Definition. Cat [leinster2016basic, 3.2.10] [tt-003G]
✍️source

We denote by \(\mathbf {Cat}\) the category of small categories and functors between them.

References

Reference. Basic category theory [leinster2016basic]

2016
Tom Leinster
https://arxiv.org/abs/1612.09375

@article{leinster2016basic,
 title = {Basic category theory},
 author = {Leinster, Tom},
 year = {2016},
 url = {https://arxiv.org/abs/1612.09375},
 journal = {arXiv preprint arXiv:1612.09375}
}

Context

Special objects and categories [tt-000G]

Definition. initial, terminal and null objects [kostecki2011introduction, 2.6] [tt-000I]

An initial object in a category \({\cal C}\) is an object \(\mathrm {0} \in \operatorname {Ob}({\cal C})\) such that for any object \(X\) in \({\cal C}\), there exists a unique arrow \(\mathrm {0} \to X\). It's also called a universal object, or a free object.

A terminal object in a category C is an object \(\mathrm {1} \in \operatorname {Ob}({\cal C})\) such that for any object \(X\) in \({\cal C}\), there exists a unique arrow \(X \to \mathrm {1}\). It's also called a final object, or a bound object.

Diagramatically,

$\text{[math]}$

A null object is an object which is both terminal and initial, confusingly, it's also called a zero object.

Lemma. Uniqueness [kostecki2011introduction, 2.7] [tt-000K]

All initial objects in a category are isomorphic.

All terminal objects in a category are isomorphic.

In other words, they are unique up to isomorphism, respectively.

Definition. element [kostecki2011introduction, 2.8, 2.9] [tt-000M]

Let \(X, S \in \operatorname {Ob}({\cal C})\).

An element or a generalized element of \(X\) at stage \(S\) (or, of shape \(S\)) is an arrow \(x : S \to X\) in \({\cal C}\), also denoted \(x \in _{S} X\).

An arrow \(1 \to X\) is called a global element of \(X\), a.k.a. a point of \(X\).

An arrow \(S \to X\), if \(S\) is not isomorphic to \(1\), is called the local element of \(X\) at stage \(S\).

An arrow \(\mathit {1}_X : X \to X\) is called the generic element of \(X\).

Remark. element [kostecki2011introduction, 2.8] [tt-002W]

In an element \(x : S \to X\), the object \(S\) is called a stage in order to express the intuition that it is a "place of view" on \(X\). In the same sense, \(S\) is also called a domain of variation, and \(X\) a variable element.

Sometimes, the term shape is used instead [leinster2016basic, 4.1.25], intuitive examples are:

when the object is a set, a generalized element of \(X\) of shape \(\mathbb N\) is a sequence in the set \(X\)
when the object is a topological space, a generalized element of \(X\) of shape \(S^1\) is a loop

In the context of studying solutions to polynomial equations, we may also call it a \(S\)-valued point in \(X\), where \(S\) is the number set where the solution is taken, e.g. the real, complex, and \(\operatorname {Spec} \mathbb F_p\)-valued solutions.

Definition. equivalent, equivalence class [kostecki2011introduction, 2.10] [tt-000N]

Two monic arrows \(x\) and \(y\) which satisfy

$\text{[math]}$ are called equivalent, which is denoted as \(x \sim y\).

The equivalence class of \(x\) is denoted as \([x]\), i.e., \([x]=\{y \mid x \sim y\}\).

Definition. subobject, Sub [kostecki2011introduction, 2.10] [tt-004J]

A subobject of any object is defined as an equivalence class of monic arrows into it.

The class of subobjects of an object \(X\) is denoted as \[ \operatorname {Sub}(X):=\{[f] \mid \operatorname {cod}(f)=X \wedge f \text { is monic }\}. \]

Definition. Set [kostecki2011introduction, 1.1, example 1] [tt-0008]

\(\mathbf {Set}\), the category of sets, consists of objects which are sets, and arrows which are functions between them. The axioms of composition, associativity and identity hold due to standard properties of sets and functions.

\(\mathbf {Set}\) has the initial object \(\varnothing \), the empty set, and the terminal object, \(\{*\}\), the singleton set.

\(\mathbf {Set}\) doesn't have a null object.

Monic arrows in \(\mathbf {Set}\) are denoted by \(f: X \hookrightarrow Y\), interpreted as an inclusion map (see also inclusion function in nLab).

Given \(X : \mathbf {Set}\), the subobjects of \(X\) are in canonical one-to-one correspondence with the subsets of \(X\).

