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.
The augmented simplex category $\Delta_a$ is the full subcategory of Cat on the free categories of finite linear directed graphs
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 $\Delta$ is the full subcategory of $\Delta_a$ (and hence of $Cat$) consisting of the free categories on finite and inhabited linear directed graphs, hence of non-empty finite linear orders or non-zero ordinals.
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 finite linearly ordered sets are unique this step is so trivial that it is often not even mentioned explicitly.
With this the objects of $\Delta$ are in bijection with natural numbers $n \in \mathbb{N}$ and one usually writes
for the object of $\Delta$ given by the category with $(n+1)$ objects. Geometrically one may think of this as the spine of the standard cellular $n$-simplex, see the discussion of simplicial sets below. In this context one also writes $\Delta[n]$ or $\Delta^n$ for the simplicial set represented by the object $[n]$: the simplicial $n$-simplex. By the Yoneda lemma one may identify the subcategory of simplicial sets on the $\Delta[n]$ with $\Delta$.
With this convention the first few objects of $\Delta$ are
etc.
The category $\Delta_a$ contains one more object, corresponding to the empty category $\emptyset$. When sticking to the above standard notation for the objects of $\Delta$, that extra object is naturally often denoted
However, in contexts where only $\Delta_a$ and not $\Delta$ plays a role, some authors prefer to start counting with 0 instead of with $-1$. Then for instance the notation
and generally
may be used.
The skeletal version of the augmented simplex category $\Delta_a$ can be presented as follows:
objects are the finite totally ordered sets $\mathbf{n} \coloneqq \{0 \lt 1 \lt \cdots \lt n-1\}$ for all $n \in \mathbb{N}$;
morphisms generated by (are all expressible as finite compositions of) the following two elementary kinds of maps
face maps: $\;\delta_i^n :\: \mathbf{n-1} \hookrightarrow \mathbf{n}\;$ is the injection whose image leaves out $i \in [n]\quad$ ($n \gt 0$ and $0 \leq i \lt n$);
degeneracy maps: $\;\sigma_i^n :\: \mathbf{n+1} \to \mathbf{n}\;$ is the surjection such that $\sigma_i(i) = \sigma_i(i+1) = i\quad$ ($n \gt 0$ and $0 \leq i \lt n$);
subject to the following relations, called the simplicial relations or simplicial identities:
The addition of natural numbers extends to a functor $\oplus : \Delta_a \times \Delta_a \to \Delta_a$ and $\oplus : \Delta \times \Delta \to \Delta$, by taking $\mathbf{m} \oplus \mathbf{n}$ to be the disjoint union of the underlying sets of $\mathbf{m}$ and $\mathbf{n}$, with the linear order that extends those on $\mathbf{m}$ and $\mathbf{n}$ by putting every element of $\mathbf{m}$ below every element of $\mathbf{n}$. This is called the ordinal sum functor. If we visualise $\mathbf{n}$ as a totally ordered set $\{0 \lt 1 \lt \cdots \lt n-1\}$, and similarly for $\mathbf{m}$, then $\mathbf{m} \oplus \mathbf{n}$ looks like
where $k^*$ denotes $k$ considered as an element of $\mathbf{n}$.
Clearly $\oplus : \Delta_a \times \Delta_a \to \Delta_a$ acts on objects as
On morphisms, given $f : \mathbf{m} \to \mathbf{m}'$ and $g : \mathbf{n} \to \mathbf{n}'$, we have
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 that does not have the unit $\mathbf{0} = [-1] = \emptyset$.
Also note that this monoidal structure is not symmetric! One does have isomorphisms $\mathbf{m} \oplus \mathbf{n} \simeq \mathbf{n+m} \simeq \mathbf{n} \oplus \mathbf{m}$ for all $m,n$, but it is easy to see that they are not bifunctorial.
Under Day convolution this monoidal structure induces the join of simplicial sets.
Being full subcategories of the 2-category $Cat$, $\Delta$ 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
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
where the counit of $\sigma_i \dashv \delta_i$ and the unit of $\delta_{i+1} \dashv \sigma_i$ are identities.
For each $n \geq 2$, the object $\mathbf{n+1}$ is given by the pushout
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 $\bot_n = \delta_{n-1}\cdots\delta_2\delta_1$ for the (morphism $\mathbf{1} \to \mathbf{n}$ corresponding to the) least element $0$ of $\mathbf{n}$, and $\top_n = \delta_0\cdots\delta_0\delta_0$ for the greatest. Then there are cospans $\mathbf{1} \to \mathbf{n} \leftarrow \mathbf{1}$ given by $\top_n$ and $\bot_n$, and each such is equivalent to the $(n-1)$ fold cospan composite (i.e. pushout) of $\mathbf{1} \to \mathbf{2} \leftarrow \mathbf{1}$ with itself. The ordinal sum $\mathbf{n} \oplus \mathbf{m}$ is given by the composite
The universal property of pushouts, together with those of the initial and terminal objects $\mathbf{0},\mathbf{1}$, then suffices to define $\oplus$ as a 2-functor.
