Characterizing Tangent Display Maps via Linear Assignments

Tangent display maps, introduced by Cruttwell and Lanfranchi [2], provide the abstract analogue of submersions inside a tangent category [1]: they single out those morphisms along which one can pull back differential bundles compatibly with connection data. In algebraic examples the tangent functor is typically far from left exact, so deciding whether a given map is “display” becomes a genuinely algebraic problem.

We work in the tangent categories constructed from a linear assignment $L$ in the sense of Ikonicoff–Lemay–Van der Linden [3]. Such an $L$ is a product-preserving endofunctor equipped with a natural isomorphism $\nu :LL\Rightarrow L$, and it induces $T(X)=X\times L(X)$ and $T(f)=f\times L(f)$. A key feature is that the iterates admit explicit decompositions

$$T^n(X)\cong X\times {(L(X))}^{2^n-1}(n\ge 0),$$

built functorially from the product comparison maps for $L$ and the idempotence isomorphisms $\nu$.

Given a pullback square

there is a canonical Beck–Chevalley comparison morphism ${\beta }_{f,q}:L(P)\to L(N){\times }_{L(M)}L(E)$. Our main result shows that, for linear-assignment tangent functors, the infinite requirement that a pullback be preserved by all iterates of $T$ collapses to a single Beck–Chevalley check at the level of $L$.

In the strongly unital (hence semi-abelian) situation of [3], $L$ is commutativization/abelianization (e.g. $T(G)=G\times G^{\mathrm{ab}}$ in $\mathrm{𝐆𝐫𝐩}$). Taking $f$ to be the zero map identifies ${\beta }_{0,q}$ with the canonical comparison $L(\ker q)\to \ker (Lq)$, giving an immediate obstruction for regular epimorphisms to be tangent display maps. We use the explicit formulas for the iterates $T^n$ to control the remaining Beck–Chevalley conditions in the definition, and we extract pullback-stable families of regular epimorphisms satisfying them, providing concrete algebraic analogues of submersions.