & \overset{\delta_0}{\to} & \mathbf{n} \\ \mathllap{\scriptsize{\delta_{n-1}}} \downarrow & & \downarrow \mathrlap{\scriptsize{\delta_0}} \\ \mathbf{n} & \underset{\delta_0}{\to} & \mathbf{n+1} }

This means that Δ a\Delta_a is generated as a 2-category by these pushouts and by taking adjoints of morphisms. Its monoidal structure is also determined in this way: for each nn, write n=δ n1δ 2δ 1\bot_n = \delta_{n-1}\cdots\delta_2\delta_1 for the (morphism 1n\mathbf{1} \to \mathbf{n} corresponding to the) least element 00 of n\mathbf{n}, and n=δ 0δ 0δ 0\top_n = \delta_0\cdots\delta_0\delta_0 for the greatest. Then there are cospans 1n1\mathbf{1} \to \mathbf{n} \leftarrow \mathbf{1} given by n\top_n and n\bot_n, and each such is equivalent to the (n1)(n-1) fold cospan composite (i.e. pushout) of 121\mathbf{1} \to \mathbf{2} \leftarrow \mathbf{1} with itself. The ordinal sum nm\mathbf{n} \oplus \mathbf{m} is given by the composite

nm n m 1 1 1 \array{ & & & & \mathbf{n} \oplus \mathbf{m} & & & & \\ & & & \nearrow & & \nwarrow & & & \\ & & \mathbf{n} & & & & \mathbf{m} & & \\ & \nearrow & & \nwarrow & & \nearrow & & \nwarrow & \\ \mathbf{1} & & & & \mathbf{1} & & & & \mathbf{1} }

The universal property of pushouts, together with those of the initial and terminal objects 0,1\mathbf{0},\mathbf{1}, then suffices to define \oplus as a 2-functor.

Universal properties

The morphisms 0δ 01σ 02\mathbf{0} \overset{\delta_0}{\to} \mathbf{1} \overset{\sigma_0}{\leftarrow} \mathbf{2} in Δ a\Delta_a make 1\mathbf{1} into a monoid object. Indeed, it is easy to see that

δ i n =iδ 0 0ni σ i n =iσ 0 1ni1 \begin{aligned} \delta^n_i & = \mathbf{i} \oplus \delta^0_0 \oplus \mathbf{n-i} \\ \sigma^n_i & = \mathbf{i} \oplus \sigma^1_0 \oplus \mathbf{n-i-1} \end{aligned}

so that the morphisms of Δ a\Delta_a are generated under \circ and \oplus by δ 0 0\delta^0_0 and σ 0 1\sigma^1_0, together with exactly the equations needed to make them the structure maps of the monoid [1][1]. The objects of Δ a\Delta_a are the elements of the free monoid generated by 1\mathbf{1} and \oplus.

Δ a\Delta_a thus becomes the universal category-equipped-with-a-monoid, in the sense that for any strict monoidal category BB, there is a bijection between monoids (M,m,e)(M,m,e) in BB and strict monoidal functors Δ aB\Delta_a \to B such that 1M\mathbf{1} \mapsto M, σ 0m\sigma_0 \mapsto m and δ 0e\delta_0 \mapsto e.

In particular, for KK a 2-category, monads in KK correspond to 2-functors BΔ aK\mathbf{B}\Delta_a \to K, where BΔ a\mathbf{B}\Delta_a is Δ a\Delta_a considered as a one-object 2-category. Because monads in KK are also the same as lax functors 1K1 \to K, this correspondence exhibits BΔ a\mathbf{B}\Delta_a as the lax morphism classifier? for the terminal category 11.

When Δ a\Delta_a is considered as a 2-category, a similar argument to the above shows that the one-object 3-category BΔ a\mathbf{B}\Delta_a classifies lax-idempotent monads: given a 3-category MM and a lax-idempotent monad tt therein, there is a unique 3-functor BΔ aM\mathbf{B}\Delta_a \to M sending [1][1] to tt, essentially because σ 0 1δ 0 1=δ 0 01\sigma^1_0 \dashv \delta^1_0 = \delta^0_0 \oplus \mathbf{1} with identity counit.

Duality with intervals

Recall that an interval is a linearly ordered set with a top and bottom element; interval maps are monotone functions which preserve top and bottom.

Parallel to the categories Δ\Delta and Δ a\Delta_a, let \nabla denote the category of finite intervals where the top and bottom elements are distinct, and let a\nabla_a denote the category of all finite intervals, including the terminal one where top and bottom coincide. Then we have concrete dualities, or equivalences of the form

Δ a op a;Δ op,\Delta_a^{op} \simeq \nabla_a; \qquad \Delta^{op} \simeq \nabla,

both induced by the ambimorphic object 2\mathbf{2}, seen as both an ordinal and an interval. In other words, we have in each case an adjoint equivalence

Int(,2) opOrd(,2)Int(-, \mathbf{2})^{op} \dashv Ord(-, \mathbf{2})

inducing the first equivalence Ord(,2):Δ a op aOrd(-, \mathbf{2}): \Delta_a^{op} \to \nabla_a, and the second equivalence by restriction.

This fact is mentioned in (Joyal), to help give some intuition for his category Θ\Theta as dual to a category of disks. See also Interval – Relation to simplices, and the section on dualities in (Wraith).


