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There is a very intimate connection between categories, simplicial sets, and topological spaces. On one hand, simplicial sets are the presheaf category on the category $\Delta$ and $\Delta$ is a canonically defined "invariant" of the theory of categories. (e.g. the machinery of Mark Weber "spits out" $\Delta$ when you "plug in" the free category monad: http://golem.ph.utexas.edu/category/2008/01/mark_weber_on_nerves_of_catego.html)

However, $\Delta$ is also linked with topological spaces. The key to this link is the functor $\Delta \to Top$ which assigns the category $[n]$ the standard n-simplex $\Delta^n$. It is this functor which produces the adjunction between the geometric realization functor and the singular nerve functor which allow you to transfer the model structure on $Top$ to $Set^{\Delta^{op}}$ so that this adjunction becomes a Quillen equivalence.

My question is the following:

Is there a deep categorical justification for the functor $\Delta \to Top$ being defined exactly how it is? If we didn't know about the standard n-simplices, how could we "cook up" such a functor? I would like a construction of this functor which is truly canonical.

The closest to an answer I've found is Drinfeld's paper http://arxiv.org/abs/math/0304064. However, this doesn't quite "nail it home" to me. First of all, the definition is just made, but not motivated. The definition shouldn't be a "guess that works", but something canonical. Moreover, if you unwind it enough, it is secretly using the fact that finite subsets of the interval with cardinality $n$ correspond to points in (the interior of) the $(n+1)$-simplex. Plus, there's some funny business going on for geometric realization of non-finite simplicial sets. (Don't get me wrong- I think it's a great paper. It just doesn't totally answer my question).

$Set^{\Delta^{op}}$ is the classifying topos for interval objects and the standard geometric realization functor $Set^{\Delta^{op}} \to Top$ is uniquely determined by its sending the generic interval to $[0,1]$. This reduces the question to "why is [0.1] the canonical interval?". Is it perhaps the unique interval object whose induced functor $Set^{\Delta^{op}} \to Top$ is both left-exact and conservative?

EDIT: I've proposed a partial answer to this below, along the lines of the above lead. I would love any feedback that anyone has on this.

2 added 438 characters in body

There is a very intimate connection between categories, simplicial sets, and topological spaces. On one hand, simplicial sets are the presheaf category on the category $\Delta$ and $\Delta$ is a canonically defined "invariant" of the theory of categories. (e.g. the machinery of Mark Weber "spits out" $\Delta$ when you "plug in" the free category monad: http://golem.ph.utexas.edu/category/2008/01/mark_weber_on_nerves_of_catego.html)

However, $\Delta$ is also linked with topological spaces. The key to this link is the functor $\Delta \to Top$ which assigns the category $[n]$ the standard n-simplex $\Delta^n$. It is this functor which produces the adjunction between the geometric realization functor and the singular nerve functor which allow you to transfer the model structure on $Top$ to $Set^{\Delta^{op}}$ so that this adjunction becomes a Quillen equivalence.

My question is the following:

Is there a deep categorical justification for the functor $\Delta \to Top$ being defined exactly how it is? If we didn't know about the standard n-simplices, how could we "cook up" such a functor? I would like a construction of this functor which is truly canonical.

The closest to an answer I've found is Drinfeld's paper http://arxiv.org/abs/math/0304064. However, this doesn't quite "nail it home" to me. First of all, the definition is just made, but not motivated. The definition shouldn't be a "guess that works", but something canonical. Moreover, if you unwind it enough, it is secretly using the fact that finite subsets of the interval with cardinality $n$ correspond to points in (the interior of) the $(n+1)$-simplex. Plus, there's some funny business going on for geometric realization of non-finite simplicial sets. (Don't get me wrong- I think it's a great paper. It just doesn't totally answer my question).

$Set^{\Delta^{op}}$ is the classifying topos for interval objects and the standard geometric realization functor $Set^{\Delta^{op}} \to Top$ is uniquely determined by its sending the generic interval to $[0,1]$. This reduces the question to "why is [0.1] the canonical interval?". Is it perhaps the unique interval object whose induced functor $Set^{\Delta^{op}} \to Top$ is both left-exact and conservative?

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# A canonical and categorical construction for geometric realization

There is a very intimate connection between categories, simplicial sets, and topological spaces. On one hand, simplicial sets are the presheaf category on the category $\Delta$ and $\Delta$ is a canonically defined "invariant" of the theory of categories. (e.g. the machinery of Mark Weber "spits out" $\Delta$ when you "plug in" the free category monad: http://golem.ph.utexas.edu/category/2008/01/mark_weber_on_nerves_of_catego.html)

However, $\Delta$ is also linked with topological spaces. The key to this link is the functor $\Delta \to Top$ which assigns the category $[n]$ the standard n-simplex $\Delta^n$. It is this functor which produces the adjunction between the geometric realization functor and the singular nerve functor which allow you to transfer the model structure on $Top$ to $Set^{\Delta^{op}}$ so that this adjunction becomes a Quillen equivalence.

My question is the following:

Is there a deep categorical justification for the functor $\Delta \to Top$ being defined exactly how it is? If we didn't know about the standard n-simplices, how could we "cook up" such a functor? I would like a construction of this functor which is truly canonical.

The closest to an answer I've found is Drinfeld's paper http://arxiv.org/abs/math/0304064. However, this doesn't quite "nail it home" to me. First of all, the definition is just made, but not motivated. The definition shouldn't be a "guess that works", but something canonical. Moreover, if you unwind it enough, it is secretly using the fact that finite subsets of the interval with cardinality $n$ correspond to points in (the interior of) the $(n+1)$-simplex. Plus, there's some funny business going on for geometric realization of non-finite simplicial sets. (Don't get me wrong- I think it's a great paper. It just doesn't totally answer my question).