nLab
simplex category

Idea

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

Definition

• The augmented simplex category ${\Delta }_a$ is the full subcategory of Cat consisting of the finite linear quivers, or equivalently the category of finite totally ordered sets, or finite ordinals, and order-preserving functions between them:

  $$\{c_0\to c_1\to \cdots \to c_n\}.$$\{c_0 \to c_1 \to \cdots \to c_n\} \,.

• The simplex category $\Delta$ is the full subcategory of ${\Delta }_a$ (and hence of $\mathrm{Cat}$) consisting of the finite inhabited linear quivers, non-empty linear orders or non-zero ordinals.

Details

First of all, it is common, convenient and without risk to use a skeleton of $\Delta$ or ${\Delta }_a$, 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 in mind, the augmented simplex category ${\Delta }_a$ can be presented as follows:

• objects are the finite totally ordered sets $[n]:=\{0<1<\cdots <n-1\}$ for all $n\in \mathbb{N}$;

• morphisms are order-preserving functions $[n]\to [m]$ – these are generated by (are all expressible as finite compositions of) the following two elementary kinds of maps

  • face maps: ${\delta }_i:={\delta }_i^n:[n-1]\hookrightarrow [n]$ is the injection whose image leaves out $i\in [n]$;

  • degeneracy maps: ${\sigma }_i:={\sigma }_i^n:[n+1]\to [n]$ is the surjection such that ${\sigma }_i(i)={\sigma }_i(i+1)=i$.

These morphisms generate ${\Delta }_a$ subject to the following relations, called the simplicial relations or simplicial identities:

$$\begin{array}{cc}{\delta }_j^{n+1}\circ {\delta }_i^n={\delta }_i^{n+1}\circ {\delta }_{j-1}^n& \mathrm{for}i<j\\ {\sigma }_j^n\circ {\sigma }_i^{n+1}={\sigma }_i^{n-1}\circ {\sigma }_{j+1}^n& \mathrm{for}i\le j\end{array}$$\array{ \delta_j^{n+1} \circ \delta_i^n = \delta_i^{n+1}\circ \delta_{j-1}^n & for i \lt j \\ \sigma_j^n \circ \sigma_i^{n+1} = \sigma_i^{n-1} \circ \sigma_{j+1}^n & for i \leq j }

$${\sigma }_j^n\circ {\delta }_i^{n+1}=\{\begin{array}{cc}{\delta }_i^n\circ {\sigma }_{j-1}^{n-1}& \mathrm{if}i<j\\ {\mathrm{Id}}_n& \mathrm{if}i=j\mathrm{or}i=j+1\\ {\delta }_{i-1}^n\circ {\sigma }_j^{n-1}& \mathrm{if}i>j+1\end{array}$$\sigma_j^n \circ \delta_i^{n+1} = \left\lbrace \array{ \delta_i^n \circ \sigma_{j-1}^{n-1} & if i \lt j \\ Id_n & if i = j or i = j+1 \\ \delta^n_{i-1} \circ \sigma_{j}^{n-1} & if i \gt j +1 } \right.

The (unaugmented) simplex category $\Delta$ is given by the full subcategory of ${\Delta }_a$ that leaves out $[0]$. Usually the objects are reindexed to start from 0, so that $[n]\in \Delta =[n+1]\in {\Delta }_a$. Authors who use the unaugmented category then often retain this numbering for ${\Delta }_a$, writing $[-1]$ for its initial object $\{\}$.

Monoidal structure

The addition of natural numbers extends to a bifunctor? $\oplus$ on both $\Delta$ and ${\Delta }_a$, by taking $[m]\oplus [n]$ to be the disjoint union of the underlying sets of $[m]$ and $[n]$, with the linear order that extends those on $[m]$ and $[n]$ by putting every element of $[m]$ below every element of $[n]$. So if we visualise $[n]$ as a totally ordered set $\{0<1<\cdots <n-1\}$, and similarly for $[m]$, then $[m]\oplus [n]$ looks like

$$[m]\oplus [n]=\{0<1<\cdots <m-1<0^{*}<1^{*}<\cdots <(n-1{)}^{*}\}$$[m] \oplus [n] = \{0 \lt 1 \lt \cdots \lt m-1 \lt 0^*\lt 1^* \lt \cdots \lt (n-1)^*\}

where $k^{*}$ denotes $k$ considered as an element of $[n]$.

Clearly $\oplus :{\Delta }_a\times {\Delta }_a\to {\Delta }_a$ acts on objects as

$$[n]\oplus [m]=[n+m],$$[n] \oplus [m] = [n + m],

On morphisms, given $f:[m]\to [m\prime ]$ and $g:[n]\to [n\prime ]$, we have

$$(f\oplus g)(i)=\{\begin{array}{cc}f(i)& \mathrm{if}0\le i\le m-1\\ m\prime +g(i-m)& \mathrm{if}m<i\le (m+n-1)\end{array}.$$(f\oplus g)(i) = \left\lbrace \array{ f(i) & if 0 \leq i \leq m - 1 \\ m' + g(i-m) & if m \lt i \leq (m+n-1) } \right. \,.

so that $f\oplus g$ can be visualised as $f$ and $g$ placed side by side.

