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I have added the following disambiguation warning:
This entry is about a notion in formal logic (type theory). For the notion in homotopy theory see at mapping telescope.
But if you can think of any adjective that could go in front of your “telescope” here, I’d suggest that you add that to the entry’s name here
“type telescope”?
Another possibility that occurs to me is “dependent context”. I kind of wish type telescopes were called that more often; it’s more descriptive of what they actually are.
Or even just “telescope (type theory)”.
I find the entry in its present state less than illuminating. For instance it is hard to discern (for me, anyways) what the first two sentences are even trying to say:
In formal logic such as type theory a telescope $\Delta$ is a finite list of terms in context, whose elements/terms/inhabitants are substitutions. One distinguishes telescopes from contexts by the fact that telescopes have elements while contexts do not have elements.
(Why would contexts not “have elements”?!)
Also the current referencing is puzzling: The first one (Winterhalter 2020) mentions the term “telescope” only a couple of times and in passing, clearly assuming that the term needs no further introdction and clearly implying that
This does indeed seem to be how the term is used in practice, e.g. in
and in most of the other references listed here.
Indeed, the second reference (Shulman) introduces (slide 5) inference rules for telescopes that look much like those for dependent pairs (just saying “substitution” for “dependent pair”).
One is left with the impression of being presented a concept meant to be novel as of $\sim$2020, but aren’t we talking about the concept that goes back to
?
There a telescope is defined to be a sequence of iterated dependent types
$(a_1 \colon A_1) ,\, \big( a_2 \colon A_2(a_1) \big) ,\, \big( a_3 \colon A_3(a_1, a_2) \big) ,\, \cdots ,\, \big( a_n \colon A_n(a_1, \cdots a_{n-1}) \big)$I suppose we can read this either as an iterated sequence of contexts, or as the corresponding dependent pair type
$(a_1 \colon A_1) \times \big( a_2 \colon A_2(a_1) \big) \times \big( a_3 \colon A_3(a_1, a_2) \big) \times \cdots \times \big( a_n \colon A_n(a_1, \cdots a_{n-1}) \big)$(in the notation of dependent functions and dependent pairs – table) and I gather we want to say “telescope” to bring out the fine distinction between the former and the latter?
If that is the case, I’d suggest to rework the entry to make this clearer. An informative lead-in sentence could go as follows:
In dependent type theory, by a telescope (of dependent types) one means [de Bruijn (1991, §3.1)] an iterated dependent type (i.e. a tower of types, each depending on the previous types) $(a_1 \colon A_1) \big( a_2 \colon A_2(a_1) \big) \big( a_3 \colon A_3(a_1, a_2) \big) \cdots \big( a_n \colon A_n(a_1, \cdots a_{n-1}) \big)$, and maybe specifically when thought of as a type context. Corresponding to this is the iterated dependent pair type $(a_1 \colon A_1) \times \big( a_2 \colon A_2(a_1) \big) \times \big( a_3 \colon A_3(a_1, a_2) \big) \times \cdots \times \big( a_n \colon A_n(a_1, \cdots a_{n-1}) \big)$ (aggregating the list of types into a single type, as usual), and many authors use “telescope” simply as shorthand for such iterated dependent pair types (e.g. [Coen & Tassi (2008), Winterhalter (2020)]). Type telescopes regarded explicitly as distinct from their iterated dependent pair types (but with much the same inference rules, after all), are also considered in [Shulman (2022)].
Something like this.
In fact, reading more closely in
it becomes clear that by “telescope” he does mean iterated dependent pair types.
Moreover, it seems de Bruijn proposed this terminology and notion already in the early 1970s, because it is already attributed to him in 1975 in:
Jeffery Zucker, Formalization of Classical Mathematics in Automath, Colloques Internationaux du Centre National de la Recherche Scientifique 249 (1975) 135-145 [web, pdf]
also in: Studies in Logic and the Foundations of Mathematics 133 (1994) 127-139 [doi:10.1016/S0049-237X(08)70202-7]
It all comes from the desire to formalize mathematical structures in dependent type theory as type telescopes:
From Zucker (1975, §10.2):
Now a general framework in which to view linear orders, or other algebraic structures, has been proposed by de Bruijn. It uses the notion of “telescope”
[…]
A telescope therefore functions like a “generalized $\sum$”.
So unless there is some major objection, I am going to completely rework and expand our entry on telescopes to reflect this.
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