Simplicial sets, the dual of delta

Question

Why do we take the dual of $\Delta$ when we consider presheaves to $\mathbb{Set}:$ $\mathbb{Set}^{\Delta^{op}}$ as simplicial sets?

Answer 1

I presume you mean the opposite of $\Delta$, i.e., $\Delta^{\text{op}}$.

A simplicial set $A=(A_n)$ must have various face maps $A_n\to A_{n-1}$ and degeneracy maps $A_n\to A_{n+1}$. These go the opposite way to the injections of faces $\Delta_{n-1}\to\Delta_n$ and collapsing maps $\Delta_{n+1}\to\Delta_n$, which occur in the category $\Delta$, so we need a simplicial map as a contravariant functor from $\Delta$ to Set.

More generally we consider presheaf categories as $\text{Set}^{C^{\text{op}}}$ since there is a Yoneda embedding $C\to\text{Set}^{C^{\text{op}}}$, and so $\text{Set}^{C^{\text{op}}}$ in effect is a natural extension of the category $C$ which is complete and cocomplete, etc.

Answer 2

Because that's the definition of what a "simplicial set" is. You could consider functors $\Delta\to\mathtt{Set}$ instead, but that's not what we call a simplicial set (we instead call such a thing a "cosimplicial set"). "Simplicial set" is just a name.

Answer 3

I want to understand the question as: why $\mathbf{Set}^{\Delta^{\rm op}}$ is of more interest than $\mathbf{Set}^{\Delta}$ for most mathematicians?

One way to answer that is the following then: because $\Delta$ constitutes basic shapes of some mathematical objects (while $\Delta^{\rm op}$ does not so much). Seems cryptic, right? Let me elaborates on that.

Given a functor $i : \mathcal A \to \mathcal E$ , you get an adjunction $$ \rho_i : \mathbf{Set}^{\mathcal A^{\rm op}} \leftrightarrows \mathcal E : \nu_i$$ where the left adjoint $\rho_i$ is given as the left Kan extension of $i$ along the Yoneda embedding $\mathcal A \to \mathbf{Set}^{\mathcal A^{\rm op}}$, and where the right adjoint $\nu_i$ is given by $X \mapsto \mathcal E\,(i(-),X)$. The left adjoint is often called the realization and the right adjoint the nerve. You really want to think of $\mathcal E$ as the category of mathematical objects that interest you, and the objects of $\mathcal A$ as "basic shapes" you know well : then the nerve of $X \in \mathcal E$ contains the informations on all the possible way to send a basic shape into the object $X$; the realization of a presheaf $F$ is the object in $\mathcal E$ obtained by gluing the basic shapes together, following the rules given by $F$.

Let's go back to $\Delta$. Take $\mathcal A = \Delta$, there is a functor $\Delta \to \mathbf{Top}$ that maps an integer $n$ to the topological simplex of dimension $n$. The machinery above gives you an adjunction: $$ \mathbf{Set}^{\Delta^{\rm op}} \leftrightarrows \mathbf{Top} $$ where the left adjoint is the geometric realization of simplicial sets and the right adjoint is the functor of singular chains. Here, the objects of interest are the topological spaces and the basic shapes are the topological simplices: the singular chains are indeed all the possible ways to put simplices into a topological space, and take a look at the usual formula for the geometric realization to understand that it is about gluing simplices together.

Another one? Take $\mathcal A = \Delta$ and consider the functor $\Delta \to \mathbf{Cat}$ that maps a integer $n$ to the category associated with $n$ seen as the poset $\{0< 1<\dots< n\}$. The machinery give you a new adjunction $$ \mathbf{Set}^{\Delta^{\rm op}} \leftrightarrows \mathbf{Cat} $$ The right adjoint is the usual nerve and the left adjoint is the "fundamental category" functor (it looks like the fundamental groupoid of a topological space, but it takes into account that simplices of simplicial sets have inherent "directions"). Again, the right adjoint is looking at shapes (here finite paths) into the mathematical objects of interest (here the categories) and the left adjoint is gluing the paths given by thinking of the simplices of a simplicial sets as compositions of morphisms.