Notation. inclusion [leinster2016basic, 0.8] [tt-000T]

In

$\text{[math]}$ the symbol \(\hookrightarrow \) is used for inclusions. It is a combination of a subset symbol \(\subset \) and an arrow.

Definition. Cat [leinster2016basic, 3.2.10] [tt-003G]
✍️source

We denote by \(\mathbf {Cat}\) the category of small categories and functors between them.

Definition. discrete category [leinster2016basic, 1.1.8] [tt-003K]

A discrete category has no arrows apart from the identity arrow, i.e. it amounts to just a class of objects.

We can regard a set as a discrete category.

Definition. terminal category [leinster2016basic, 4.1.6] [tt-003F]

A terminal category, denoted \(\mathbf {1}\), has only one object, denoted \(\mathrm {*}\), and only the identity arrow, denoted \(\mathit {1}\).

$\text{[math]}$

Definition. opposite category [kostecki2011introduction, 2.5] [tt-000H]

A category is called the opposite category of \({\cal C}\), denoted \(C^{op}\), iff

(reversion of arrows) \[\operatorname {Ob}({\cal C}^{op}) = \operatorname {Ob}({\cal C})\] \[\operatorname {Arr}({\cal C}^{op}) \ni f: Y \to X \Longleftrightarrow \operatorname {Arr}({\cal C}) \ni f: X \to Y\]
(reversion of composition) $\text{[math]}$

Definition. product category [leinster2016basic, 1.1.11] [tt-0048]

Given categories \({\cal C}\) and \({\cal D}\), there is a product category, denoted \({\cal C} \times {\cal D}\), in which

an object is a pair \((X, Y)\)
an arrow \((X, Y) \to \left (X', Y'\right )\) is a pair \((f, g)\)
the composition is given by \[(f_1, g_1) \mathbin {\bullet } (f_2, g_2) = (f_1 \mathbin {\bullet } f_2, g_1 \mathbin {\bullet } g_2)\]
the identity on \((X, Y)\), denoted \(\mathit {1}_{(X, Y)}\) is \((\mathit {1}_X, \mathit {1}_Y)\)

where \(X \in {\cal C}\), \(Y \in {\cal D}\), \(f: X \to X' \in {\cal C}\), and \(g: Y \to Y' \in {\cal D}\).

Definition. (full) subcategory [leinster2016basic, 1.2.18] [tt-002Y]

Let \({\cal C}\) be a category. A subcategory \({\cal S}\) of \({\cal C}\) consists of a subclass \(\operatorname {Ob}({\cal S})\) of \(\operatorname {Ob}({\cal C})\) together with, for each \(S, S' \in \operatorname {Ob}({\cal S})\), a subclass \({\cal S}\left (S, S'\right )\) of \({\cal C}\left (S, S'\right )\), such that \({\cal S}\) is closed under composition and identities.

It is a full subcategory if \({\cal S}\left (S, S'\right )={\cal C}\left (S, S'\right )\) for all \(S, S' \in \operatorname {Ob}({\cal S})\).

Backlinks

Definition. Ob, Arr [leinster2016basic, 4.1.6] [tt-003U]

Given a small category \({\cal C}\), there is a functor \(\operatorname {Ob} : \mathbf {Cat} \to \mathbf {Set}\) that sends \({\cal C}\) to its set of objects where \(\mathbf {Cat}\) is the category of small categories. Thus, \[ \mathscr {H}^{\mathrm {1}}({\cal C}) \cong \operatorname {Ob}({\cal C}) \] where \(\mathscr {H}\) is a Yoneda embedding functor.

This isomorphism is natural in \({\cal C}\); hence \(\operatorname {Ob} \cong \mathbf {Cat}(\mathrm {1}, -)\) where \(\mathbf {Cat}(\mathrm {1}, -)\) is a covariant hom-functor.

Functor \(\operatorname {Ob}\) is representable. Similarly, the functor \(\operatorname {Arr} : \mathbf {Cat} \to \mathbf {Set}\) sending a small category to its set of arrows is representable.

Related

Definition. (locally) small, hom-set [kostecki2011introduction, 1.3] [tt-000A]

A category C is called locally small iff for any of its objects \(X, Y\) , the collection of arrows from \(X\) to \(Y\) is a set, called a hom-set, denoted \[\operatorname {Hom}_{{\cal C}}(X, Y)\]

A category is called small iff the collection of all its arrows is a set.