I am confused about some notation in the quick way of constructing the geometric realization of a simplicial set.

Consider the simplex category $\Delta \downarrow X$ of a simplicial set $X$.

The objects are the maps (natural transformations) $\sigma:\Delta ^n \to X$ where $\Delta^n= \Delta(-, [n]): \Delta^{op} \to \textbf{Set}$ is the simplicial set.

An arrow between $\sigma: \Delta^n \to X$ to $\tau: \Delta^m \to X$ is the map (natural transformation) $\theta : \Delta^n \to \Delta^m$ induced by the map $\theta*: [m] \to [n]$ in $\Delta^{op}$ such that the following holds:

$$\tau \circ \theta = \sigma$$

The Geometric Realization of a simplicial set $X$ is defined as:

$$|X| = \lim_{\rightarrow \\ {\Delta^n\to X} \\ in \Delta\downarrow X}|\Delta^n|$$

I am confused about the notation here about what the diagram $F$ of this colimit $\lim F$ is exactly.


1 Answer 1


The diagram $F:\Delta\downarrow X\to \mathrm{Top}$ sends $(\sigma:\Delta^n\to X)$ to $|\Delta^n|$ and sends a map in $\Delta\downarrow X$ induced by $\theta:\Delta^m\to \Delta^n$ to $|\theta|$. In other words, there is a natural projection functor $p:\Delta\downarrow X\to \Delta$, and $F$ is the composition of $p$ with the canonical functor $\Delta\to\mathrm{Top}$.

  • $\begingroup$ The structure of that colimit - the composition of some function with a projection out of a comma category - is strikingly similar to the colimit formula for the left Kan extension. Is there a connection? $\endgroup$
    – FShrike
    Commented Oct 29, 2022 at 15:36
  • 1
    $\begingroup$ @FShrike Yep! The geometric realization is the left Kan extension of its restriction to the simplices. More generally the same is true for any cocontinuous functor in simplicial sets, or indeed on any presheaf category. $\endgroup$ Commented Oct 29, 2022 at 17:38
  • $\begingroup$ Thank you for responding, I think that makes sense as a consequence of the fact every presheaf is the colimit of representables, indexed by the category of elements $\endgroup$
    – FShrike
    Commented Oct 29, 2022 at 20:02
  • $\begingroup$ That leads me to a different question, if you don’t mind my asking. Does it then make sense to define the geometric realisation as the unique cocontinuous extension of the obvious action on the standard simplices? $\endgroup$
    – FShrike
    Commented Oct 29, 2022 at 20:14
  • $\begingroup$ @FShrike Yep, any cocontinuous functor on a presheaf category is uniquely determined by its restriction to the representables. $\endgroup$ Commented Oct 30, 2022 at 2:45

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