The morphisms $\mathbf{0} \overset{\delta_0}{\to} \mathbf{1} \overset{\sigma_0}{\leftarrow} \mathbf{2}$ in $\Delta_a$ make $\mathbf{1}$ into a monoid object. Indeed, it is easy to see that
so that the morphisms of $\Delta_a$ are generated under $\circ$ and $\oplus$ by $\delta^0_0$ and $\sigma^1_0$, 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 $\mathbf{1}$ 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 $\mathbf{1} \mapsto M$, $\sigma_0 \mapsto m$ and $\delta_0 \mapsto e$.
In particular, for $K$ a 2-category, monads in $K$ correspond to 2-functors $\mathbf{B}\Delta_a \to K$, where $\mathbf{B}\Delta_a$ 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 $\mathbf{B}\Delta_a$ 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 $\mathbf{B}\Delta_a$ classifies lax-idempotent monads: given a 3-category $M$ and a lax-idempotent monad $t$ therein, there is a unique 3-functor $\mathbf{B}\Delta_a \to M$ sending $[1]$ to $t$, essentially because $\sigma^1_0 \dashv \delta^1_0 = \delta^0_0 \oplus \mathbf{1}$ with identity counit.
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 $\Delta$ and $\Delta_a$, let $\nabla$ denote the category of finite intervals where the top and bottom elements are distinct, and let $\nabla_a$ 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
both induced by the ambimorphic object $\mathbf{2}$, seen as both an ordinal and an interval. In other words, we have in each case an adjoint equivalence
inducing the first equivalence $Ord(-, \mathbf{2}): \Delta_a^{op} \to \nabla_a$, and the second equivalence by restriction.
This fact is mentioned in (Joyal), to help give some intuition for his category $\Theta$ 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 $f : \mathbf{m} \to \mathbf{n}$ in $\Delta_a$ is completely specified by fixing $k$ elements of $\mathbf{n}$ as the image of $f$, together with a composition of $\mathbf{m}$ into $k$ parts, each part denoting a non-empty, contiguous subset of elements of $\mathbf{m}$ sharing their value of $f$. That is, each such composition is given by a collection of $k$ interval parts $[0,i_1], [i_1 + 1, i_2], \ldots, [i_{k-1}+1, m-1]$, determined by a $(k-1)$-element subset $\{i_1, \ldots, i_{k-1}\}$ of an $(m-1)$-element set $\{0, \ldots, m-2\}$. Hence, there are a total of
different morphisms of type $\mathbf{m} \to \mathbf{n}$ in $\Delta_a$, where we obtain the expression on the right by applying the Chu–Vandermonde identity. For example, there are
different morphisms $\mathbf{3} \to \mathbf{2}$, corresponding to the four functions
A more direct bijective proof of the identity $|\Delta_a(\mathbf{m},\mathbf{n})| = \binom{n+m-1}{m}$ is also possible: see this comment on the nForum.
As some interesting special cases, taking $m=n$ gives the number of monotone endofunctions on $\mathbf{n}$ (OEIS sequence A088218, or A001700 if we consider endomorphisms $[n] \to [n] \in \Delta$), while taking $m=2$ gives the triangular numbers (OEIS sequence A000217).
Presheaves on $\Delta$ are simplicial sets. 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 $Y([n]) =: \Delta^n$. So in particular we have $(\Delta^n)[m] = Hom_{\Delta}([m],[n])$ and hence $\Delta^n[m]$ is a finite set with $\binom{n+m+1}{n}$ elements.
The face and degeneracy maps and the relation they satisfy are geometrically best understood in terms of the full and faithful image under $Y$ 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.
There are important standard functors from $\Delta$ to other categories which realize $[n]$ as a concrete model of the standard $n$-simplex.
The functor $\Delta[-] : \Delta \to$ sSet (the Yoneda embedding) realizes $[n]$ as a simplicial set.
The functor $|\cdot| : \Delta \to$ Top
sends $[n]$ to the standard topological $n$-simplex $[n] \mapsto \{x_0 \leq x_1 \leq \cdots \leq x_n \leq 1\}\subset \mathbb{R}^{n}$. This functor induced geometric realization of simplicial sets.
The functor $O : \Delta \to Str\omega Cat$ sends $[n]$ to the $n$th oriental. This induces simplicial nerves of omega-categories.
Under the functor $Str \omega Cat \to Cat$ which discards all higher morphisms and identifies all 1-morphisms that are connected by a 2-morphisms, this becomes again the identification of $\Delta$ with the full subbcategory of $Cat$ on linear quivers that we started the above definition with
geometric shapes for higher structures
simplex category
See the references at simplicial set.
See also:
Section VII.5 of Categories for the Working Mathematician
Section II.2 of P. Gabriel and M. Zisman, Calculus of Fractions and Homotopy Theory. Springer, 1967
The relation to intervals and the generalization to the cell category is due to
A discussion of the opposite categories of $\Delta, \Delta_a$ and related categories can be found here: