부산대학교 도서관

A single axiomatization of tangent categories enables one to deal with tangent bundles appearing in multiple contexts simultaneously. Furthermore, many definitions and constructions in differential geometry are closely linked to the tangent bundle, where vector fields, the Lie bracket, connections, and differential forms can all be viewed as certain maps in the category of smooth manifolds, with the domain or codomain being the tangent bundle, or other bundles related to it. Thus a notion of tangent categories provides an axiomatic framework for understanding tangent bundles and differential operations occurring in differential geometry, algebraic geometry, categories in homotopy theory and computer science, giving a generalized and a unified insight into vector fields and their Lie bracket, vector bundles, and connections, to name a few. \par G.~S.~H. Cruttwell and R.~B.~B. Lucyshyn-Wright investigate differential and sector forms in tangent categories, where sector forms form symmetric cosimplicial objects, providing these categories with a rich structure. More precisely, multilinear, but not necessarily alternating, maps from the $n$-th iterate $T^nM$ of the tangent bundle of $M$ to some coefficient object $E$ in a tangent category are known as sector forms. The exterior derivative of singular forms (which are multilinear alternating maps $T^nM\rightarrow R$) works for sector forms, so in addition to the complex of singular forms, tangent categories have complexes of sector forms. In the category of smooth manifolds, the resulting complex of sector forms has a subcomplex isomorphic to the de~Rham complex of differential forms, which may be identified with alternating sector forms. \par Sector forms have more structure beyond what can be obtained from cochain complexes; more precisely, for each $n$, there are $n+1$ ``derivative'' or co-face operations taking sector $n$-forms to sector $(n+1)$-forms (Theorem 7.7, page 904). Also, there are $n-1$ codegeneracy operations taking sector $n$-forms to sector $(n-1)$-forms and there are $n-1$ symmetry operations taking sector $n$-forms to sector $n$-forms (Proposition 7.3, page 900). In addition, there is a functor on the category of finite cardinals, providing the structure of an augmented symmetric cosimplicial object of sector forms (Theorem 7.7, page 904). \par Examples of tangent categories include finite-dimensional smooth manifolds with the usual tangent bundle structure, and infinitesimally and vertically linear objects in any model of synthetic differential geometry, where if $D$ is the object of square-zero infinitesimals, then $TM\coloneq M^D$. Another example arises from abstract homotopy theory; since every Cartesian differential category is a tangent category, K. Bauer, B. Johnson, C. Osborne, E. Riehl, and A. Tebbe used this insight to prove the existence of certain higher-order chain rules for abelian functor calculus.