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Let 𝒞 be a category having finite products and a terminal...

Let $\mathcal{C}$ be a category having finite products and a terminal object $\Omega$. A group object in $\mathcal{C}$ is a quadruple $(G, \mu, \eta, \epsilon)$, where $G$ is an object in $\mathcal{C}, \mu: G \prod G \rightarrow G, \eta: G \rightarrow G$, and $\epsilon: \Omega \rightarrow G$ are morphisms, so that the following diagrams commute: Associativity: where $\omega: G \rightarrow \Omega$ is the unique morphism to the terminal object. (i) Prove that a group object in Sets is a group. (ii) Prove that a group object in Groups is an abelian group. (iii) Define a morphism between group objects in a category $\mathcal{C}$, and prove that all the group objects form a subcategory of $\mathcal{C}$. (iv) Define the dual notion cogroup object, and prove the dual of (iii).

Let $\mathcal{C}$ be a category having finite products and a terminal object $\Omega$. A group object in $\mathcal{C}$ is a quadruple $(G, \mu, \eta, \epsilon)$, where $G$ is an object in $\mathcal{C}, \mu: G \prod G \rightarrow G, \eta: G \rightarrow G$, and $\epsilon: \Omega \rightarrow G$ are morphisms, so that the following diagrams commute:
Associativity:
where $\omega: G \rightarrow \Omega$ is the unique morphism to the terminal object.
(i) Prove that a group object in Sets is a group.
(ii) Prove that a group object in Groups is an abelian group.
(iii) Define a morphism between group objects in a category $\mathcal{C}$, and prove that all the group objects form a subcategory of $\mathcal{C}$.
(iv) Define the dual notion cogroup object, and prove the dual of (iii).

Key Concepts

Category Theory

Category theory provides a framework for abstracting mathematical concepts through objects and morphisms. It emphasizes the relationships and mappings between objects rather than their internal structure, allowing for a uniform treatment of diverse mathematical structures. Fundamental notions such as limits, colimits, products, and subcategories emerge naturally within this framework.

Group Object

A group object in a category is an abstraction of the group concept that is defined by equipping an object with morphisms corresponding to the group operations. These operations include a multiplication map, an identity (or unit) map, and an inverse map. The axioms of a group—associativity, identity, and inverses—are expressed through commutative diagrams, allowing the concept of a group to be internalized in any category with the necessary limits.

Terminal Object

A terminal object in a category is one that has a unique morphism coming from any other object in the category. Its uniqueness property makes it a convenient tool for defining certain structures, such as the identity element of a group object, since this element can be seen as arising from a morphism from the terminal object.

Finite Products

Finite products in a category generalize the Cartesian product of sets. They provide a mechanism for combining objects in a way that mimics the Cartesian product's universal property. The existence of finite products is crucial for defining the multiplication morphism of a group object, as it requires forming a product of the object with itself.

Diagram Commutativity

Commutative diagrams are used in category theory to express equalities between composed morphisms. In the context of group objects, these diagrams capture the group axioms by ensuring that different ways of composing the structural morphisms yield the same result, hence formalizing the requirements of associativity, the existence of an identity element, and the existence of inverses.

Morphism of Group Objects

A morphism between group objects is an arrow in the category that preserves the group structure. This means that the morphism must commute with the multiplication, identity, and inverse maps of the group objects. Such structure-preserving maps are analogous to group homomorphisms in the traditional algebraic sense.

Subcategory

A subcategory is a collection of objects and morphisms within a larger category that itself forms a category. In the case of group objects, showing that morphisms between them form a subcategory involves verifying that the composition of structure-preserving maps remains structure-preserving and that identity morphisms exist and behave appropriately.

Duality and Cogroup Objects

Duality in category theory involves reversing the arrows of the definitions and proofs, leading to the concept of a cogroup object. A cogroup object is defined similarly to a group object but with the roles of products and coproducts reversed, and the diagrams are dualized. This duality reflects the fundamental symmetry in category theory between limiting and colimiting constructions.

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