**Category of Manifolds**
**Definition**
In mathematics, the category of manifolds is a category whose objects are manifolds—topological spaces locally resembling Euclidean space—and whose morphisms are structure-preserving maps between these manifolds, such as smooth maps in the case of smooth manifolds. This categorical framework allows for the systematic study of manifolds and their relationships using the language and tools of category theory.
—
## Category of Manifolds
The concept of a *category of manifolds* arises naturally in differential geometry and topology, where manifolds serve as fundamental objects of study. By organizing manifolds and the maps between them into a category, mathematicians can apply categorical methods to analyze manifold structures, morphisms, and their compositions. This article explores the definition, examples, properties, and significance of categories of manifolds, focusing primarily on smooth manifolds but also touching on other manifold categories.
—
### 1. Introduction to Manifolds
A **manifold** is a topological space that locally resembles Euclidean space (mathbb{R}^n). More precisely, an (n)-dimensional manifold is a Hausdorff, second-countable topological space in which every point has a neighborhood homeomorphic to an open subset of (mathbb{R}^n). Manifolds serve as the setting for much of modern geometry and physics, providing a generalization of curves and surfaces to higher dimensions.
Manifolds can be endowed with additional structures, such as differentiable (smooth), complex, or Riemannian structures, which refine the notion of allowable maps and morphisms between them.
—
### 2. Categories in Mathematics
A **category** consists of a collection of objects and morphisms (arrows) between these objects, satisfying two axioms: composition of morphisms is associative, and each object has an identity morphism. Categories provide a unifying language to describe mathematical structures and their relationships.
In the context of manifolds, the objects are manifolds themselves, and the morphisms are maps preserving the relevant structure (e.g., continuous, smooth, or holomorphic maps).
—
### 3. The Category of Topological Manifolds
The most basic category of manifolds is the category of **topological manifolds**, denoted (mathbf{TopMan}).
– **Objects:** Topological manifolds of fixed or varying dimension.
– **Morphisms:** Continuous maps between these manifolds.
This category captures the topological aspects of manifolds but does not incorporate differentiable or other geometric structures.
**Properties:**
– Morphisms are continuous functions.
– Composition and identity morphisms are given by function composition and identity maps.
– The category is large, as it includes manifolds of all dimensions.
– It is not generally closed under limits and colimits in the category-theoretic sense, but certain constructions like products and coproducts exist.
—
### 4. The Category of Smooth Manifolds
The most studied and important category in differential geometry is the category of **smooth manifolds**, often denoted (mathbf{Diff}) or (mathbf{Man}).
– **Objects:** Smooth manifolds, i.e., manifolds equipped with a maximal smooth atlas.
– **Morphisms:** Smooth maps between manifolds, i.e., maps that pull back smooth functions to smooth functions.
This category reflects the differentiable structure of manifolds and is central to differential geometry, topology, and mathematical physics.
#### 4.1. Objects: Smooth Manifolds
A smooth manifold is a topological manifold with an atlas whose transition maps are infinitely differentiable. This structure allows for calculus on manifolds, including notions of tangent spaces, vector fields, differential forms, and more.
#### 4.2. Morphisms: Smooth Maps
A smooth map (f: M to N) between smooth manifolds is a continuous map such that for any charts ((U, varphi)) on (M) and ((V, psi)) on (N), the map (psi circ f circ varphi^{-1}) is smooth as a map between open subsets of Euclidean spaces.
#### 4.3. Properties of (mathbf{Diff})
– **Composition:** The composition of smooth maps is smooth.
– **Identity:** The identity map on any smooth manifold is smooth.
– **Limits and Colimits:** The category (mathbf{Diff}) has finite products (e.g., Cartesian products of manifolds) but lacks certain colimits, such as general quotients, unless additional structure is imposed.
– **Subcategories:** Important subcategories include the category of compact smooth manifolds, oriented manifolds, or manifolds with boundary.
—
### 5. Other Categories of Manifolds
Beyond topological and smooth manifolds, other categories arise by imposing additional structures or relaxing conditions.
#### 5.1. Category of Complex Manifolds
– **Objects:** Complex manifolds, i.e., manifolds modeled on (mathbb{C}^n) with holomorphic transition maps.
– **Morphisms:** Holomorphic maps.
This category is fundamental in complex geometry and several complex variables.
#### 5.2. Category of Riemannian Manifolds
– **Objects:** Smooth manifolds equipped with a Riemannian metric.
– **Morphisms:** Isometries or smooth maps preserving the metric structure.
This category is more restrictive and often considered in geometric analysis.
#### 5.3. Category of Manifolds with Boundary
– **Objects:** Manifolds that locally look like (mathbb{R}^n_+ = {x in mathbb{R}^n : x_n geq 0}).
– **Morphisms:** Smooth maps respecting the boundary structure.
—
### 6. Functors and Natural Transformations in Categories of Manifolds
The categorical viewpoint allows the use of functors and natural transformations to relate different categories of manifolds or to associate algebraic invariants to manifolds.
#### 6.1. Tangent Bundle Functor
The **tangent bundle** construction defines a functor (T: mathbf{Diff} to mathbf{Diff}), assigning to each smooth manifold (M) its tangent bundle (TM), and to each smooth map (f: M to N) its differential (Tf: TM to TN).
This functor preserves composition and identities, reflecting the differential structure categorically.
#### 6.2. Other Functors
– **Homology and Cohomology Functors:** Assign algebraic invariants to manifolds.
– **Frame Bundle Functor:** Assigns to each manifold its principal frame bundle.
—
### 7. Limits, Colimits, and Constructions in Categories of Manifolds
While categories of manifolds have some categorical limits and colimits, they are generally not complete or cocomplete.
#### 7.1. Products
The product of two manifolds (M) and (N) is their Cartesian product (M times N), which is again a manifold with the product smooth structure.
#### 7.2. Coproducts
Disjoint unions serve as coproducts in categories of manifolds.
#### 7.3. Quotients and Pushouts
Quotients by group actions or identifications often fail to be manifolds, so pushouts and general colimits may not exist within the category.
—
### 8. Enriched and Higher Categories of Manifolds
Recent developments in higher category theory and homotopy theory have led to enriched or higher categories of manifolds, where morphisms themselves have additional structure or where manifolds are considered up to homotopy equivalence.
—
### 9. Applications and Significance
The categorical approach to manifolds provides a unifying framework for:
– Studying manifold invariants via functors.
– Formalizing constructions such as fiber bundles, sheaves, and stacks.
– Connecting differential geometry with algebraic topology and algebraic geometry.
– Facilitating the use of abstract categorical tools in geometry and physics.
—
### 10. Summary
The category of manifolds, particularly the category of smooth manifolds, is a fundamental structure in modern mathematics. It organizes manifolds and smooth maps into a framework that supports the application of category theory to geometry and topology. While categories of manifolds have limitations regarding completeness and cocompleteness, they remain central to many areas of research and provide a language for expressing and analyzing manifold-related concepts systematically.
—
**Meta Description:**
The category of manifolds is a mathematical framework organizing manifolds and structure-preserving maps into a category, enabling the study of manifold properties through category theory. This article explores various categories of manifolds, including smooth, topological, and complex manifolds, and their categorical properties.