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It is well known that a category can be defined as a certain simplicial set obtained by iterated fibred products which satisfies the internal horn filler condition; moreover, requiring horn filling for all horns (i.e., the Kan condition) one obtains the notion of groupoid. Then both the notions of category and groupoid should have an internalization in any category where one is able to arrange things in a way to have the required fibered products, and to state the horn filling condition.
This is what happens, e.g., when one defines Lie groupoids imposing that the source and target maps are submersions. Similarly one has a notion of Lie category, which by some reason seems to be less widely known of the more particular notion of Lie groupoid (maybe this is not surprising.. after all I suspect the notion of category is less known of that of group..). Another classical example are topological categories and groupoids.
Moving from categories and groupoids to oo-categories and oo-groupoids, one should have a similar internal simplicial object based notion of, e.g., Lie oo-groupoid. However, in the nLab the oo-sheaf point of view seems to be largely preferred to the internal Kan object point of view. Why is it so? is the oo-sheaf version just more general and powerful or there are problems with the internal version? I’m asking this since at internal infinity-groupoid it is said that a classical example of the internal Kan complex definition of oo-groupoid are Lie oo-groupoids, but then at Lie infinity-groupoid there is no trace of the internal Kan complex definition.
The short answer to your question is : geometric stack and Structured Spaces.
The longer answer is: $\infty$-stacks on a site form a kind of completion of $\infty$-groupoids internal to the site (“geometric $\infty$-stacks”) that has lots of nice abstract properties which the latter is lacking. So you want to be formulating concepts in the larger context of all $\infty$-stacks and check in examples that the general nonsense spits out objects that happen to actually be “geometric $\infty$-stack”s.
There are some ways to formalize this idea: one is Lurie’s notion of Structured Spaces. That provides a formalism for finding in a large $\infty$-topos those objects that are modeled on a certain geometry (for structured (infinity,1)-toposes).
This choice of geometry however still is a choice. One could ask if given an $(\infty,1)$-topos there is a canonical notion of geometric objects inside it associated with it. One way to answer this is suggested by Bertrand Toen in his work that is summarized and linked at rational homotopy theory in an (infinity,1)-topos: there one effectively notices that if the $(\infty,1)$-topos has a singled-out line or path object, then this induces a notion of geometric $\infty$-stack.
To see these issues in action, an instructive article to look at is André Henriques’ “Integrating $L_\infty$-algebras”. He integrates to $\infty$-groupids internal to Banach manifolds, but continuously uses the fact that he can step outside this this to general simplicial presheaves.
From this perspective a single main point that I tried to highlight at Lie infinity-groupoid is: if we admit straight away that our geometric $\infty$-groupoids are sitting inside a larger structure that forms an $(\infty,1)$-topos, then we can make full use of some nice abstract constructions in there, that help us think about the special nice objects that we are actually interested in.
For instance: trying to understand how there is a canonica Maurer-Cartan $L_\infty$-algebra valued form on a (geometric!) $\infty$-group can be quite a mind-bender. But if we allow ourselves to step outside the restricted world of geometric $\infty$-groupoids and also allow non-geometric ones, then we find that there is a general abstract construction that spits out the answer. So then we can just follow the tao, turn the crank and work out what the answer is, restricted to the geoemtric objects that we are interested in.
I have edited the Idea-section of Lie infinity-groupoid somewhat in an attempt to reflect the above discussion. But there is certainly still much room for improvement.
Concernng “Lie categories”: the idea is explicitly envoked and treated a bit in Toby Bartels’ thesis on 2-bundles in terms of categories internal to diffeological spaces. There they are called 2-spaces .
Nowadays, we know what the $(\infty,1)$-category of $(\infty,2)$-sheaves on CartSp is, and I think this is the right context to study Lie $(\infty,1)$-categories.
mmm… something blinking in the obscurity.. so you are saying that I should always adopt the relative point of view: groupoid always means groupoid over a site (i.e. stack), and groupods which are represented internally deserve the special name geometric groupoid. so, for instance, a Lie groupoid presents a geometric groupoid over Diff (or over CartSp).
very neat, indeed :)
Yes. It’s just Grothendieck’s age-old insight once again: if you want to study some type of objects that don’t form a nice categeory, throw in generalized such objects such that the totality does form a nice category. Then develop your concepts int the nice category of generalized objects and only after you apply the general machinery check if you land back in nice objects. If you don’t, then the construction in question does not exist for nice objects anyway, but then you already have in hands the next best generalized solution.
I think the discussion of de Rham objects are Lie infinity-groupoid is a good example: for understanding differential geometry based on a geometric $\infty$-Lie groups $G$ it is very useful to have the generalized $\infty$-Lie groupoid $\mathbf{\flat}_{dR}\mathbf{B}G$ in hands and particularly it is useful to have the canonical morphism $G \to \mathbf{\flat}_{dR} \mathbf{B}G$. This encodes information about the geometric $G$ in terms of the “non-geometric” (non-concrete) $\mathbf{\flat}_{dR}\mathbf{B}G$. Since we need that morphism to extract that informaton, it is useful to have both objects regarded in the same context. So for understanding the geometric $G$ it is good to study it in the more general context of not-necessarily geometric objects.
Similarly one has a notion of Lie category, which by some reason seems to be less widely known of the more particular notion of Lie groupoid
Ehresmann introduced these at the same time as Lie groupoids (but at that time they were called differentiable categories and groupoids - the rename happened I believe in the 1980s), but they didn’t seem to catch on. The big result from that paper was that the category of transitive, locally trivial Lie groupoids is equivalent to that of principal bundles (incidentally, Michael Murray studied this result when on an undergraduate summer scholarship way back when…)
David, please don’t forget to put that bit of info into the Lab…
So the right setting for oo-Lie theory is that of oo-stacks over Diff.. but then the right setting for ordinary Lie theory should be general differential stacks rather than representable ones.. mmm.. makes sense: if I think of a Lie group $G$ as a presheaf over Diff (or better some extension of this containing infinitesimal spaces), then the Lie algebra of $G$ is just $G(Spec \mathbb{K}[\epsilon]/(\epsilon^2))$.
So “true” Lie theory relates stacks on infinitesimal spaces and stacks on Cartesian spaces. Then one can investigate representability issues, and this gives Lie algebras and Lie groups.
Extremely clear, indeed.
Yes. But obtaining the Lie algebra is not just restriction to infinitesimal test spaces. I think you mean to look at morphisms $Spec(\mathbb{L}[\epsilon]/(\epsilon^2)) \to G$ that send the unique point of the infinitesimal interval to the origin of $G$.
That gives the Lie algebra as a vector space, yes. One can also get the Lie algebra including its algebra structure in a nice way.
Roughly: start with the delooping groupoid $\mathbf{B}G$ which has a single object, and then restrict it to the infinitesimal neightbourhood of that object.
The details are in the examples-section at infinity-Lie algebroid.
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