Category of manifolds

Category of manifolds
Name	Category of manifolds
Type	Category
Objects	Smooth manifolds, differentiable manifolds, topological manifolds
Morphisms	Smooth maps, diffeomorphisms, embeddings, immersions

Category of manifolds

The Category of manifolds is the categorical framework in which objects are manifolds and morphisms are structure-preserving maps; it situates manifold theory alongside categorical constructions used in modern mathematics. It connects classical figures and institutions in differential topology and geometry such as Henri Poincaré, Élie Cartan, John Milnor, Mikhail Gromov, René Thom, and research centers like the Princeton University, Institut des Hautes Études Scientifiques, and Clay Mathematics Institute. This categorical viewpoint interacts with foundational works including Differential Topology, Topology of Manifolds, Foundations of Differential Geometry, Morse Theory, and Hodge Theory.

Definition and basic properties

In the standard formulation the objects are smooth n-dimensional manifolds modeled on Euclidean spaces used in Bernhard Riemann's and Carl Friedrich Gauss's developments, and morphisms are smooth maps compatible with atlases as in the work of Andrey Kolmogorov and Andrey Nikolaevich; typical properties include local Euclidean charts, Hausdorff and second countable conditions influenced by conventions from Élie Cartan and Leray; these properties ensure compatibility with constructions in Algebraic Topology, Differential Geometry, Symplectic Geometry, Gauge Theory, and with invariants studied by Atiyah–Singer index theorem contributors like Michael Atiyah and Isadore Singer. The category admits identity morphisms given by identity diffeomorphisms studied by William Thurston and composition of smooth maps following standards in texts by Shlomo Sternberg and John Lee. Manifold categories are often enriched over categories used by Alexander Grothendieck and interact with toposes considered by William Lawvere and André Joyal.

Examples and common subcategories

Prominent objects include spheres and projective spaces central to the work of Henri Poincaré and Lefschetz such as the n-sphere, real projective space, complex projective space, and tori appearing in studies by Chern and Weil; Lie groups like SO(n), SU(n), GL(n,R), and Sp(n) form subcategories linking to representation theory in research by Harish-Chandra and Élie Cartan. Other subcategories include compact manifolds as in the classifications pursued by William Thurston and Michael Freedman, oriented manifolds used in works by Andrew Wiles and Simon Donaldson, spin manifolds central to contributions by Edward Witten and Nigel Hitchin, and symplectic manifolds foundational to Alan Weinstein and Maxim Kontsevich; singular strata link to stratified spaces examined by Goresky–MacPherson and Mark Goresky. Low-dimensional subcategories involve 1-, 2-, and 3-manifolds studied by Carl Gauss, Bernhard Riemann, William Thurston, Grigori Perelman, and Henri Poincaré, while high-dimensional techniques bring in surgery theory from C.T.C. Wall and classification results involving Kervaire–Milnor and Borel.

Morphisms and smooth maps

Morphisms in the category are smooth maps, immersions, embeddings, diffeomorphisms, and submersions used in analyses by René Thom and Stephen Smale. Important classes include proper maps considered in works by Jean Leray and Bott; transversality theorems of René Thom and Stephen Smale control genericity of morphisms and underpin intersection-theoretic constructions by Raoul Bott and Michael Atiyah. Embeddings relate to theorems by Whitney and extension results associated with Hassler Whitney's imbedding theorem; isotopies and concordances studied by William Browder and Barry Mazur characterize homotopy-theoretic behavior of morphisms. Diffeomorphism groups investigated by John Mather and Dennis Sullivan provide automorphism objects, while equivariant maps connect with symmetry groups like Weyl groups and representation-theoretic themes in the work of Harish-Chandra.

Functorial constructions and products

Functoriality appears via tangent and cotangent bundle functors central to Élie Cartan and Shlomo Sternberg; the tangent functor, cotangent functor, frame bundle, and jet bundle constructions link to concepts advanced by Jean-Pierre Serre and André Weil. Products give Cartesian products of manifolds underlying work by Élie Cartan and Hermann Weyl; Lie group products relate to structure theory from Élie Cartan and Sophus Lie. Functors to Top and to categories used by Alexander Grothendieck carry manifold objects into topological spaces and homotopy types studied by Daniel Quillen and J. Peter May. Other functorial constructions include mapping spaces featured in research by Graeme Segal and Michael Boardman, fiber bundles in the tradition of Steenrod and Charles Ehresmann, and classifying spaces connected to Quillen and Daniel Quillen's algebraic K-theory circle of ideas.

Categorical limits, colimits, and fiber products

Limits and colimits in the manifold category are subtle: finite products exist as Cartesian products studied by Hermann Weyl and Élie Cartan, but arbitrary limits may fail to be manifolds, an issue addressed by counterexamples from John Milnor and René Thom. Fiber products (pullbacks) exist under transversality assumptions from theorems by René Thom and Stephen Smale; pushouts often require gluing data and collaring theorems like those of R. C. Kirby and L.C. Glaser to yield manifolds. Quotients by group actions produce orbifolds and stacks studied by Ieke Moerdijk and Konstantin Tolmachov and formalized by Maxim Kontsevich and Pierre Deligne; these constructions connect to surgery and classification methods of C.T.C. Wall and invariants from Atiyah–Singer theory.

Relations to other categories (Top, Diff, Topological manifolds)

The Category of manifolds embeds into Top via the forgetful functor emphasized in classical texts by L.E.J. Brouwer and Poincaré; it relates to Diff as a full subcategory when morphisms are restricted to diffeomorphisms, echoing frameworks used by John Mather and Dennis Sullivan. Comparisons with the category of topological manifolds bring in results by Morris Kline and classification breakthroughs by Michael Freedman and Grigori Perelman; the passage from smooth to piecewise-linear categories intersects work by C. P. Rourke and B. J. Sanderson and the triangulation conjecture historically associated with Kirby–Siebenmann and Casson. Modern viewpoints involving stacks, Lie groupoids, and differentiable stacks tie to research centers such as Institut des Hautes Études Scientifiques and contributors like Ieke Moerdijk and André Henriques, linking manifold categories to broader categorical and homotopical paradigms.

Category:Manifolds