Category Theory

Expository Study Notes on Category Theory

Abstract

These notes follow a textbook-style roadmap through category theory, moving from concrete definitions to abstract machinery and onward to cross-domain applications. I open with the core anatomy of a category: objects, hom-sets, associative composition and identity arrows; emphasis is placed on locally-small categories and the dual-category trick that instantly doubles one’s supply of examples. Functors enter next as structure-preserving translations between categories, preparing the ground for natural transformations—“maps of functors” whose commutative-diagram discipline is illustrated with explicit calculations. The Yoneda Lemma is then derived step-by-step, showing how every fobject lives vicariously through its representable functor and why “proof by probing” becomes a legitimate strategy. With this embedding tool in hand I explore limits and colimits (products, equalizers, pull-/push-outs) and spell out their universal properties diagram-by-diagram; adjoint functors appear soon after as “optimal solution providers”, exemplified by free groups and the Stone–Čech compactification. A final chapter bridges to computer science: algebraic data types as algebras for endofunctors, concurrent processes as coalgebras, and how monadic composition captures side effects—culminating in the Beck-style monadicity theorem. Throughout, definitions, theorems and exercises alternate with motivational remarks, mirroring the rhythm of a lecture course and keeping the abstract landscape navigable.

Keywords

Natural Transformation, Functor, Adjoint Functor, Limits

1 Category Theory

1.1 Code Examples

1.1.1 Haskell Implementation

-- CategoryDemo.hs: Haskell中的函子实例示例
import Data.Functor (Functor(..))

-- 自定义数据类型模拟"树"结构
data Tree a = Leaf a | Node (Tree a) (Tree a) deriving (Show)

-- 为Tree实现Functor实例（函子）
instance Functor Tree where
    -- fmap: (a→b) → Tree a → Tree b（态射映射）
    fmap f (Leaf x) = Leaf (f x)
    fmap f (Node l r) = Node (fmap f l) (fmap f r)

-- 测试：函子保持态射复合
main :: IO ()
main = do
    let tree = Node (Leaf 1) (Node (Leaf 2) (Leaf 3))
        f = (+1)    -- 态射f: Int→Int
        g = (*2)    -- 态射g: Int→Int
    -- 验证 fmap (g . f) = fmap g . fmap f（函子性质）
    print $ fmap (g . f) tree  -- Node (Leaf 4) (Node (Leaf 6) (Leaf 8))
    print $ (fmap g . fmap f) tree  -- 与上一行结果相同

1.1.2 Python Implementation

# demo.py: 范畴学基础概念的Python模拟实现
from typing import Generic, TypeVar, Callable, List

# 类型变量用于模拟范畴中的对象类型
A = TypeVar('A')
B = TypeVar('B')
C = TypeVar('C')


class Morphism(Generic[A, B]):
    """模拟范畴中的态射（morphism）"""
    def __init__(self, dom: A, codom: B, func: Callable[[A], B]):
        self.dom = dom  # 定义域对象
        self.codom = codom  # 陪域对象
        self.func = func  # 态射对应的函数实现

    def __call__(self, x: A) -> B:
        return self.func(x)

    def __matmul__(self, other: 'Morphism[C, A]') -> 'Morphism[C, B]':
        """用 @ 运算符模拟态射复合 (g @ f 表示 g ∘ f)"""
        if other.codom != self.dom:
            raise ValueError("态射复合定义域不匹配")
        return Morphism(other.dom, self.codom, lambda x: self.func(other.func(x)))


class Category(Generic[A]):
    """模拟一个范畴（category）"""
    def __init__(self, objects: List[A], morphisms: List[Morphism]):
        self.objects = objects
        self.morphisms = morphisms

    def identity(self, obj: A) -> Morphism[A, A]:
        """获取对象上的单位态射"""
        return Morphism(obj, obj, lambda x: x)


class Functor(Generic[A, B]):
    """模拟函子（functor）：从范畴C到范畴D的映射"""
    def __init__(self, 
                 source: Category[A], 
                 target: Category[B],
                 map_obj: Callable[[A], B],
                 map_morph: Callable[[Morphism[A, A]], Morphism[B, B]]):
        self.source = source  # 源范畴
        self.target = target  # 目标范畴
        self.map_obj = map_obj  # 对象映射
        self.map_morph = map_morph  # 态射映射

    def __call__(self, obj: A) -> B:
        """函子作用于对象"""
        return self.map_obj(obj)

    def apply_morph(self, morph: Morphism[A, A]) -> Morphism[B, B]:
        """函子作用于态射"""
        return self.map_morph(morph)


