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added to simplicial object a section on the canonical simplicial enrichment and tensoring of $D^{\Delta^{op}}$ for $D$ having colimits and limits.
I just noticed this in simplicial object as the Idea:
A simplicial object $X$ in a category $C$ is a collection $\{X_n\}_{n \in \mathbb{N}}$ of objects in $C$ that behave as if $X_n$ were an $n$-dimensional simplex internal to $C$.
As someone who works a lot with simplicial objects this does not correspond to my idea of a simplicial object and quite frankly I do not understand the meaning of ‘simplex internal to $C$’. Does anyone else feel like I do? I think that a couple of instances of simplcial objects would give the idea better.
I have modified the sentence. But if anyone has more leisure for it than I currently do, feel free to have a go at it.
Thanks that is already much better. I have added in a bit more. I am wondering if there is s specific example that is general enough to be ’of use’ for a large number of people (other than simplicial sets). I will add one slightly later if I can.
The entry lists a bunch of Examples. Are you looking for something else?
I am wondering if there should be a specific example (e.g. the smooth singular complex of a Lie group can be considered in several settings as giving a simplicial group, simplicial topological group, etc. but that example fits better elsewhere). It may not be needed as there are other examples in other entries. Perhaps ‘If you want a specific example of this type of construction look at …’ might be worth putting somewhere. I am in two minds about it.
added pointer to:
Added:
The original definition of simplicial objects, maps between them, and homotopies of such maps is due to Daniel M. Kan:
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