11.1 Definition and examples of monoidal categories
3 min read•july 23, 2024
categories extend regular categories by adding a and . These new elements allow us to combine objects and morphisms in interesting ways, creating richer structures.
The tensor product lets us multiply objects, while the unit object acts like a neutral element. This setup mirrors familiar mathematical structures, making monoidal categories a powerful tool for understanding various algebraic systems.
Monoidal Categories
Components of monoidal categories
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Objects the objects of the underlying category C
Morphisms the morphisms of the underlying category C
Tensor product a bifunctor ⊗:C×C→C assigns to each pair of objects (A,B) an object A⊗B
Also defined on morphisms for f:A→A′ and g:B→B′, there is a f⊗g:A⊗B→A′⊗B′
Unit object an object I∈C serves as an identity for the tensor product
Coherence conditions the tensor product and unit object satisfy
a natural isomorphism αA,B,C:(A⊗B)⊗C→A⊗(B⊗C) for all objects A,B,C
Left unit a natural isomorphism λA:I⊗A→A for all objects A
Right unit a natural isomorphism ρA:A⊗I→A for all objects A
Examples of monoidal categories
Category of sets Set with the Cartesian product × as the tensor product and the singleton set {∗} as the unit object
For sets A and B, the Cartesian product A×B is the set of ordered pairs (a,b) with a∈A and b∈B
The singleton set {∗} satisfies A×{∗}≅A≅{∗}×A for all sets A
Category of vector spaces [Vect](https://www.fiveableKeyTerm:vect)K over a field K with the tensor product ⊗K and the field K as the unit object
For vector spaces V and W, the tensor product V⊗KW is the vector space generated by elements of the form v⊗w with v∈V and w∈W, subject to certain relations
The field K satisfies V⊗KK≅V≅K⊗KV for all vector spaces V
Constraints in monoidal categories
Associativity constraint the natural isomorphism αA,B,C:(A⊗B)⊗C→A⊗(B⊗C) ensures the tensor product is associative up to isomorphism
Allows for unambiguous expressions like A⊗B⊗C without specifying parentheses
Satisfies the pentagon identity, which ensures coherence when reassociating tensor products of four objects
Unit constraints the natural isomorphisms λA:I⊗A→A and ρA:A⊗I→A ensure the unit object I behaves as an identity for the tensor product
Allows for expressions like I⊗A and A⊗I to be identified with A
Satisfy the triangle identity, which ensures coherence when combining the associativity and unit constraints
Interactions in monoidal categories
Tensor product is a bifunctor compatible with composition of morphisms
For morphisms f:A→A′, g:B→B′, h:A′→A′′, and k:B′→B′′, we have (h⊗k)∘(f⊗g)=(h∘f)⊗(k∘g)
Unit object I interacts with morphisms
For any morphism f:A→B, there are morphisms f⊗idI:A⊗I→B⊗I and idI⊗f:I⊗A→I⊗B
These morphisms are compatible with the unit constraints ρB∘(f⊗idI)=f∘ρA and λB∘(idI⊗f)=f∘λA
Key Terms to Review (19)
Associativity: Associativity is a property of certain operations that states that the grouping of operations does not affect the final result. In the context of category theory, this property is crucial for understanding how morphisms can be composed without ambiguity, leading to a consistent framework for manipulating objects and morphisms within categories.
Braiding Theorem: The Braiding Theorem is a result in category theory that provides conditions under which two objects in a monoidal category can be interchanged or 'braided' without affecting the overall structure. This theorem plays a crucial role in understanding how morphisms and tensor products interact in monoidal categories, revealing deeper symmetries and relationships between objects.
Cat: In category theory, a 'cat' refers to a category, which is a mathematical structure consisting of objects and morphisms (arrows) that define relationships between those objects. Categories can be used to abstractly represent various mathematical concepts and structures, enabling mathematicians to study their properties and relationships in a unified way. A category must satisfy specific axioms such as the existence of identity morphisms and the associativity of composition.
Categorical: In mathematics, particularly in category theory, 'categorical' refers to properties or structures that can be defined and understood in the context of categories. This concept emphasizes the relationships and mappings between objects rather than focusing solely on the objects themselves. By looking at how objects interact through morphisms, we can gain insights into their nature and the operations defined within a category, making it a crucial aspect when studying more complex structures like monoidal categories.
Category of modules: The category of modules is a mathematical structure where objects are modules over a ring and morphisms are module homomorphisms. This framework allows for the study of algebraic structures in a categorical context, highlighting relationships and transformations between different modules. It plays a crucial role in various areas of mathematics, including representation theory and homological algebra.
