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Stream: theory: mathematics

Topic: 'Infinitesimal 2-morphisms' for Lie algebras?

Tobias Fritz (Aug 27 2025 at 07:08):

With Grp denoting the category of groups, it's a standard fact (often given as an exercise) that Grp really is a 2-category, where a 2-morphism between group homomorphisms $\phi, \psi : G \to H$ is an element $h \in H$ with $\phi(g) = h \psi(g) h^{-1}$ for all $g \in G$. Upon considering the groups as categories and the homomorphisms as functors, these 2-morphisms are precisely the natural transformations.

Tobias Fritz (Aug 27 2025 at 07:09):

My question now is: do we get a similar structure for Lie algebras, but with 'infinitesimal 2-morphisms' instead, and what exactly does this mean? In more detail, consider the naturality equation $\phi(g) = h \psi(g) h^{-1}$ for Lie groups, specialized to elements of the form $g = e^x$ and $h = e^y$ where $x$ and $y$ live in the Lie algebra. Then abusing notation a bit by using the same notation for the associated Lie algebra homomorphisms as well, this means $\phi(x) = \psi(x) + [y, \psi(x)] + O(y^2)$.

Tobias Fritz (Aug 27 2025 at 07:09):

So if we now genuinely work with Lie algebras $\mathfrak{g}$ and $\mathfrak{h}$ and a Lie algebra homomorphism $\psi : \mathfrak{g} \to \mathfrak{h}$, then these considerations indicate that for every $y \in \mathfrak{h}$, the map $x \mapsto [y, \psi(x)]$ can be thought of as an 'infinitesimal 2-morphism' pointing away from $\psi$ in the direction of another homomorphism. But I don't know what kind of 2-categorical structure this would be!

Tobias Fritz (Aug 27 2025 at 07:18):

Maybe what I'm looking for is some definition of 'set with tangent vectors' which the category of Lie algebras would be enriched over. This sounds like the right notion might be Lie algebroids, but I'm not really satisfied with this because I don't want to assume that the ground field of the Lie algebras is $\mathbb{R}$ or $\mathbb{C}$, and even then I doubt that the hom-sets would be manifolds to begin with. But possibly one can dualize and consider an enrichment over Lie-Rinehart algebras instead?

Tobias Fritz (Aug 27 2025 at 07:18):

Or perhaps what I'm looking for is a Lie algebra version of Sweedler's measuring coalgebra, so that the category of Lie algebras would be enriched over (Lie?) coalgebras. But I don't have enough intuition about measuring coalgebras to know whether this is going in the right direction.

Tobias Fritz (Aug 27 2025 at 07:22):

Here's a closely related question which applies much more broadly. Homomorphisms between many algebraic structures have a notion of formal deformation. What kind of enrichment or 2-categorical structure lets us keep track of these deformations?

Paolo Perrone (Aug 27 2025 at 07:37):

In Lie algebra cohomology, similarly to what happens for groups, the first cohomology group is the space of derivations modulo inner derivations, where inner derivations are exactly the Lie brackets that you are considering.
That seems to go somewhat in the direction of your question, but of course, this leads to further questions:

• Does this homology theory actually comes from a higher category?
• Is there a model structure anywhere?
• Is this an instance of some sort of Dold-Kan correspondence?
• Is there a Lie algebra equivalent of the Eilenberg-MacLane space of a group?

(All the questions above are however way out of my expertise, I wouldn't even know how to state them properly.)

Tobias Fritz (Aug 27 2025 at 08:10):

Thanks! That sounds like a useful direction to look into as well. At Chevalley-Eilenberg cochain complex, we have:

A finite dimensional Lie algebra $\mathfrak{g}$ or degreewise finite-dimensional $L_\infty$ algebra $\mathfrak{g}$ is encoded in a differential $D:\lor^\bullet \mathfrak{g} \to \lor^\bullet \mathfrak{g}$ on the cofree co-commutative coalgebra generated by $\mathfrak{g}$.

The dual of this is a differential graded algebra $(\wedge^\bullet \mathfrak{g}^*,d)$. The underlying cochain complex (forgetting the monoidal structure) is the Chevalley-Eilenberg cochain complex.

There is in fact a bijection between quasi-free cochain differential graded algebras in non-negative degree and $L_\infty$ algebras.

This indicates that what I'm looking for may be the enrichment of dg-algebras over dg-coalgebras.

Martti Karvonen (Aug 27 2025 at 08:47):

The example of groups sits in a wider context I studied with Pieter Hofstra here. In a nutshell, inner automorphisms (which can be understood categorically as elements of the isotropy group) endow any category whatsoever with 2-cells, and more generally, so does any crossed $\mathbf{C}$-module (a (co)presheaf $G$ of groups with homomorphisms $G(A)\to Aut(A,A)$ for all objects satisfying some conditions). I don't know if you have a crossed $C$-module structure lying around here, but if you did, you'd get your $2$-cells from there.