It is easy to see now that $({\Delta }_a,\oplus ,[0])$ is a strict monoidal category.

It is important to note that this tensor does not give a monoidal structure to $\Delta$, as it does not have a unit. The unit that would be needed would be the empty ordinal, and, as that is there in ${\Delta }_a$, that does become a monoidal category.

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

$\Delta$ and ${\Delta }_a$ as 2-categories

Being full subcategories of the 2-category $\mathrm{Cat}$, $\Delta$ and ${\Delta }_a$ are themselves 2-categories: their 2-cells? $f\Rightarrow g$ are given by the pointwise order on monotone functions. Equivalently, they are generated under (vertical and horizontal) composition by the inequalities

$${\delta }_{i+1}^n\le {\delta }_i^n{\sigma }_i^n\le {\sigma }_{i+1}^n.$$\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 $n$ there is a string of adjunctions

$${\delta }_{n-1}^n⊣{\sigma }_{n-2}^n⊣{\delta }_{n-2}^n⊣\cdots ⊣{\delta }_1^n⊣{\sigma }_0^n⊣{\delta }_0^n$$\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 ${\sigma }_i⊣{\delta }_i$ and the unit of ${\delta }_{i+1}⊣{\sigma }_i$ are identities.

For each $n\ge 2$, the object $[n+1]$ is given by the pushout

$$\begin{array}{ccc}[n-1]& \stackrel{{\delta }_0}{\to }& [n]\\ {\delta }_{n-1}\downarrow & & \downarrow {\delta }_0\\ [n]& \underset{{\delta }_0}{\to }& [n+1]\end{array}$$\array{ [n-1] & \overset{\delta_0}{\to} & [n] \\ \mathllap{\scriptsize{\delta_{n-1}}} \downarrow & & \downarrow \mathrlap{\scriptsize{\delta_0}} \\ [n] & \underset{\delta_0}{\to} & [n+1] }

This means that ${\Delta }_a$ 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 $n$, write ${\perp }_n={\delta }_{n-1}\cdots {\delta }_2{\delta }_1$ for the (morphism $[1]\to [n]$ corresponding to the) least element $0$ of $[n]$, and ${\top }_n={\delta }_0\cdots {\delta }_0{\delta }_0$ for the greatest. Then there are cospans $[1]\to [n]\leftarrow [1]$ given by ${\top }_n$ and ${\perp }_n$, and each such is equivalent to the $(n-1)$ fold cospan composite (i.e. pushout) of $[1]\to [2]\leftarrow [1]$ with itself. The ordinal sum $[n]\oplus [m]$ is given by the composite

$$\begin{array}{ccccccccc}& & & & [n]\oplus [m]& & & & \\ & & & \nearrow & & \nwarrow & & & \\ & & [n]& & & & [m]& & \\ & \nearrow & & \nwarrow & & \nearrow & & \nwarrow & \\ [1]& & & & [1]& & & & [1]\end{array}$$\array{ & & & & [n] \oplus [m] & & & & \\ & & & \nearrow & & \nwarrow & & & \\ & & [n] & & & & [m] & & \\ & \nearrow & & \nwarrow & & \nearrow & & \nwarrow & \\ [1] & & & & [1] & & & & [1] }

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

Universal properties

The morphisms $[0]\stackrel{{\delta }_0}{\to }[1]\stackrel{{\sigma }_0}{\leftarrow }[2]$ in ${\Delta }_a$ make $[1]$ into a monoid object. Indeed, it is easy to see that

$$\begin{array}{rl}{\delta }_i^n& =i\oplus {\delta }_0^0\oplus (n-i)\\ {\sigma }_i^n& =i\oplus {\sigma }_0^1\oplus (n-i-1)\end{array}$$\begin{aligned} \delta^n_i & = i \oplus \delta^0_0 \oplus (n-i) \\ \sigma^n_i & = i \oplus \sigma^1_0 \oplus (n-i-1) \end{aligned}

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

${\Delta }_a$ thus becomes the universal category-equipped-with-a-monoid, in the sense that for any strict monoidal category $B$, there is a bijection between monoids $(M,m,e)$ in $B$ and strict monoidal functors ${\Delta }_a\to B$ such that $[1]\mapsto M$, ${\sigma }_0\mapsto m$ and ${\delta }_0\mapsto e$.

In particular, for $K$ a 2-category, monads in $K$ correspond to 2-functors $B{\Delta }_a\to K$, where $B{\Delta }_a$ is ${\Delta }_a$ considered as a one-object 2-category. Because monads in $K$ are also the same as lax functors $1\to K$, this correspondence exhibits $B{\Delta }_a$ as the lax morphism classifier? for the terminal category $1$.