Coherence Theorem: The coherence theorem is a fundamental result in category theory that asserts the uniqueness of morphisms between objects when certain conditions are met, particularly in the context of monoidal categories and symmetric monoidal categories. It provides a way to demonstrate that different diagrams or constructions that appear to define the same morphism or natural transformation are indeed equivalent. This theorem plays a crucial role in ensuring that the operations and structures defined in these categories behave consistently and predictably.
Enriched Category: An enriched category is a generalization of the concept of a category, where the hom-sets between objects are replaced by objects in a suitable monoidal category. This means that instead of having just sets of morphisms, we have a structure that allows us to measure the relationships between objects in a more nuanced way. The hom-objects can provide additional information and properties, enriching our understanding of the relationships and interactions within the category.
Left Unitor: A left unitor is a morphism in a monoidal category that acts as an identity for the tensor product when paired with the unit object. It provides a way to 'unit' elements of the category with respect to the tensor product, ensuring that the structure behaves well when combining objects. This morphism is essential for establishing the coherence conditions that define monoidal categories.
Monoidal: A monoidal category is a type of category equipped with a tensor product that allows for the combination of objects and morphisms in a coherent way. This structure includes an identity object, which acts as a neutral element for the tensor product, and the associativity and unitality constraints that ensure the coherence of this combination. Monoidal categories are important because they provide a framework for understanding various algebraic structures, including vector spaces and categories of modules.
Monoidal Functor: A monoidal functor is a structure-preserving map between two monoidal categories that respects the tensor product and the unit object. It provides a way to relate different categories with their own tensor operations while ensuring that the essential properties of these operations are maintained. This connection is crucial for understanding how various mathematical structures interact and can be transformed into one another, and it plays a significant role in establishing coherence conditions across different categories.
Morphism: A morphism is a structure-preserving map between two objects in a category, reflecting the relationships between those objects. Morphisms can represent functions, arrows, or transformations that connect different mathematical structures, serving as a foundational concept in category theory that emphasizes relationships rather than individual elements.
Natural transformation: A natural transformation is a way of transforming one functor into another while preserving the structure of the categories involved. It consists of a family of morphisms that connect the objects in one category to their images in another category, ensuring that the relationships between the objects are maintained across different mappings. This concept ties together various important aspects of category theory, allowing mathematicians to relate different structures in a coherent manner.
Right unitor: In a monoidal category, a right unitor is a specific morphism that acts as a unit for the tensor product on the right side. It connects objects through an isomorphism that demonstrates how the tensor product of an object with the unit object behaves like the original object. This concept highlights the structural role of units in monoidal categories, essential for understanding how objects interact under the tensor operation.
Strict monoidal category: A strict monoidal category is a type of monoidal category where the tensor product is associative and the unit object behaves strictly as an identity for the tensor product. This means that the natural isomorphisms used to express associativity and identity are actually equalities, simplifying the structure of the category. In this framework, morphisms respect both the associativity of the tensor product and the unital properties without requiring any coherence conditions.
Strong monoidal functor: A strong monoidal functor is a type of functor between two monoidal categories that not only preserves the structure of the categories but also comes equipped with a way to handle the tensor products and unit objects consistently. It acts on objects and morphisms while maintaining coherence with the monoidal structure, essentially making it compatible with the tensor operation of both categories. This compatibility means that the functor respects the tensor product and the identity object in a way that aligns their behaviors in both source and target categories.
Symmetric monoidal category: A symmetric monoidal category is a special type of category equipped with a tensor product that allows for the combination of objects and morphisms, along with an identity object, all while satisfying certain coherence conditions. It extends the idea of monoidal categories by introducing a symmetry that allows for the interchange of objects in the tensor product without affecting the outcome, which is crucial for many applications in both mathematics and theoretical computer science.
Tensor Product: The tensor product is an operation that combines two objects from a category to produce a new object in a way that captures bilinear relationships. This operation is central to the structure of monoidal categories, where it allows for the composition of objects and morphisms while maintaining the coherence and associativity properties required by the category's structure.
Unit object: A unit object is a specific type of object in category theory that serves as an identity element for the tensor product in a monoidal category. It acts as a neutral element, meaning when it is combined with any other object through the tensor product, it returns that other object unchanged. This concept is crucial for understanding the structure and operations within monoidal categories, including their coherence and symmetry properties.
Vect: In category theory, 'vect' typically refers to the category of vector spaces over a field, with linear transformations as morphisms. This category is fundamental in understanding structures in mathematics where linearity is a key feature, allowing for connections to various concepts such as functors, adjunctions, and monoidal categories.