ora-0027
1.15 Tangent categories and LINCS
The algebraic probes used below come from synthetic differential geometry. Fix a ground field \(k\), usually \(\mathbb R\).
A Weil algebra over \(k\) is a finite-dimensional commutative local \(k\)-algebra \(W\) whose residue field is \(k\). Equivalently, there is an augmentation \(\epsilon _W:W\to k\) with nilpotent maximal ideal
for some finite \(r\). The infinitesimal object represented contravariantly by \(W\) is its Weil probe \(\mathbb D_W\). For an object \(X\), the corresponding Weil prolongation is denoted
whenever the indicated internal hom exists; more generally, \(T^W\) denotes the associated prolongation functor.
The basic example is the algebra of dual numbers
Its probe is written \(\mathbb D=\mathbb D_{W_1}\). A map \(\mathbb D\to X\) based at \(x\) represents a first-order tangent vector at \(x\), and \(T^{W_1}X\) is the ordinary tangent object \(TX\). Larger Weil algebras encode several infinitesimal directions, higher jets, or prescribed relations among infinitesimals. The nilpotence is what truncates Taylor expansion to finite order.
The notation \(\mathbb D_W\) is genuinely synthetic. For example, the only ordinary real number satisfying \(d^2=0\) is zero, but the internal object \(\mathbb D=\{ d\mid d^2=0\} \) in a suitable smooth topos has enough generalized elements to detect derivatives [ Kock , 2006 ] . Thus a Weil probe is not being treated as a small subset of classical points; it is a test object through which infinitesimal variation is observed.
A tangent category equips a category \(\mathcal C\) with an endofunctor \(T:\mathcal C\to \mathcal C\) and natural maps abstracting the zero section, projection, addition, lift, and canonical flip of ordinary tangent bundles [ Cockett and Cruttwell , 2014 ] . These data satisfy coherence axioms ensuring that infinitesimal variation composes.