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In something I am writing I needed the notion of a (n+1)-tuple and thought ’how can that be done elegantly?’ I then realised that there is not actually a definition in the lab. As this is probably most useful in the logical, type theoretic and constructive settings I will write just a stub and leave the experts in those areas to add in suitable commentary and details. I needed the idea that I call the support of a tuple, that is its set of component elements. Does anyone know of a better term for that?
Gee, I’m not sure what you want! An $(n+1)$-tuple is a section of a map $X \to [n+1]$? Or, if all the components of the tuple are understood as belonging to the same set $Y$, a section of the projection $\pi: Y \times [n+1] \to [n+1]$, which can in turn be identified with a map $[n+1] \to Y$.
The support of an $(n+1)$-tuple $f: [n+1] \to Y$ is the image $i(f) \hookrightarrow Y$?
Sorry if I’m saying something dumb…
I was just wondering if someone had met the idea elsewhere. I looks as if support is quite a good term for it so I will leave it at that. The construct is just needed to construct the functor from the category of simplicial complexers to that of simplicial sets.
wondering if someone had met the idea elsewhere
What is “the idea” that you have in mind here? Is it the term “support” as indicated in the entry tuple?
I am not fully sure what you mean by it.
the term I used was support for a n-tuple of entries from a set $X$, as being the underlying set of elements or the set of components of the ’tuple’. It is a convenience for handling the functor from $Simp.Comp$ to $SSet$ given by sending a simplicial complex $K$ to the simplcial set, $K^{simp}$ with $\sigma = \langle v_0,\ldots, v_n\rangle$ being an $n$-simplex of $K^{simp}$ if the ’support’ of $\sigma$, thus the actual vertices used, forms a simplex of $K$. Thus if $K$ is the simplicial complex 1-simplex, then $K^{simp}$ has a 2-simplex that looks like $\langle 1,0,1\rangle$, which is not degenerate. It is a simplex because its support is the set $\{0,1\}$ which is a simplex of $K$.
I can’t help feeling that this notion must have a name (other than the overused ’underlying set’).
This occurs in the paper by Abels and Holz that I mentioned in another thread, but their description is not 100% to my liking, so …. . (This leads on to orbihedra and triangulable orbifolds.)
Since I usually think of an $n$-tuple as a function from $[n] \coloneqq \{0, \ldots, n - 1\}$, I often call the support the range.
I expanded the discussion of formalisation of tuples, which is not always an obvious generalisation of the fomalisation of ordered pairs. (Try to generalise $(a, b) \coloneqq \{\{a\}, \{a, b\}\}$ to triples, for example.) Actually, I expanded all of it.
Great. I was hoping someone in the know would do that. Thanks.
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