# 示例：集合范畴（Set）的模拟
if __name__ == "__main__":
    # 1. 定义集合范畴中的对象（集合）
    set_objects = [int, str, list]  # 以Python类型模拟集合对象

    # 2. 定义态射（函数）
    # 态射f: int → str（整数转字符串）
    f = Morphism(int, str, lambda x: str(x))
    # 态射g: str → list（字符串转字符列表）
    g = Morphism(str, list, lambda s: list(s))
    # 复合态射g ∘ f: int → list
    h = g @ f

    # 3. 构建集合范畴
    set_category = Category(set_objects, [f, g, h])

    # 4. 定义列表函子（List Functor）：Set → Set
    # 对象映射：将集合A映射到列表集合List[A]
    def list_map_obj(a: type) -> type:
        return List[a]  # 用Python的List类型模拟

    # 态射映射：将函数f: A→B映射到f*: List[A]→List[B]（逐元素应用f）
    def list_map_morph(morph: Morphism[A, B]) -> Morphism[List[A], List[B]]:
        return Morphism(
            List[morph.dom], 
            List[morph.codom],
            lambda lst: [morph(x) for x in lst]
        )

    # 实例化列表函子
    list_functor = Functor(
        source=set_category,
        target=set_category,
        map_obj=list_map_obj,
        map_morph=list_map_morph
    )

    # 测试：函子作用于对象
    print("列表函子作用于int对象：", list_functor(int))  # 输出 List[int]

    # 测试：函子作用于态射f（int→str）
    mapped_f = list_functor.apply_morph(f)
    test_list = [1, 2, 3]
    print("函子作用于态射后的结果：", mapped_f(test_list))  # 输出 ['1', '2', '3']

1.2 Foundational Definitions & Theorems

1.2.1 Theorem: Hom-Functor Preserves Limits

Theorem 1 For any locally small category \(\mathcal{C}\) and any object \(A \in \mathcal{C}\), the functor \(\mathcal{C}(A, -)\) preserves limits.

See Thm. Theorem 1 (Lane 1998).

1.2.2 Lemma: Full & Faithful Functors Reflect Isomorphisms

Lemma 1 Every full and faithful functor reflects isomorphisms.

See Lem. Lemma 1 .

1.2.3 Definition: Adjunction

Definition 1 An adjunction consists of a pair of functors \(L \colon \mathcal{C} \rightleftarrows \mathcal{D} \colon R\) together with a natural isomorphism \(\mathcal{D}(LA, B) \cong \mathcal{C}(A, RB)\).

1.2.4 Corollary: Completeness of Functor Categories

Corollary 1 Any functor category \([\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]\) is complete.

See Cor. Corollary 1 .

1.2.5 Proposition: Left Adjoints Preserve Colimits

Left adjoints preserve colimits.

See Prop. (prop-reflect?) (Lane 1998).

1.3 Examples & Remarks

1.3.1 Example: Yoneda Embedding for Sets

The functor \(\mathbf{Set}(1, -)\) is naturally isomorphic to the identity.

See Ex. (ex-yoneda?) .

1.3.2 Remark: Non-Surjectivity of Yoneda Embedding

Remark 1. The Yoneda embedding is not surjective on objects in general.

See Rem. Remark 1.

1.4 Figures

See Fig. (fig:picture1?).

1.5 Introduction

1.5.1 Natural transformation

Let \(f \in L^1(X\times Y)\). Then \[ \int_X\!\int_Y f(x,y)\,dy\,dx = \int_Y\!\int_X f(x,y)\,dx\,dy. \]

Definition 2 A group is a set \(G\) with a binary operation \(\cdot\) satisfying: 1. Associativity: \((a\cdot b)\cdot c=a\cdot(b\cdot c)\); 2. Identity: \(\exists e\in G\) such that \(e\cdot a=a\cdot e=a\); 3. Inverses: \(\forall a\in G\ \exists a^{-1}\in G\) such that \(a\cdot a^{-1}=a^{-1}\cdot a=e\).

The set \(\mathbb Z\) with addition is a group.

Lemma 2 The product of two integers is even if at least one factor is even.

Theorem 2 There are infinitely many prime numbers.

