# Intuition and analogue of Wraith axiom from synthetic differential geometry

In synthetic differential geometry, an object $$M$$ verifies the Wraith axiom if for all functions $$\tau:D\times D\to M$$ which are constant on the axes $$\tau(d,0)=\tau(0,d)=\tau(0,0)$$ for all $$d\in D$$, there's a unique factorization through the multiplication map, i.e there's a unique function $$t:D\to M$$ such that $$t(d_1\cdot d_2)=\tau(d_1,d_2)$$.

What is the geometric/physical intuition behind this axiom? What is the analogue in the category of smooth manifolds?

Update. Following the answers I think I should add some motivation. Lavendhomme's book defines the commutator of vector fields as $$\tau(d_1,d_2)=Y_{-d_2}X_{-d_1}Y_{d_2}X_{d_1}$$. Because this is constant on axes, it factors through the multiplication map to give a vector field $$t$$ charaterized by $$t(d_1d_2)=\tau(d_1,d_2)$$. I understand $$t$$ is desirable since it is a vector field, but I don't know how to geometrically interpret its characterizing property. For instance, why not consider the vector field $$\tau(d_1,d_1)$$ given by precomposing the diagonal? This motivated my question.

Axiom W is about the behaviour of the second tangent bundle - it ensures that the vertical bundle of the tangent bundle, $$V(M) \subseteq T\circ T(M)$$, where $$V(M) = T(p)^{-1}(0)$$, decomposes as the pullback of the projection $$p_M: T(M) \to M$$ along itself. The map $$[\bullet, M]:[D,M] \to [D \times D, M]$$ would be written fiberwise as $$\ell(v) = \frac{d}{dt}_{t = 0} (vt)$$. If you look at Robin Cockett and Geoff Cruttwell's first paper on tangent categories, you can see they spend a fair bit of time talking about the universality of the vertical lift and its relationship with the Lie bracket.

You may also want to look at the definition of the core of a double vector bundle - axiom W can also be read as saying the core of the second tangent bundle of M is $$TM$$.

Edit: If you look at the answer here, you can see how the lift from the fibred product $$TM \times_M TM \to T^2(M)$$ is written. Using infinitesimals, you would write $$\gamma,\beta:D\to M, \gamma(0)=\beta(0)$$ is sent to the map $$D\times D \to M$$ which is given by $$d_1,d_2 \mapsto \gamma(d_1) + \beta(d_1d_2)$$. The condition that this be the kernel of $$T(p)$$ is provable from property W (and a good exercise).

The square root business you mentioned looks to be how you would show property $$W$$ holds in the category of smooth manifolds (when rewritten as the universality of the vertical lift from the tangent category axioms).

• Dear Ben, thanks for this answer. I don't quite see how the Wraith axiom gives the canonical isomorphism between the vertical bundle and its kernel pair. Indeed the classical analogue is straightforward. It seems much more blatantly relevant for the Lie bracket due to the square root business, which is much more confusing. The paper by Cockett and Cruttwell is fairly impenetrable for me, so I would enormously appreciate some more details on the geometric explanation! Jan 18 '20 at 10:32
• Hi Arrow, I've expanded my answer a bit. Hopefully that helps. Jan 18 '20 at 18:49
• Dear Ben, I asked a follow up question here. Nov 9 '20 at 22:37

An analogue for smooth manifolds $$M$$ is: For all continuous bilinear functions $$\tau: \mathbb{R}\times\mathbb{R} \rightarrow TM_p$$ which are constant on the axes $$\tau(r,0)=\tau(0,r)=\tau(0,0)$$ for all $$r \in \mathbb{R}$$, there's a unique factorization through the multiplication map, i.e. there's a unique function $$t: \mathbb{R}\rightarrow TM_p$$ such that $$t(r_1 \cdot r_2) =\tau(r_1, r_2)$$.

So the geometric intuition is that all functions are continuous, and are linear at infinitesimal scales.

• Dear Matt, the Kock-Lawvere axiom in SDG asserts that things are linear at an infinitesimal scale. My question is "what is the geometric meaning of factoring through the multiplication map?". Any bilinear map is constant (zero) on the axes, and $t(r)=\tau(r,1)=\tau(1,r)$. I'm just not sure how to think of this... (Maybe there's nothing to say.) Jan 17 '20 at 22:19
• It's more interesting if you generalize to continuous bi-affine functions (satisfying $f(2x)+f(2y)=2f(x+y)$), but that still doesn't leave a lot to say. Jan 17 '20 at 22:55
• I'm not quite sure about this (although I may be misunderstanding your notation). If $TM_p$ is the tangent space above a point $p$, this is just stating $R \otimes R \cong R$, which isn't what the Wraith axiom is capturing. Jan 18 '20 at 4:20
• @BenMacAdam I updated the question with some motivation. Jan 18 '20 at 9:39