When ${\Delta }_a$ is considered as a 2-category, a similar argument to the above shows that the one-object 3-category $B{\Delta }_a$ classifies lax-idempotent monads: given a 3-category $M$ and a lax-idempotent monad $t$ therein, there is a unique 3-functor $B{\Delta }_a\to M$ sending $[1]$ to $t$, essentially because ${\sigma }_0^1⊣{\delta }_0^1={\delta }_0^0\oplus [1]$ with identity counit.

Remarks

Simplicial sets

Presheaves on $\Delta$ are simplicial sets, and presheaves on ${\Delta }_a$ are augmented simplicial sets.

Under the Yoneda embedding $Y:\Delta \to$ SSet the object $[n]$ induces the standard simplicial $n$-simplex $Y([n])=:{\Delta }^n$.

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

• the face map $Y({\delta }_i):{\Delta }^{n-1}\to {\Delta }^n$ injects the standard simplicial $(n-1)$-simplex as the $i$th face into the standard simplicial $n$-simplex;

• the degeneracy map $Y({\sigma }_i):{\Delta }^{n+1}\to {\Delta }^n$ projects the standard simplicial $(n+1)$-simplex onto the standard simplicial $n$-simplex by collapsing its vertex number $i$ onto the face opposite to it.

realization and nerve

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

• The functor

  $$\mid \cdot \mid :\Delta \to \mathrm{Top}$$|\cdot| : \Delta \to Top

  sends $[n]$ to the standard topological $n$-simplex $[n]\mapsto \{x_0\le x_1\le \cdots \le x_n\le 1\}\subset {ℝ}^n$. This functor induced geometric realization of simplicial sets.

• The functor

  $$O:\Delta \to \mathrm{Str}\omega \mathrm{Cat}$$O : \Delta \to Str\omega Cat

  sends $[n]$ to the $n$th oriental. This induces simplicial nerves of omega-categories.

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

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

References

see the references at simplicial set.

---

Discussion

An earlier version of this entry started the following discussion:

Todd says: Historically, and especially for algebraic topologists, $\Delta$ refers to the category of nonempty totally ordered sets and order-preserving maps; an adjective like “augmented” would be attached to “simplicial object” if they wanted to refer to a contravariant functor coming out of what is being defined here as $\Delta$. While I understand arguments why one might wish to redefine $\Delta$ this way, there are also countervailing arguments (cf. Tom’s Café post How I Came to Love the Nerve Construction); in any event, given the weight of history, the “sometimes” strikes me as inappropriate understatement. I think more discussion is called for before we appropriate $\Delta$ for the lesser-used concept, and rename the more commonly used one as $\Delta$ with a dot over it (do other people use that notation?).

Urs: I have tried to change the entry accordingly now.

Todd: Having said all of the above, I myself prefer (on conceptual grounds) to treat the category of all finite ordinals (Ross Street used to call it “algebraists’ $\Delta$”) as the primary concept, and the category of finite nonempty ordinals (“topologists’ $\Delta$”) as secondary. It’s just that I worried about introducing confusion, since my own opinion may well be a minority opinion.

Mike: I think there are many good reasons, including but not limited to tradition, to give the unadorned name to the topologists’ $\Delta$. In my experience, in most applications to homotopy theory, the topologists’ $\Delta$ is the important object, both conceptually and mathematically, with its augmented version playing at most a technical role occasionally. And while the augmented $\Delta$ does have a cute universal property as a monoidal category, the category of simplicial sets (presheaves on the unaugmented $\Delta$) also has a good universal property: it is the classifying topos for linear orders.

Todd: Not meaning to nitpick, Mike, since you raise good points, but I believe “linear order” in your sense should mean “linear order with distinct top and bottom”. (See for instance the discussion in Mac Lane-Moerdijk.) Presheaves on the augmented $\Delta$ would give the classifying topos for “linear orders with top and bottom” (not necessarily distinct). (If you want just plain linear orders, I think you’d use presheaves on ${\Delta }_a^{\mathrm{op}}$.)

Mike: Yes, of course.

Toby: I certainly prefer the algebraists' $\Delta$; it's part of my general preference for not ignoring the empty set. (Mike's example, with Todd's correction, only serves to confirm my opinion.) Seeing the universal construction of $\Delta$, I made the article consistent by picking my favourite, which fit that construction.

I didn't know a good notation for the topologists' unaugmented $\Delta$, so I just used a dot as my standard notation for deleting the basepoint: if $X$ is a pointed set with point $x$, then $\dot{X}:=X\setminus \{x\}$. (I think that I first saw this in point-set topology to turn a neighbourhood of a point into a deleted neighbourhood.) It is by no means sacred.

category: category