Proof. Assume finitely many primes \(p_1,\dots,p_n\).
Consider \(P=p_1\cdots p_n+1\).
\(P>1\) has a prime divisor \(p\), but \(p\) divides neither \(P-1\) nor \(1\), contradiction.

Corollary 2 For every integer \(n\ge 2\) there exists a prime \(p>n\).

Remark 2. Euclid’s proof is constructive: it gives an upper bound on the next prime.

Definition (Index set \(\left(A_i\right)_{i \in I}\) ) \(A: I \rightarrow X, i \mapsto A_i\) Definition (Product over the index set \(\left(A_i\right)_{i \in I}\) ) \[ \prod_{i \in I} A_i:=\left\{a: I \rightarrow \bigcup_{i \in I} A_i \mid \forall i \in I, a(i) \in A_i\right\} \]

Definition (Semigroup \((S, *)\) )

\[ *: S \times S \rightarrow S \]

• Axiom 1 (Closure)

\[ \forall a, b \in S, a * b \in S \]

• Axiom 2 (Associativity)

\[ \forall a, b, c \in S,(a * b) * c=a *(b * c) \]

Example (Semigroup)

\[ (\mathbb{N},+), \mathbb{N}=\{1,2,3, \ldots\}, 0 \notin \mathbb{N} \]

Definition (Commutative semigroup)

\[ \forall a, b \in S, a * b=b * a \]

Definition (Non-commutative semigroup)

\[ \left\{\begin{array}{l} \forall a, b, c \in S,(a * b) * c=a *(b * c) \\ \exists x, y \in S, x * y \neq y * x \end{array}\right. \] 查看是否更新

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1.6 物理推导

真空中的麦克斯韦方程组（微分形式） [ \[\begin{aligned} \nabla \cdot \mathbf{E} &= \frac{\rho}{\varepsilon_0}, \\ \nabla \cdot \mathbf{B} &= 0, \\ \nabla \times \mathbf{E} &= -\frac{\partial \mathbf{B}}{\partial t}, \\ \nabla \times \mathbf{B} &= \mu_0\mathbf{J} + \mu_0\varepsilon_0\frac{\partial \mathbf{E}}{\partial t}. \end{aligned}\]

]

这里放入你希望先保存到盒子再复用的内容，例如一张 TikZ 图或复杂表格。

导出 PDF 时，上方两段会被分别包裹为

而在 HTML 里它们只是带对应 class 的 <div>，方便你自行加 CSS。

2 Category Theory

2.1 1. Foundational Definitions of Categories

Definition 3 A category \(\mathcal{C}\) consists of the following data: 1. A class \(\text{ob}(\mathcal{C})\) whose elements are called objects; 2. For every pair of objects \(A, B \in \text{ob}(\mathcal{C})\), a set \(\text{hom}_{\mathcal{C}}(A, B)\) (called morphisms from \(A\) to \(B\)); 3. For every triple of objects \(A, B, C \in \text{ob}(\mathcal{C})\), a composition function: \[\text{hom}_{\mathcal{C}}(A, B) \times \text{hom}_{\mathcal{C}}(B, C) \to \text{hom}_{\mathcal{C}}(A, C), \quad (f, g) \mapsto g \circ f\] satisfying three axioms: - (Disjointness) Morphism sets are disjoint (each morphism uniquely determines its domain and codomain); - (Associativity) For all \(f \in \text{hom}(A,B)\), \(g \in \text{hom}(B,C)\), \(h \in \text{hom}(C,D)\), \((h \circ g) \circ f = h \circ (g \circ f)\); - (Identity) For every object \(A\), there exists a unique identity morphism \(\text{id}_A \in \text{hom}(A,A)\) such that \(f \circ \text{id}_A = f\) (for all \(f \in \text{hom}(A,B)\)) and \(\text{id}_A \circ g = g\) (for all \(g \in \text{hom}(C,A)\)).

See Def. Definition 3. For context on “locally small categories” (where \(\text{hom}_{\mathcal{C}}(A,B)\) is a set for all \(A,B\)) and “small categories” (where \(\text{ob}(\mathcal{C})\) is a set), refer to (algant2019?).

2.2 2. Dual Categories and Isomorphisms

Definition 4 The dual category \(\mathcal{C}^{\text{op}}\) of a category \(\mathcal{C}\) has: - The same objects as \(\mathcal{C}\) (i.e., \(\text{ob}(\mathcal{C}^{\text{op}}) = \text{ob}(\mathcal{C})\)); - Morphisms \(\text{hom}_{\mathcal{C}^{\text{op}}}(A, B) = \text{hom}_{\mathcal{C}}(B, A)\) (all morphisms are reversed); - Composition \(\circ_{\text{op}}\) defined by \(f \circ_{\text{op}} g = g \circ f\) (where \(\circ\) is composition in \(\mathcal{C}\)).

See Def. Definition 4. This construction embodies the duality principle: any theorem about \(\mathcal{C}\) translates to a theorem about \(\mathcal{C}^{\text{op}}\) by reversing morphisms (e.g., “products in \(\mathcal{C}\)” become “coproducts in \(\mathcal{C}^{\text{op}}\)”).

Definition 5 A morphism \(f \in \text{hom}_{\mathcal{C}}(A, B)\) is an isomorphism if there exists a morphism \(g \in \text{hom}_{\mathcal{C}}(B, A)\) (called its inverse) such that \(g \circ f = \text{id}_A\) and \(f \circ g = \text{id}_B\). If such an \(f\) exists, \(A\) and \(B\) are isomorphic (written \(A \cong B\)).

See Def. Definition 5. The inverse of an isomorphism is unique (Exercise: Prove this using identity morphism properties (algant2019?)).

2.3 3. Natural Transformations (Morphisms of Functors)

Definition 6 Let \(F, G: \mathcal{C} \to \mathcal{D}\) be two functors between categories \(\mathcal{C}\) and \(\mathcal{D}\). A natural transformation \(\eta: F \Rightarrow G\) consists of: 1. For every object \(X \in \text{ob}(\mathcal{C})\), a component morphism \(\eta_X \in \text{hom}_{\mathcal{D}}(F(X), G(X))\); 2. A commutativity condition: For every morphism \(f \in \text{hom}_{\mathcal{C}}(X, Y)\) in \(\mathcal{C}\), the diagram \[\begin{CD} F(X) @>{\eta_X}>> G(X) \\ @V{F(f)}VV @VV{G(f)}V \\ F(Y) @>{\eta_Y}>> G(Y) \end{CD}\] commutes (i.e., \(\eta_Y \circ F(f) = G(f) \circ \eta_X\)).

See Def. Definition 6. Examples include the double dual transformation for vector spaces (associating each vector space \(V\) to its double dual \(V^{**}\)) and the Hurewicz homomorphism in algebraic topology (naturaltrans2025?).

2.4 4. Adjoint Functors and Universal Properties

Definition 7 An adjunction between categories \(\mathcal{C}\) and \(\mathcal{D}\) consists of: 1. A pair of functors \(F: \mathcal{D} \to \mathcal{C}\) (left adjoint) and \(G: \mathcal{C} \to \mathcal{D}\) (right adjoint); 2. For all \(C \in \text{ob}(\mathcal{C})\) and \(D \in \text{ob}(\mathcal{D})\), a natural bijection \[\text{hom}_{\mathcal{C}}(F(D), C) \cong \text{hom}_{\mathcal{D}}(D, G(C)).\] Naturality means the bijection is consistent across all objects \(C\) (via functors \(\mathcal{C}(F(-), C)\) and \(\mathcal{D}(-, G(C))\)) and \(D\) (via functors \(\mathcal{C}(F(D), -)\) and \(\mathcal{D}(D, G(-))\)). We write \(F \dashv G\).

See Def. Definition 7. Adjunctions generalize “weak equivalences” between categories and arise from universal constructions: - Algebra: The free group functor \(F: \text{Set} \to \text{Grp}\) (associating a set to its free group) is left adjoint to the forgetful functor \(G: \text{Grp} \to \text{Set}\) (adjoint2025?); - Topology: The Stone–Čech compactification functor is left adjoint to the inclusion functor of compact Hausdorff spaces into topological spaces (adjoint2025?).

2.5 5. Computer Science Applications

Category theory provides a foundational framework for computer science, as highlighted in (tcd2014?): - Algebraic Data Types: Many data types (e.g., lists, trees) are algebras for endofunctors (functors from a category to itself); - Computational Processes: Dynamical systems and concurrent processes are modeled as coalgebras for endofunctors; - Foundational Structures: Cartesian closed categories (with exponential objects) correspond to typed λ-calculi, the basis of functional programming.

The list data type (over a set \(X\)) is an algebra for the endofunctor \(F(Y) = 1 + X \times Y\) (where \(1\) is the terminal object). The algebra structure maps \(\text{inl}(\star)\) to the empty list and \(\text{inr}(x, y)\) to the list \(x :: y\) (cons operation).

See Ex. (ex-list-algebra?).

2.5.1 Definition (n-category)

An n-category is an algebraic structure that generalizes ordinary categories (1-categories). In an n-category: - There are objects (0-morphisms). - For any two objects (a, b), there is an ((n-1))-category ((a, b)) whose objects are called 1-morphisms from (a) to (b). - This structure continues recursively up to k-morphisms for (0 k n). - For (k < n), all k-morphisms commute strictly; while composition of the highest-dimensional n-morphisms only requires weak commutativity, typically characterized by coherence isomorphisms.

When (n = ), we obtain an ∞-category.

2.5.2 Theorem (Equivalent characterizations of ∞-categories)

An ∞-category () can be equivalently characterized by one of the following structures:

1. As a quasi-category: A simplicial set satisfying the inner Kan condition. This is the primary model adopted by B. Lurie in his book Higher Topos Theory.
2. As a complete Segal space: A simplicial object of spaces satisfying the Segal condition and completeness condition.
3. As a Segal category: A weakened simplicial category where Segal maps are equivalences.

These models are Quillen equivalent, thus all describing the same notion of “∞-category”.

2.5.3 Lemma (n-category structure of functor categories)

Let () and () be n-categories. Then all n-functors from () to () and their n-natural transformations form a new n-category, denoted (^) or ((, )), called the functor n-category or exponential n-category.

This lemma ensures that the “mapping space” between n-categories is itself an n-category, forming the basis for constructions like adjunctions in higher category theory.

2.5.4 Proposition (n-groupoids and homotopy n-types)

An n-groupoid is an n-category where all morphisms (in all dimensions) are equivalences.

In homotopy theory, there is a deep connection between n-groupoids and homotopy n-types. Specifically, the homotopy n-type of a topological space is completely determined by its fundamental n-groupoid. This is the content of the Homotopy Hypothesis, which asserts: “The category of n-groupoids is equivalent to the category of homotopy n-types.”

2.5.5 Corollary (1-categories as special cases of ∞-categories)

Directly from the theorem “Equivalent characterizations of ∞-categories”, we deduce:

Any ordinary 1-category () can be regarded as a special case of an ∞-category.

Proof sketch: Treat objects of () as 0-morphisms, 1-morphisms as 1-morphisms, and in all dimensions above 1, include only identity morphisms (or equivalently, no non-trivial morphisms). The resulting quasi-category is a discrete ∞-category that fully encodes the information of the original 1-category.

Remark (Remark (On the importance of “weak” structures)). A core distinction between higher category theory and ordinary category theory lies in the treatment of “commutativity”. In 1-categories, we talk about commutative diagrams, requiring equalities like (f g = h) to hold strictly.

However, in higher-dimensional settings, strict commutativity is too restrictive—there are almost no interesting examples. Thus, the essence of higher category theory is to replace it with weak commutativity. This means we no longer require (f g) and (h) to be strictly equal, but instead allow a higher-dimensional morphism (e.g., a 2-morphism (: f g h)), which itself is an equivalence.

This “weakening” idea is the soul of the subject, allowing us to use categorical language to describe flexible mathematical objects like space homotopies and algebraic deformations.

2.5.6 Example (Geometric instance of an ∞-groupoid)

A crucial example of an ∞-groupoid comes from topology:

Let (X) be a topological space.

We can construct an ∞-groupoid (_(X)), called the fundamental ∞-groupoid of (X), as follows: - 0-morphisms (objects): Points (x X). - 1-morphisms: Paths (: x y) connecting points (x) and (y) in (X). - 2-morphisms: Homotopies (H: ) between paths. - 3-morphisms: Homotopies between homotopies, and so on…

In this ∞-groupoid, all morphisms are homotopy equivalences, perfectly capturing all homotopy information of the space (X). When (X) is a single point, (_(X)) is the trivial ∞-category containing only identity morphisms.

Proof (Proof (Proposition: Homotopy interpretation of n-groupoids)). We sketch the proof of the proposition “n-groupoids and homotopy n-types”.

(Sketch) Let () be an n-groupoid. We need to construct a topological space (B) (called the classifying space or geometric realization of ()) such that its homotopy n-type corresponds to ().

1. Construction:
   • Start with the objects of (), treating each object as a point.
   • For each 1-morphism (f: a b), attach an interval ([0,1]) with endpoints connected to (a) and (b).
   • For each 2-morphism (: f g), attach a square ([0,1] ), gluing its boundary according to the source and target of ().
   • Continue recursively, attaching a k-dimensional cube for each k-morphism and gluing according to its source and target.
2. Verification:
   • The fundamental groupoid (_1(B)) of the space (B) constructed this way is the 1-truncation of () (i.e., ignoring all morphisms above dimension 1).
   • Higher homotopy groups (_k(B, x)) are given by equivalence classes of k-morphisms in () based at (x).
   • Since () is an n-groupoid, all homotopy groups above dimension (n) are trivial, so (B) is a homotopy n-type.

Conversely, the fundamental n-groupoid of any homotopy n-type is an n-groupoid. This establishes the equivalence between the two. ()

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	google-scholar	Google Scholar
	google-scholar-square	Google Scholar (方形)
	hal	HAL
	hal-square	HAL (方形)
	hypothesis	Hypothesis
	hypothesis-square	Hypothesis (方形)
	ideas-repec	IDEAS/RePEc
	ideas-repec-square	IDEAS/RePEc (方形)
	ieee	IEEE Xplore
	ieee-square	IEEE Xplore (方形)
	impactstory	ImpactStory
	impactstory-square	ImpactStory (方形)
	inaturalist	iNaturalist
	inaturalist-square	iNaturalist (方形)
	inpn	INPN
	inpn-square	INPN (方形)
	inspire	INSPIRE
	inspire-square	INSPIRE (方形)
	isidore	Isidore
	isidore-square	Isidore (方形)
	jstage	J-STAGE
	jstage-square	J-STAGE (方形)
	jstor	JSTOR
	jstor-square	JSTOR (方形)
	lattes	Lattes
	lattes-square	Lattes (方形)
	license	License
	license-square	License (方形)
	linkedin	LinkedIn
	linkedin-square	LinkedIn Square
	manuscript	Manuscript
	manuscript-square	Manuscript (方形)
	mathoverflow	MathOverflow
	mathoverflow-square	MathOverflow (方形)
	medrxiv	medRxiv
	medrxiv-square	medRxiv (方形)
	mendeley	Mendeley
	mendeley-square	Mendeley (方形)
	microsoft	Microsoft
	moodle	Moodle
	moodle-square	Moodle (方形)
	mtmt	MTMT
	mtmt-square	MTMT (方形)
	nature	Nature
	nature-square	Nature (方形)
	nakala	Nakala
	nakala-square	Nakala (方形)
	npg	Nature Publishing Group
	npg-square	Nature Publishing Group (方形)
	obp	Open Book Publishers
	obp-square	Open Book Publishers (方形)
	open-access	Open Access
	open-access-square	Open Access (方形)
	open-data	Open Data
	open-data-square	Open Data (方形)
	open-materials	Open Materials
	open-materials-square	Open Materials (方形)
	openedition	OpenEdition
	openedition-square	OpenEdition (方形)
	orcid	ORCID
	orcid-square	ORCID (方形)
	osf	OSF
	osf-square	OSF (方形)
	overleaf	Overleaf
	overleaf-square	Overleaf (方形)
	paho	PAHO
	paho-square	PAHO (方形)
	peerj	PeerJ
	peerj-square	PeerJ (方形)
	philpapers	PhilPapers
	philpapers-square	PhilPapers (方形)
	piazza	Piazza
	piazza-square	Piazza (方形)
	plos	PLOS
	plos-square	PLOS (方形)
	poster	Poster
	poster-square	Poster (方形)
	preprint	Preprint
	preprint-square	Preprint (方形)
	preregistered	Preregistered
	preregistered-square	Preregistered (方形)
	proceedings	Proceedings
	proceedings-square	Proceedings (方形)
	protocols	Protocols
	protocols-square	Protocols (方形)
	psyarxiv	PsyArXiv
	psyarxiv-square	PsyArXiv (方形)
	publons	Publons
	publons-square	Publons (方形)
	pubmed	PubMed
	pubmed-square	PubMed (方形)
	pubmedcentral	PubMed Central
	pubmedcentral-square	PubMed Central (方形)
	pubpeer	PubPeer
	pubpeer-square	PubPeer (方形)
	pub	Publication
	pub-square	Publication (方形)
	pnas	PNAS
	pnas-square	PNAS (方形)
	researcherid	ResearcherID
	researcherid-square	ResearcherID (方形)
	researchgate	ResearchGate
	researchgate-square	ResearchGate (方形)
	ror	ROR
	ror-square	ROR (方形)
	rsc	Royal Society of Chemistry
	rsc-square	Royal Society of Chemistry (方形)
	sagepub	SAGE Publications
	sagepub-square	SAGE Publications (方形)
	sci-hub	Sci-Hub
	sci-hub-square	Sci-Hub (方形)
	scirate	SciRate
	scirate-square	SciRate (方形)
	scopus	Scopus
	scopus-square	Scopus (方形)
	science	Science
	science-square	Science (方形)
	sciencedirect	ScienceDirect
	sciencedirect-square	ScienceDirect (方形)
	semantic-scholar	Semantic Scholar
	semantic-scholar-square	Semantic Scholar (方形)
	slides	Slides
	slides-square	Slides (方形)
	springer	Springer
	springer-square	Springer (方形)
	springeropen	SpringerOpen
	springeropen-square	SpringerOpen (方形)
	stackoverflow	Stack Overflow
	stackoverflow-square	Stack Overflow (方形)
	ssrn	SSRN
	ssrn-square	SSRN (方形)
	tandfonline	Taylor & Francis Online
	tandfonline-square	Taylor & Francis Online (方形)
	taylorfrancis	Taylor & Francis
	taylorfrancis-square	Taylor & Francis (方形)
	theses	Theses
	theses-square	Theses (方形)
	thesis	Thesis
	thesis-square	Thesis (方形)
	video	Video
	video-square	Video (方形)
	wiley	Wiley
	wiley-square	Wiley (方形)
	webofscience	Web of Science
	webofscience-square	Web of Science (方形)
	zenodo	Zenodo
	zenodo-square	Zenodo (方形)
	zotero	Zotero
	zotero-square	Zotero (方形)

