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I am hereby moving the following old Discussion box from interval object to here
Urs Schreiber: this is really old discussion by now. We might want to start putting dates on discussions. In principle it can be seen from the entry history, but readers glancing at this here hardly will. Maybe discussions like this here are better had at the forum after all.
We had originally started discussing the notion of interval objects at homotopy but then moved it to this entry here. The above entry grew out of the following discussion we had, together with discussion at Trimble n-category.
Urs: Let me chat a bit about what I am looking for here. It seems very useful to have a good notion of what it means in a context like a closed category of fibrant objects to say that path objects are compatibly corepresented.
By this should be meant: there exists an object $I$ such that
for $B$ any other object, $[I,B]$ is a path object;
and such that $I$ has some structure and property which makes it “nice”.
In something I am thinking about the main point of $I$ being nice is that it can model compositon: it must be possible to put two intervals end-to-end and get an interval of twice the length. In some private notes here I suggest that:
a “category with interval object” should be
with a compatible structure of a category of fibrant objects
and equipped with an internal co-categoy on $\sigma, \tau : pt \to I$ for $I$ the interval object;
such that $I$ co-represents path objects, in that for all objects $B$, $[I,B]$ is a path object for $B$.
I think there are a bunch of obvious examples: all familiar models of higher groupoids (Kan complexes, $\omega$-groupoids etc.) with the interval object being the obvious cellular interval $\{a \stackrel{\simeq}{\to} c\}$.
I also describe one class of applications which I think this is needed/useful for: recall how Kenneth Brown in section 4 of his article on category of fibrant objects (see theorems recalled there and reference given there) describes fiber bundles in the abstract homotopy theory of a pointed category of fibrant objects. This is pretty restrictive. In order to describe things like $\infty$-vector bundles in an context of enriched homotopy theory one must drop this assumption of the ambient category being pointed. The structure of it being a category with an interval object is just the necessary extra structure to still allow to talk of (principal and associated) fiber bundles in abstract homotopy theory. It seems.
Comments are very welcome.
Todd: The original “Trimblean” definition for weak $n$-categories (I called them “flabby” $n$-categories) crucially used the fact that in a nice category $Top$, we have a highly nontrivial $Top$-operad where the components have the form $\hom_{Top}(I, I^{\vee n})$, where $X \vee Y$ here denotes the cospan composite of two bipointed spaces (each seen as a cospan from the one-point space to itself), and the hom here is the internal hom between cospans.
My comment is that the only thing that stops one from generalizing this to general (monoidal closed) model categories is that “usually” $I$ doesn’t seem to be “nice” in your sense here, and so one doesn’t get an interesting (nontrivial) operad when my machine is applied to the interval object. But I’m generally on the lookout for this sort of thing, and would be very interested in hearing from others if they have interesting examples of this.
to be continued in the next comment
continuation from previous comment
Urs Schreiber: Thanks, Todd. I should have listed the examples I had in mind: I was thinking about strict omega-category here, where the 1st oriental $G_1 = I = \{a \to b\}$ should naturally be an internal co-category, where co-composition is the functor which sends $a \to b$ to the composite $a_1 \to (b_1 = a_2) \to a_2$.
More generally, there are, I think $n$ different co-category structures on the standard $n$-globe, with co-source and co-target given by the two injections of the standard $(k \lt n)$-globe.
The composition operations in the internal hom $\omega$-category $hom(C,D)$ in strict $\omega$-categories can, I’d think, then be thought of as coming from the image of these co-categories under $Hom(C\otimes -, D)$.
A description of what I just tried to say with the illuminating diagrams is on p. 35 here. Hope I got this right. Please let me know if I am mixed up.
Todd: It seems to me there might be some trickiness about which hom you want. The thing you’re proposing sounds like it would work to describe the hom for $\omega$-Cat as a cartesian closed category, but I’m not sure off the bat how it will play out with respect to the Crans-Gray monoidal biclosed structure. I’d have to think about it more carefully, but there’s something a little “thin” about the strict co-category structure on say the category 2 (as an interval co-category in $Cat$) which makes me wonder.
(After an email from Urs:) I think Urs is right after all – this should work fine for either monoidal structure.
Urs: Also by email, Todd points out that of course more generally, we want our interval objects to form internal co-categories only up to coherent homotopy, because otherwise the example of $G_1$ in strict $\omega$-categories is likely to be essentially the only good example. We want internal homotopy co-categories.
I need to learn more about how one would go about systematically defining concepts internal to a model category up to homotopy. What are the available tools for handling higher coherent homotopies in an arbitrary model category?
My understanding is that Todd is going to write an entry on the Trimble definition of $\infty$-categories, and that this issue appears there in some guise. So maybe I’ll just wait for Todd’s entry to appear…
end of forwarded discussion
I am hereby moving the following old discussion box from interval object to here:
+–{.query} Might there be two notions of interval object, one in a closed category such that $[I,B]$ is a path object, and one in a monoidal category such that $I \otimes B$ is a cylinder object? (And then a stronger notion, combining these, in a closed monoidal category.) —Toby
Urs: True, depending on application, one may be able to and want to drop some assumptions here. We might eventually give a layered definition, which adds assumptions step by step.
But, on the other hand, the main purpose of the interval object here, which goes beyond the idea for instance in a cylinder functor is that we want to induce for any object $B$ on the internal hom-object $(B_0 \leftarrow [I,B]\rightarrow B_0)$ the structure of a (homotopy coherent- or $A_\infty$-) internal category. Namely the fundamental category $\Pi_1(B)$. To get that we need both the closed and the monoidal and the homotopical structure.
Toby: Actually, I wasn't so much thinking about weakening the requirements on $V$ (although admittedly I did phrase my question to include that). My main point that they are, naïvely, two concepts: one that uses the closed structure to give path objects, and one that uses the monoidal structure to give cylinder objects. So will we have interval objects, co-interval objects, and bi-interval objects?
=–
There are some inconsiastencies in interval object: In the section “Berger-Moerdijk segment object” a segment object is defined in a monoidal model category. Then later in this section there is the sentence “If V is equipped with the structure of a model category then a segment object is an interval in V if…”. Then in the next section “Intervals for Trimble $\omega$-categories” we find the sentence “The following definition of category with interval object aims to abstract this construction away from V= Top to other closed monoidal homotopical categories.” - but above this section already have been given various definitions where $V$ is not Top. At this point is also being said that this definition will be motivated “further below” (probably this refers to section “Fundamental little 1-cubes space induced from an interval”) but parts of this motivation are given in the idea section (the second bulleted point).
Yeah, the page talks about different formalizations of interval objects in different contexts. A little editing could certainly improve this.
Are you looking into re-editing the entry? Let me know if you need help.
Yes, I am just editing directed object and then I will look at interval object.
Among some minor edits I transferred the sub sections on “fundamental (∞,1)-categories induced from interval objects” from interval object to fundamental (infinity,1)-category.
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