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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"?
• What is your philosophy of simplicial sets?

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?

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

Since ordering is only used for homology (?), I will ignore it for now and consider $\Gamma = \mathbf{FinSetSkel}$, the category of finite ordinals, and functions instead. This category can be viewed as the category of standard $n$-simplices and 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 $\Gamma$, the presheaf category $[\Gamma^{op},\mathbf{Set}]$. The realisation functor $R : \Gamma \to \mathbf{Top}$ 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 $\Gamma$ (or $\Delta$) special in this way, that fails for say cubical sets?
• What is your philosophy of simplicial sets?

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"?
• What is your philosophy of simplicial sets?