Set of All Morphisms in a Category

The set of all morphisms $ \text{mor}(\mathcal{C}) $ contains every morphism between any two objects within the category $ \mathcal{C} $.

Formally, this collection of morphisms is defined as:

\[ \text{mor}(\mathcal{C}) = \bigcup_{(A, B) \in \text{Ob}(\mathcal{C}) \times \text{Ob}(\mathcal{C})} \text{Hom}(A, B) \]

Here, $ \text{Ob}(\mathcal{C}) $ denotes the set of all objects in $ \mathcal{C} $, while $ \text{Hom}(A, B) $ comprises all morphisms from $ A $ to $ B $.

This set $ \text{mor}(\mathcal{C}) $ captures every conceivable morphism within the category, including both endomorphisms and identities specific to each object.

Identity Morphisms
Each object $ A $ within $ \mathcal{C} $ is associated with an identity morphism $ \text{id}_A $ located in $ \text{Hom}(A, A) $, which is included in $ \text{mor}(\mathcal{C}) $.
Composition of Morphisms
The set $ \text{mor}(\mathcal{C}) $ also encompasses morphisms that can be sequentially combined, adhering to the category’s rules for composition, and ensuring associativity and identity are maintained.

Depending on the category’s scale, the collection of all morphisms, Mor(C), can either be a set or a proper class:

Set: If the category qualifies as "small", meaning its collection of objects and morphisms can be encapsulated in a single set by the rules of set theory, then Mor(C) forms a set.
Proper Class: In "large" categories, where the sheer number of objects or morphisms precludes them from fitting into a single set (such as the category of all sets, where each set is an object and any conceivable function between sets is a morphism), Mor(C) constitutes a proper class.

In instances where both the morphism class Mor(C) and the object class Obj(C) are sets, the category is described as a "small category".

The architecture of Mor(C) plays a pivotal role in the analysis of the category. It offers a holistic framework for exploring relationships between objects, understanding the category’s structural properties, and delving into internal dynamics like symmetries and transformations. Furthermore, it allows mathematicians to apply these category properties to address challenges across various fields.

Example

Let's consider a category $ \mathcal{C} $ with three objects $ A $, $ B $, and $ C $, along with three morphisms $ f $, $ g $, and $ h $ that facilitate intriguing compositions.

Each morphism uniquely connects different objects, enabling significant compositional possibilities.

The category includes:

$$ Obj(C) = \{A, B, C \} $$

It features three morphisms:

$ f: A \to B $ - A morphism connecting $ A $ to $ B $.
$ g: B \to C $ - A morphism linking $ B $ to $ C $.
$ h: A \to C $ - A direct morphism from $ A $ to $ C $.

These morphisms allow for the creation of a new composition of morphisms:

$ g \circ f: A \to C $

This composition effectively maps $ A $ directly to $ C $ via $ B $.

In addition, each object in the category is equipped with an identity morphism:

$ \text{id}_A: A \to A $
$ \text{id }_B: B \to B $
$ \text{id}_C: C \to C $

Taking all these elements into account, the set of all morphisms in $ \mathcal{C} $ includes:

The direct morphisms $ f $, $ g $, and $ h $.
A composed morphism $ g \circ f $.
The identity morphisms $ \text{id}_A $, $ \text{id}_B $, and $ \text{id}_C $.

Thus, $ \text{mor}(\mathcal{C}) $ is comprised of the following elements:

$$ \text{mor}(\mathcal{C}) = \{ f, g, h, g \circ f, \text{id}_A, \text{id}_B, \text{id}_C \} $$

This example constructs a straightforward category that vividly illustrates the composition of the set of all morphisms within a category.