The simplex category encodes one of the main geometric shapes for higher structures. Its objects are the standard cellular -simplices. It is also called the simplicial category, but that term is ambiguous.
The simplex category is the full subcategory of (and hence of ) 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 or , 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 are in bijection with natural numbers and one usually writes
for the object of given by the category with objects. Geometrically one may think of this as the spine of the standard cellular -simplex, see the discussion of simplicial sets below. In this context one also writes or for the simplicial set represented by the object : the simplicial -simplex. By the Yoneda lemma one may identify the subcategory of simplicial sets on the with .
With this convention the first few objects of are
The category contains one more object, corresponding to the empty category . When sticking to the above standard notation for the objects of , that extra object is naturally often denoted
However, in contexts where only and not plays a role, some authors prefer to start counting with 0 instead of with . Then for instance the notation
may be used.
The skeletal version of the augmented simplex category can be presented as follows:
objects are the finite totally ordered sets for all ;
morphisms generated by (are all expressible as finite compositions of) the following two elementary kinds of maps
face maps: is the injection whose image leaves out ( and );
degeneracy maps: is the surjection such that ( and );
subject to the following relations, called the simplicial relations or simplicial identities:
The addition of natural numbers extends to a functor and , by taking to be the disjoint union of the underlying sets of and , with the linear order that extends those on and by putting every element of below every element of . This is called the ordinal sum functor. If we visualise as a totally ordered set , and similarly for , then looks like
where denotes considered as an element of .
Clearly acts on objects as
On morphisms, given and , we have
so that can be visualised as and placed side by side.
It is easy to see now that is a strict monoidal category.
It is important to note that this tensor does not give a monoidal structure to , as that category does not contain the unit .
Being full subcategories of the 2-category , and are themselves 2-categories: their 2-cells 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 extends to a 2-functor in the obvious way.
For each there is a string of adjunctions
where the counit of and the unit of are identities.
For each , the object is given by the pushout
This means that 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 , write for the (morphism corresponding to the) least element of , and for the greatest. Then there are cospans given by and , and each such is equivalent to the fold cospan composite (i.e. pushout) of with itself. The ordinal sum is given by the composite
The universal property of pushouts, together with those of the initial and terminal objects , then suffices to define as a 2-functor.
The morphisms in make into a monoid object. Indeed, it is easy to see that
so that the morphisms of are generated under and by and , together with exactly the equations needed to make them the structure maps of the monoid . The objects of are the elements of the free monoid generated by and .
thus becomes the universal category-equipped-with-a-monoid, in the sense that for any strict monoidal category , there is a bijection between monoids in and strict monoidal functors such that , and .
In particular, for a 2-category, monads in correspond to 2-functors , where is considered as a one-object 2-category. Because monads in are also the same as lax functors , this correspondence exhibits as the lax morphism classifier? for the terminal category .
When is considered as a 2-category, a similar argument to the above shows that the one-object 3-category classifies lax-idempotent monads: given a 3-category and a lax-idempotent monad therein, there is a unique 3-functor sending to , essentially because with identity counit.
Parallel to the categories and , let denote the category of finite intervals where the top and bottom elements are distinct, and let 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 , seen as both an ordinal and an interval. In other words, we have in each case an adjoint equivalence
inducing the first equivalence , and the second equivalence by restriction.
As an order-preserving function between finite ordinals, any morphism in is completely specified by fixing elements of as the image of , together with a composition of into parts, each part denoting a non-empty, contiguous subset of elements of sharing their value of . That is, each such composition is given by a collection of interval parts , determined by a -element subset of an -element set . Hence, there are a total of
different morphisms of type in , where we obtain the expression on the right by applying the Chu–Vandermonde identity. For example, there are
different morphisms , corresponding to the four functions
As some interesting special cases, taking gives the number of monotone endofunctions on (OEIS sequence A088218, or A001700 if we consider endomorphisms ), while taking gives the triangular numbers (OEIS sequence A000217).
the face map injects the standard simplicial -simplex as the th face into the standard simplicial -simplex;
the degeneracy map projects the standard simplicial -simplex onto the standard simplicial -simplex by collapsing its vertex number onto the face opposite to it.
The functor Top
Under the functor which discards all higher morphisms and identifies all 1-morphisms that are connected by a 2-morphisms, this becomes again the identification of with the full subbcategory of on linear quivers that we started the above definition with
See the references at simplicial set.
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
A discussion of the opposite categories of and related categories can be found here: