Reference. 2-Dimensional Categories [johnson-yau-2d-categories-2020]

June 17, 2020
Johnson, Niles, Yau, Donald
https://arxiv.org/abs/2002.06055v3

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Definition. Isomorphism (morphisms) [002s]

November 29, 2025

A morphism \(f : X \to Y\) is called an Isomorphism if there exists a morphism \(g : Y \to X\) such that the following hold (1):

\[ g \circ f = 1_X \quad f \circ g = 1_Y \]

Sometimes an isomorphism is also denoted

\[ X \xrightarrow {\cong } Y \]

Definition. Category [002m]

November 27, 2025

There are a few ways you can define a category. In the most basic intuitive sense a category consists of a collection of things called objects and binary relationships (or transitions) between those objects called morphisms (or arrows). We can combine these relationships by composing them, and for each object there is an identity morphism that acts as a neutral element for composition. (1)

In the context of quivers a category can be defined as a quiver with a rule saying for how we can compose two edges that fit together to get a new edge. Furthermore, each vertex (object) has an edge starting and ending at that vertex (the identity morphism). The classical definition is something like this:

The data

November 27, 2025

A category \(C\) consists of:

A collection (or class (2)) of objects, denoted as \(\text {Ob}(C)\) or \(C_0\).
A collection (or set (2)) of morphisms (or arrows), denoted \(C_1\) or \(C(x, y)\) for \(x, y \in \text {Ob}(C)\).
- For every morphism \(f \in C(x, y)\), there are two associated objects: the source (or domain) \(x\) and the target (or co-domain) \(y\). In standard function notation, we write \(f: x \to y\) where \(x = \text {dom}(f)\) and \(y = \text {cod}(f)\). NLab has a nice convention where it denotes the source \(s\) of a morphism as \(s(f)\) and the target \(t\) as \(t(f)\).
- For every pair of morphisms \(f \in C(x, y)\) and \(g \in C(y, z)\) (s.t. \(t(f) = s(g)\) i.e. the morphisms type check), there is a composition morphism \(g \circ f \in C(x, z)\). Written out we can denote this as: \[ C(x, y) \times C(y, z) \to C(x, z) \] in diagrammatic order this is often written as \(f; g\) we can equivalently use a more graphical notation:
- For every object \(x \in \text {Ob}\) there is an identity morphism: \[ (\text {id}_x : x \to x) \in C(x, x) \]
Note: Some additional notations for morphisms include \(\text {hom}(x, y)\), \(\text {hom}_C(x, y)\) or \(C_1(x, y)\). Additionally, people use the notation \(\text {Mor(C)}\) to denote the following disjoint union \[ \text {Mor}(C) = \bigsqcup _{x, y \in \text {Ob}(C)} C(x, y) \] Which just expresses the idea that the collection of all morphisms in a category is made up of the morphisms between each pair of objects.

The axioms

November 27, 2025

The above are often called data of a category. In addition to this data, a category must satisfy the following axioms or (conditions):

Morphisms need to be associative which means that for every triple of morphisms \(f \in C(w, x)\), \(g \in C(x, y)\), and \(h \in C(y, z)\) the following holds: \[ h \circ (g \circ f) = (h \circ g) \circ f \]
For each morphism \(f \in C(x, y)\) the identity morphisms act as neutral elements for composition: \[ \text {id}_y \circ f = f = f \circ \text {id}_x \] This is also known as the left and right unit laws or just unity in general.

Remarks

November 27, 2025

A category such as the one described above is often also called a 1-category to distinguish it from higher categories such as 2-categories, n-categories.

Definition. Grothendieck universe [002q]

November 27, 2025

Motivation

November 27, 2025

In set theory (along with other theories which consider collections of stuff) there is an issue which you try to talk about the collection of collections, as it gives rise to certain paradoxes by the recursive nature of it. One way out is to consider a hierarchy of notions of collections, where a "collection of collections" is not in fact considered a "collection" itself but rather some notion of a higher increment of collection.

This is the idea behind universes, where the universe is a collection of all set's which intuitively can be thought of as a "set of all sets" but is not itself a set but one increment above it in the hierarchy.

Definition

November 27, 2025

A Grothendieck universe is a set (in ZFC) \(\mathcal U\) with the following properties (1):

(transitivity) If \(x \in \mathcal U\) and \(y \in x\), then \(y \in \mathcal U\).
(power set) If \(x \in \mathcal U\), then \(\mathcal P(x) \in \mathcal U\) where \(\mathcal P(x)\) is the powerset (set of all subsets) of \(x\).
(unions) If \(I \in \mathcal U\) and \(x_i \in \mathcal U\) for each \(i \in I\), then the union \[ \bigcup _{i \in I} x_i \in \mathcal U \]
(natural numbers) The set of natural numbers \(\mathbb {N} \in \mathcal U\).

Additional properties of \(\mathcal U\) can be enumerated as follows:

If \(x, y \in \mathcal U\) then \(\{x, y\} \in \mathcal U\).
If \(x \in \mathcal U\) and \(y \subset x\), then \(y \in \mathcal U\).
If \(I \in \mathcal U\) and \(x_i \in \mathcal U\) for each \(i \in I\), we have the cartesian product: \[ \prod _{i \in I} x_i \in \mathcal U \]
If \(I \in \mathcal U\) and \(x_i \in \mathcal U\) for each \(i \in I\), we have the disjoint union: \[\coprod _{i \in I} x_i \in \mathcal U \]
If \(x, y \in \mathcal U\), then the function set from \(x\) to \(y\) is also in \(\mathcal U\): \[ y^x = \{ f : x \to y \} \in \mathcal U \]

An aside on classes and sets

November 27, 2025

In a very general sense a class is any collection of sets defined by a property that all its members share. So we can express it it as a predicate on some universe \(\mathcal U\):

\[ A : \mathcal U \to \texttt {Prop} \]

As an example in the case of an unsorted set theory (such as ZFC) in which all objects are sets, we define a class \(C\) as a proposition or truth value in the context of some free variable \(A\), the notation used in (5) is:

\[ \Gamma , A \mid \phi \vdash C : \texttt {Prop} \]

This expresses that in the context of assumptions \(\Gamma \) extended with the variable \(A\) the proposition \(\phi \) defines a class \(C\). A more classical, albiet probably less formal, way to read this might be:

\[ C = \{ x \in \mathcal U \mid \phi (x)\} \]

Furthermore we say that a class \(X\) is a set if and only if there exists a class \(Y \in \mathcal U\) such that \(X \in Y\). A proper class is a class which is not a set (6).

Axiom of Universes

November 27, 2025

The common theme amongst these properties is that the aim to express closure of the set \(\mathcal U\) under various standard set-theoretic operations. This allows us to treat \(\mathcal U\) as a universe of sets in which we can carry out normal set-theoretic constructions without leaving the universe.

The Axiom of Universes is the general convention which is adopted in category theory and other fields which states that for every set \(x\) there exists a Grothendieck universe \(\mathcal U\) such that \(x \in \mathcal U\).

An element of \(\mathcal U\) is by convention called a \(\mathcal U\)-small set or simply a small set when the universe is clear from context or apparently also just a set (1).

A subset of \(\mathcal U\) is called \(\mathcal U\)-moderate or simply moderate when the universe is clear from context.

The general line of reasoning, at least in ZFC is that for every set \(x\) we assume there exists a Grothendieck universe \(\mathcal U\) such that \(x \in \mathcal U\). Accordinly for every universe \(\mathcal U\) there exists a larger universe \(\mathcal U'\) such that \(\mathcal U \in \mathcal U'\), and therefore also \(U \subseteq U'\). Thus under this assumption with have the guarantee of a hierarchy of universes (4):

\[ \mathcal U_0 \in \mathcal U_1 \in \mathcal U_2 \in \cdots \]

Which are also cumulative in the sense that

\[ \mathcal U_0 \subseteq \mathcal U_1 \subseteq \mathcal U_2 \subseteq \cdots \]

Remarks

November 27, 2025

An additional point made in (2) is that \(\mathcal U\) denotes a pure set, that is, all elements of \(\mathcal U\) are sets and there are no urelements (elements which are not sets). Formally we say that a set \(x\) is pure if for every finite sequence or chain (3):
\[ x_0 = x \in x_1 \in x_2 \in \cdots \in x_n \]
each \(x_i\) is a set. Another way this is intuitively thought of is that every object reachable from \(x\) by finitely many membership steps is also a set.

Since we are typically working in the context of ZFC set theory which only allows for sets as elements, all sets are pure by default hence the distinction is sometimes omitted. Theories such as ZFA (Zermelo-Fraenkel set theory with atoms) allow for urelements which do not qualify as sets but can be members of sets.
The countable set property is sometimes (7) also replaces with
\[ \emptyset \in \mathcal U \quad \text { or } \quad \omega \in \mathcal U \]
Where \(\omega \) is the set of all finite (von Neumann) ordinals.
Subset's of \(\mathcal U\) seem to have a somewhat confusing naming convention. So nlab says they are called moderate though this is sometimes also overloaded with the ambiguous term large. They mention that under the assumption that set is equivalent to small set then moderate clearly aligns with proper classes (as those are definitionally distinct from sets). I assume the convention taken in at least (1) is that they use they essentially use the term class to mean proper class and by implication moderate set relative to some universe.

My assumption is that people use class and proper class interchangeably mainly because a proper class actually refers to an object distinct from a set, whereas a class can make it ambiguous whether you mean a set (which is a class) or a proper class (which is not a set).