부산대학교 도서관

A single axiomatization of tangent categories enables one to deal withtangent bundles appearing in multiple contexts simultaneously.Furthermore, many definitions and constructions in differentialgeometry are closely linked to the tangent bundle, where vector fields,the Lie bracket, connections, and differential forms can all be viewedas certain maps in the category of smooth manifolds, with the domain orcodomain being the tangent bundle, or other bundles related to it. Thusa notion of tangent categories provides an axiomatic framework forunderstanding tangent bundles and differential operations occurring indifferential geometry, algebraic geometry, categories in homotopytheory and computer science, giving a generalized and a unified insightinto vector fields and their Lie bracket, vector bundles, andconnections, to name a few.\par G.~S.~H. Cruttwell and R.~B.~B. Lucyshyn-Wright investigatedifferential and sector forms in tangent categories, where sector formsform symmetric cosimplicial objects, providing these categories with arich structure. More precisely, multilinear, but not necessarilyalternating, maps from the $n$-th iterate $T^nM$ of the tangent bundleof $M$ to some coefficient object $E$ in a tangent category are knownas sector forms. The exterior derivative of singular forms (which aremultilinear alternating maps $T^nM\rightarrow R$) works for sectorforms, so in addition to the complex of singular forms, tangentcategories have complexes of sector forms. In the category of smoothmanifolds, the resulting complex of sector forms has a subcomplexisomorphic to the de~Rham complex of differential forms, which may beidentified with alternating sector forms.\par Sector forms have more structure beyond what can be obtained fromcochain 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 $(n-1)$-formsand there are $n-1$ symmetry operations taking sector $n$-forms tosector $n$-forms (Proposition 7.3, page 900). In addition, there is afunctor on the category of finite cardinals, providing the structure ofan augmented symmetric cosimplicial object of sector forms (Theorem7.7, page 904).\par Examples of tangent categories include finite-dimensional smoothmanifolds with the usual tangent bundle structure, and infinitesimallyand vertically linear objects in any model of synthetic differentialgeometry, where if $D$ is the object of square-zero infinitesimals,then $TM\coloneq M^D$. Another example arises from abstract homotopytheory; since every Cartesian differential category is a tangentcategory, K. Bauer, B. Johnson, C. Osborne, E. Riehl, and A. Tebbe usedthis insight to prove the existence of certain higher-order chain rulesfor abelian functor calculus.