I have been wondering lately what makes simplicial sets 'tick'.


**Edited**


The category $\Delta$can be viewed as the category of standard $n$-simplices and order preserving simplicial maps. The goal of simplicial sets is to build spaces out of these building blocks by gluing, and allow maps to be defined simplex by simplex, so it makes sense to take the free cocompletion of $\Delta$, the presheaf category $[\Delta^{op},\mathbf{Set}]$. The realisation functor $R : \Delta\to \mathbf{Top}$ can be readily extended to the cocompletion, so as to make $\hat{R}$ preserve colimits.


So my questions are:


* How do we intuitively understand why $\hat{R}$ preserves finite products? (I understand that there are some subtleties with $k$-ification)
* What makes $\Delta$ special in this way, that fails for say $\Gamma=\mathbf{FinSetSkel}$, ie. "symmetric simplicial sets" and cubical sets?
* What is your philosophy of simplicial sets?



What's special about the Simplex category?
Intuitively, I see the product-preservation or indeed finite limit preservation of geometric realization $\hat{R}: [\Delta^{op}, \mathbf{Set}] \to \mathbf{kSpace}$ as lifting (through the forgetful functor $U: \mathbf{kSpace} \to \mathbf{Set}$) a more basic left exact left adjoint $[\Delta^{op}, \mathbf{Set}] \to \mathbf{Set}$. The basic result is that such left exact left adjoints can be seen to correspond to linear orders with distinct top and bottom, aka *intervals*, and the specific interval being used for geometric realization is the standard unit interval $I = [0, 1]$. 


There is a famous result from topos theory called Diaconescu's theorem, where left exact left adjoints $[C^{op}, \mathbf{Set}] \to \mathbf{Set}$ correspond (in the sense of an equivalence of categories) to filtered colimits of representables $C \to \mathbf{Set}$. In the case $C = \Delta$, to understand filtered colimits of representables $\hom\_\Delta([n], -)$, it helps to know that the category of finite intervals is dual to the category of finite nonempty linear orders (the equivalence $\Delta^{op} \to \text{FinInt}$ takes $[n] = \{0 < 1 < \ldots < n\}$ to the interval $\hom\_\Delta([n], \{0 < 1\})$, with $n+2$ elements). For example, take the 
restricted representable $\hom\_{\text{Int}}(i-, I)$, restricting along the subcategory $i: \text{FinInt} \hookrightarrow \text{Int}$. Like any interval, the standard unit interval $I$ can be represented as a filtered colimit of the diagram of its finite subintervals and inclusions between them, say $I = \text{colim}\_{d \in D}\; I\_d$ where $I\_d$ is an interval with $n\_d + 2$ elements. Then the functor $\hom\_{\text{Int}}(i-, I)$ is a filtered colimit of representables of $\Delta$: 


$$\begin{array}{lll}
\hom\_{\text{Int}}(i-, I) & = & \hom\_{\text{Int}}(i-, \text{colim}\_d\; I\_d): \text{FinInt}^{op} \to \mathbf{Set} \\ 
 & \cong & \text{colim}\_d\; \hom\_{\text{Int}}(i-, I\_d) \\
& \cong & \text{colim}\_d \hom\_\Delta([n\_d], -): \Delta \to \mathbf{Set}
\end{array}$$ 


(to get to the second line, note that for any finite subinterval $J$, the functor $\hom\_{\text{Int}}(J, -)$ preserves filtered colimits, i.e., $J$ is finitely presentable in the category of intervals). 


Summarizing, there are three main ingredients, all of which should be considered soft and conceptual: 


* $\Delta$ is dual to the category of finite intervals;
* The Ind-completion of the category of finite intervals is the category of all intervals, in which finite intervals are the finitely presentable objects;
* This implies that filtered colimits of representables $\Delta \to \mathbf{Set}$, or left exact left adjoints $[\Delta^{op}, \mathbf{Set}] \to \mathbf{Set}$, are classified by intervals, and we pick the standard interval to get (the underlying set of) geometric realization to be left exact.


In particular, the canonical continuous map 


$$\hat{R}(\hom\_\Delta(-, [m]) \times \hom\_\Delta(-, [n])) \to \hat{R}(\hom\_\Delta(-, [m])) \times \hat{R}(\hom\_\Delta(-, [n]))$$ 


is a bijection at the underlying set level. Coupled with a brief argument that the left side is compact and the right side is Hausdorff, this map is a homeomorphism. 


Then one finishes off with a formal calculation where general simplicial sets $X$ are colimits of representables: 


$$\begin{array}{lll} 
\hat{R}(X \times Y) & \cong & \hat{R}(\text{colim}\_d \; \hom(-, [n\_d]) \times \text{colim}\_e \; \hom(-, [n\_e])) \\ 
 & \cong & \hat{R}(\text{colim}\_{d, e} \; \hom(-, [n\_d]) \times \hom(-, [n\_e])) \\ 
 & \cong & \text{colim}\_{d, e} \; \hat{R}(\hom(-, [n\_d]) \times \hom(-, [n\_e])) \\ 
 & \cong & \text{colim}\_{d, e} \; \hat{R}(\hom(-, [n\_d])) \times \hat{R}(\hom(-, [n\_e])) \\ 
 & \cong & \text{colim}\_d \; \hat{R}(\hom(-, [n\_d])) \times \text{colim}\_e \; \hat{R}(\hom(-, [n\_e])) \\ 
 & \cong & \hat{R}(\text{colim}\_d \; \hom(-, [n\_d)) \times \hat{R}(\text{colim}\_e \; \hom(-, [n\_e])) \\ 
 & \cong & \hat{R}(X) \times \hat{R}(Y)
\end{array} 
$$ 


where the crucial step is to the fifth line; this is where we use the fact that $\mathbf{kSpace}$ is cartesian closed, so that the cartesian product in $\mathbf{kSpace}$ preserves colimits in each of its two arguments (which is not the case for $\mathbf{Top}$). (The second line is similar; we use the fact that simplicial sets is cartesian closed.) 


Anyway, I hope the crucial role of the *order* of combinatorial simplices is apparent here. I don't think I have a single philosophy of simplicial sets, because simplicial sets has many hearts. For example, the category of finite ordinals, including the empty one, is the initial monoidal category with a monoid; this is incredibly important for bar constructions. But for another, see this lovely [blog post](http://golem.ph.utexas.edu/category/2008/01/mark_weber_on_nerves_of_catego.html) by Tom Leinster. 


