Start a new discussion

Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

Discussion Tag Cloud

Vanilla 1.1.10 is a product of Lussumo. More Information: Documentation, Community Support.

• CommentRowNumber1.
• CommentAuthorUrs
• CommentTimeJul 1st 2017

I needed a way to point to the topological interval $[0,1]$ regarded as an interval object for the use in homotopy theory. Neither the entry interval nor the entry interval object seemed specific enough for this purpose, so I created topological interval.

• CommentRowNumber2.
• CommentAuthorTodd_Trimble
• CommentTimeJul 1st 2017

I added Freyd’s characterization. (This is stated elsewhere, but also belongs here.)

• CommentRowNumber3.
• CommentAuthorUrs
• CommentTimeJul 1st 2017

Okay, thanks. Just for completeness, to make it the topological interval, I have added mentioning of the induced order topology to the statement of the theorem.

• CommentRowNumber4.
• CommentAuthorTodd_Trimble
• CommentTimeJul 1st 2017

In some sense “interval” is a red herring because the functor could just as well be defined for bipointed topological spaces and one would get $[0, 1]$ with its standard topology. Maybe it’s a matter of taste how one puts it, but I used intervals because the article is “topological interval”.

• CommentRowNumber5.
• CommentAuthorUrs
• CommentTimeJul 1st 2017

Okay, but then we should not say

as an object of the category $Int$ of intervals (linear orders with distinct top and bottom elements)

but

as an object of the category $Int$ of intervals (linear orders with distinct top and bottom elements) equipped with their order topology

• CommentRowNumber6.
• CommentAuthorTodd_Trimble
• CommentTimeJul 1st 2017

Not necessarily, because then we are speaking of the “category of topological intervals” or something. That’s more structural baggage than we actually need to carry around.

I probably could have made my point better by going minimal and removing both words, “topology” and “interval”. Just define $[0, 1]$ as the terminal $F$-coalgebra in the category of sets equipped with a distinct first and second point. Then the interval structure you want can be coinductively defined. More at coalgebra of the real interval.

• CommentRowNumber7.
• CommentAuthorUrs
• CommentTimeJul 2nd 2017

All I want is that the entry titled “topological interval” says explicitly how this terminal coalgebra is a topological space, either way.

• CommentRowNumber8.
• CommentAuthorTodd_Trimble
• CommentTimeJul 2nd 2017

Okay, I reworded that section and hopefully it’s satisfactory now.

• CommentRowNumber9.
• CommentAuthorUrs
• CommentTimeJul 2nd 2017

Okay, thanks!