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 simplex category $\Delta$ is the full subcategory of Cat consisting of the finite inhabited linear quivers.

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

• Equivalently: $\Delta$ is the category of finite inhabited totally ordered sets and order-preserving functions between them.

Detailed description

In more details, $\Delta$ looks as follows.

First of all, it is common, convenient and without risk to use a skeleton of $\Delta$, 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.

Often the simplex category is defined to be the following skeleton:

• objects are the finite totally ordered sets $[n]:=\{0<1<\cdots <n\}$ 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 missing $i\in [n]$;

  • 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 morphism generate $\Delta$ subject to the following relations, called the simplicial relations

$$\begin{array}{cc}{\delta }_j^{n+1}\circ {\delta }_i^n={\delta }_i^{n+1}\circ {\delta }_{j-1}^n& \mathrm{for}i<j\\ {\sigma }_j^n\circ {\delta }_i^{n+1}={\sigma }_i^{n-1}\circ {\sigma }_{j+1}^n& \mathrm{for}i\le j\end{array}$$\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 \delta_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.

Remarks

simplicial sets

Presheaves on $\Delta$ are 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.

augmented simplex category

• Often, it is highly convenient to extend the category $\Delta$ to contain also the empty totally ordered set $[-1]:=\varnothing$. This is called the augmented simplex category ${\Delta }_a$, and presheaves on it are augmented simplicial sets. The category ${\Delta }_a$ may be characterized as the initial strict monoidal category equipped with a monoid; the monoidal product is ordinal sum, and the monoidal unit is the empty ordinal. This style of definition also opens up the possibility of using string diagrams to visualize the structure of ${\Delta }_a$ and of the cube category $\square$.

monoidal structure

The idea of addition of natural numbers extends, and adapts, to give a tensor product-type functor on both $\Delta$ and ${\Delta }_a$. If we visualise an object, $[n]$ of $\Delta$, as above, as a totally ordered set $\{0<1<\cdots <n\}$, then from two such $[m]$ and $[n]$, we can form a new one by making all the elements of $[n]$, strictly greater than those in $[m]$. (It can be convenient to adopt the following notational device here. Add an ${}^{*}$ to each element of $[n]$ just as a label (a different font would serve the same purpose), so we form a new ordinal $[m]+[n]=\{0<1<\cdots <m<0^{*}<1^{*}<\cdots <n^{*}\}$.) Clearly we have

$$+:\Delta \times \Delta \to \Delta$$+ : \Delta \times \Delta \to \Delta

acts on objects

$$[n]+[m]=[n+m+1],$$[n] + [m] = [n + m + 1],

but that is not really where the subtleties are apparent, they are clearer with the morphisms. On morphisms given $f:[m]\to [m\prime ]$ and $g:[n]\to [n\prime ]$, we have

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

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.

Tim: a neat notation for the ordinal sum is $\oplus$ read as ‘o plus’ with ‘o’ recalling ‘ordinal’ and ‘plus …. . In editing the above I did not change the pre-existing notation, but would have liked to!

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.

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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