As an order-preserving function between finite ordinals, any morphism f:mnf : \mathbf{m} \to \mathbf{n} in Δ a\Delta_a is completely specified by fixing kk elements of n\mathbf{n} as the image of ff, together with a composition of m\mathbf{m} into kk parts, each part denoting a non-empty, contiguous subset of elements of m\mathbf{m} sharing their value of ff. That is, each such composition is given by a collection of kk interval parts [0,i 1],[i 1+1,i 2],,[i k1+1,m1][0,i_1], [i_1 + 1, i_2], \ldots, [i_{k-1}+1, m-1], determined by a (k1)(k-1)-element subset {i 1,,i k1}\{i_1, \ldots, i_{k-1}\} of an (m1)(m-1)-element set {0,,m2}\{0, \ldots, m-2\}. Hence, there are a total of

k(nk)(m1k1)= k(nk)(m1mk)=(n+m1m) \sum_k \binom{n}{k} \binom{m-1}{k-1} = \sum_k \binom{n}{k} \binom{m-1}{m-k} = \binom{n+m-1}{m}

different morphisms of type mn\mathbf{m} \to \mathbf{n} in Δ a\Delta_a, where we obtain the expression on the right by applying the Chu–Vandermonde identity. For example, there are

(21)(20)+(22)(21)=2+2=4=(43) \binom{2}{1}\binom{2}{0} + \binom{2}{2}\binom{2}{1} = 2+2 = 4 = \binom{4}{3}

different morphisms 32\mathbf{3} \to \mathbf{2}, corresponding to the four functions

  1. f(0)=f(1)=0,f(2)=1f(0) = f(1) = 0, f(2) = 1
  2. f(0)=0,f(1)=f(2)=1f(0) = 0, f(1) = f(2) = 1
  3. f(0)=f(1)=f(2)=0f(0) = f(1) = f(2) = 0
  4. f(0)=f(1)=f(2)=1f(0) = f(1) = f(2) = 1

A more direct bijective proof of the identity |Δ a(m,n)|=(n+m1m)|\Delta_a(\mathbf{m},\mathbf{n})| = \binom{n+m-1}{m} is also possible: see this comment on the nForum.

As some interesting special cases, taking m=nm=n gives the number of monotone endofunctions on n\mathbf{n} (OEIS sequence A088218, or A001700 if we consider endomorphisms [n][n]Δ[n] \to [n] \in \Delta), while taking m=2m=2 gives the triangular numbers (OEIS sequence A000217).

Simplicial sets

Presheaves on Δ\Delta are simplicial sets. Presheaves on Δ a\Delta_a are augmented simplicial sets.

Under the Yoneda embedding Y:ΔY : \Delta \to SSet the object [n][n] induces the standard simplicial nn-simplex Y([n])=:Δ nY([n]) =: \Delta^n. So in particular we have (Δ n)[m]=Hom Δ([m],[n])(\Delta^n)[m] = Hom_{\Delta}([m],[n]) and hence Δ n[m]\Delta^n[m] is a finite set with (n+m+1n)\binom{n+m+1}{n} elements.

The face and degeneracy maps and the relation they satisfy are geometrically best understood in terms of the full and faithful image under YY in SSet:

  • the face map Y(δ i):Δ n1Δ nY(\delta_i) : \Delta^{n-1} \to \Delta^{n} injects the standard simplicial (n1)(n-1)-simplex as the iith face into the standard simplicial nn-simplex;

  • the degeneracy map Y(σ i):Δ n+1Δ nY(\sigma_i) : \Delta^{n+1} \to \Delta^{n} projects the standard simplicial (n+1)(n+1)-simplex onto the standard simplicial nn-simplex by collapsing its vertex number ii onto the face opposite to it.

Realization and nerve

There are important standard functors from Δ\Delta to other categories which realize [n][n] as a concrete model of the standard nn-simplex.

  • The functor Δ[]:Δ\Delta[-] : \Delta \to sSet (the Yoneda embedding) realizes [n][n] as a simplicial set.

  • The functor ||:Δ|\cdot| : \Delta \to Top

    sends [n][n] to the standard topological nn-simplex [n]{x 0x 1x n1} n[n] \mapsto \{x_0 \leq x_1 \leq \cdots \leq x_n \leq 1\}\subset \mathbb{R}^{n}. This functor induced geometric realization of simplicial sets.

  • The functor O:ΔStrωCatO : \Delta \to Str\omega Cat sends [n][n] to the nnth oriental. This induces simplicial nerves of omega-categories.

    Under the functor StrωCatCatStr \omega Cat \to Cat which discards all higher morphisms and identifies all 1-morphisms that are connected by a 2-morphisms, this becomes again the identification of Δ\Delta with the full subbcategory of CatCat on linear quivers that we started the above definition with

    [n]{01n}. [n] \mapsto \{0 \to 1 \to \cdots \to n\} \,.


See the references at simplicial set.

See also:

The relation to intervals and the generalization to the cell category is due to

  • André Joyal, Disks, duality and Θ\Theta-categories, preprint, 1997 (pdf)

A discussion of the opposite categories of Δ,Δ a\Delta, \Delta_a and related categories can be found here:

  • Gavin C. Wraith, Using the generic interval, Cah. Top. Géom. Diff. Cat. XXXIV 4 (1993) pp.259-266. (pdf)

category: category

Revised on September 7, 2015 02:34:23 by Noam Zeilberger (