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A simplicial set is by definition a contravariant functor $X:\mathbf{\Delta^{op}}\to\mathbf{Set}$ from category of simplices (whose objects are finite totally ordered sets and morphisms are non-decreasing functions) to category of sets. Intuitively this is a bunch of simplices glued together to form a partially ordered set (or a higher dimensional directed graph).

Suppose we have another functor $F:\mathbf{\Delta}\to\mathbf{C},$ can we extend this to a functor $\check{F}:\mathbf{sSet}\to\mathbf{C},$ from category of simplicial sets? If so, is this extension unique?

Please provide a proof or a reference for your answer.

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  • $\begingroup$ I suppose so, as long as $\bf C$ has the right kind of limits or colimits. $\endgroup$ Commented Nov 29, 2019 at 20:45

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Very generally, if $\mathbf{D}$ is any small category and $\mathbf{C}$ is any cocomplete category, then any functor $\mathbf{D}\to\mathbf{C}$ extends uniquely (up to natural isomorphism) to a cocontinuous functor $\mathbf{Set}^{\mathbf{D}^{op}}\to\mathbf{C},$ considering $\mathbf{D}$ as a subcategory of the presheaf category $\mathbf{Set}^{\mathbf{D}^{op}}$ via the Yoneda embedding. In other words, $\mathbf{Set}^{\mathbf{D}^{op}}$ is the "free cocompletion" of $\mathbf{D}$.

To sketch the proof, every object of $\mathbf{Set}^{\mathbf{D}^{op}}$ can be written canonically as a colimit of objects of $\mathbf{D}$ (i.e., representable presheaves), essentially the same way as every simplicial set is the colimit of its simplices. This then gives a unique possible formula for the cocontinuous extension of the functor, and then you just have to check that this formula actually works. You can find more details and related discussion on nLab: https://ncatlab.org/nlab/show/co-Yoneda+lemma and https://ncatlab.org/nlab/show/free+cocompletion.

In the case of simplicial sets, this is just the "obvious" construction: thinking of objects in the image of $F$ as "simplices" in $\mathbf{C}$, you glue together these simplices using colimits to get a "geometric realization" of any simplicial set in $\mathbf{C}$.

(If you don't require $\mathbf{C}$ to be cocomplete there is no reason to expect an extension to exist, and if you don't require the extension to be cocontinuous then there is no reason to expect it be unique. I don't know counterexamples off the top of my head but probably they aren't too hard to cook up.)

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  • $\begingroup$ Thank you very much for your detailed explanation. $\endgroup$
    – Bumblebee
    Commented Nov 30, 2019 at 20:59

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