Martti Karvonen (Aug 27 2025 at 08:49):

In that kind of setting, you can take your 1-cell $f\colon A\to B$, an element of $G(B)$, and these define uniquely a 2-cell out of $f$. This smells a bit like the "infinitesimal 2-morphism pointing away from $\psi$ in the direction of another homomorphism" you mentioned above.

Tobias Fritz (Aug 27 2025 at 08:55):

Ah yes, I remember that paper! It's definitely similar in flavour, but technically I think that it doesn't apply because infinitesimal automorphisms are not actually automorphisms, similar to how a tangent vector on a manifold is an infinitesimal path but not an actual path. Right?

Martti Karvonen (Aug 27 2025 at 09:02):

For this to work, one doesn't need an actual automorphism, as long as there is a way to extract an honest one in a nice way. That said, I guess that fails here too, so that an infinitesimal automorphism does not induce an actual automorphism in any reasonable way?

Graham Manuell (Aug 27 2025 at 09:18):

Groups and Lie-algebras are both action-representable semiabelian categories. In such categories there is a universal split extension with a given kernel. In the case of groups, split extensions with kernel $K$ and cokernel $H$ correspond to homorphisms from $H$ to $\mathrm{Aut}(K)$ and the universal split extension is $K \to \mathrm{Hol}(K) \to \mathrm{Aut}(K)$ where $\mathrm{Hol}(K)$ is the holomorph of $K$. This split exact sequence gives $\mathrm{Aut}(K)$ the structure of an internal category / a crossed module. The 2-cells are precisely the 2-morphisms from the 2-category of groups. If instead of considering homomorphisms from $H$ to $\mathrm{Aut}(K)$, you equip $H$ with the structure of a discrete internal category and consider internal anafunctors from it this internal category, you classify all extensions instead of split extensions. A similar thing works with Lie algebras where $\mathrm{Aut}(K)$ is replaced with the Lie algebra of derivations $\mathrm{Der}(K)$. I think this is also how to make the link with Lie algebra cohomology that @Paolo Perrone mentioned.

Tobias Fritz (Aug 27 2025 at 09:30):

Martti Karvonen said:

For this to work, one doesn't need an actual automorphism, as long as there is a way to extract an honest one in a nice way. That said, I guess that fails here too, so that an infinitesimal automorphism does not induce an actual automorphism in any reasonable way?

That's right. A map of the form $[y,-]$ on a Lie algebra is not an inner automorphism, but rather an inner derivation; the Jacobi identity is precisely the Leibniz rule! Over $\mathbb{R}$ or $\mathbb{C}$, one can exponentiate such a map to a 1-parameter family of actual automorphisms. But I think that a good structural account of what's going on should work over any ground field. Hence we can't expect to have genuine inner automorphisms.

Tobias Fritz (Aug 27 2025 at 09:41):

Graham Manuell said:

A similar thing works with Lie algebras where $\mathrm{Aut}(K)$ is replaced with the Lie algebra of derivations $\mathrm{Der}(K)$.

Interesting, thanks! So what does this suggest what the category of Lie algebras is enriched over, by analogy with the category of groups being enriched in groupoids?

Graham Manuell (Aug 27 2025 at 10:02):

I'm not completely sure. The internal categories are the analogues of the monoidal categories / 1-object 2-categories at each object, but something more would need to be done to allow for multiple objects. There is also the issue of this approach only being able to deal with isomorphisms of groups instead of general group homomorphisms.

David Michael Roberts (Aug 27 2025 at 10:20):

Given the 2-category of groups is really the 2-category of one-object groupoids, one might consider what 2-category of algebroids there is, and then specialise to the one-object case.

Tobias Fritz (Aug 27 2025 at 10:24):

Actually I've been trying to do the opposite, @David Michael Roberts :sweat_smile: I'd like to understand the category of Lie algebroids, and figured that perhaps starting with the single-object case might be easier!

Tobias Fritz (Aug 27 2025 at 10:26):

But really my question is what the category of Lie algebroids is enriched in, assuming that one wants to have infinitesimal gauge transformations as something like 'infinitesimal 2-morphisms'.

David Michael Roberts (Aug 27 2025 at 11:37):

It occurs to me that someone somewhere should have studied the/a functor LieGpd -> LieAlgbrd and checked if it extended to a 2-functor. In that case, what is it? Also, you can consider LieGpd as being enriched over diffeological groupods, and I wonder if this helps...