3 Default icons

图标预览	图标名称	对应网站/用途	使用代码
	address-book	地址簿
	address-card	地址卡
	adjust	调整
	air-freshener	空气清新剂
	airplane	飞机
	alarm	闹钟
	align-center	居中对齐
	align-justify	两端对齐
	align-left	左对齐
	align-right	右对齐
	all-inclusive	全包
	anchor	锚
	angle-double-down	双向下箭头
	angle-double-left	双向左箭头
	angle-double-right	双向右箭头
	angle-double-up	双向上箭头
	angle-down	向下箭头
	angle-left	向左箭头
	angle-right	向右箭头
	angle-up	向上箭头
	angry	生气
	apple-alt	苹果（简化）
	archive	归档
	arrow-alt-circle-down	圆形向下箭头
	arrow-alt-circle-left	圆形向左箭头
	arrow-alt-circle-right	圆形向右箭头
	arrow-alt-circle-up	圆形向上箭头
	arrow-circle-down	圆形向下箭头（实心）
	arrow-circle-left	圆形向左箭头（实心）
	arrow-circle-right	圆形向右箭头（实心）
	arrow-circle-up	圆形向上箭头（实心）
	arrow-down	向下箭头（实心）
	arrow-left	向左箭头（实心）
	arrow-right	向右箭头（实心）
	arrow-up	向上箭头（实心）
	arrows	双向箭头
	arrows-alt	双向箭头（扩展）
	arrows-h	水平双向箭头
	arrows-v	垂直双向箭头
	artstation	ArtStation（艺术平台）
	assistive-listening-systems	助听系统
	asterisk	星号
	at	(符号?)
	audio-description	音频描述
	autumn	秋天
	award	奖项
	baby	婴儿
	baby-carriage	婴儿车
	backspace	退格键
	backpack	背包
	balance-scale	天平
	ball-pool	球池
	ban	禁止
	band-aid	创可贴
	barcode	条形码
	bars	菜单（ bars）
	baseball-ball	棒球
	basketball-ball	篮球
	bath	浴缸
	battery-empty	电池（空）
	battery-full	电池（满）
	battery-half	电池（半满）
	battery-quarter	电池（四分之一）
	battery-three-quarters	电池（四分之三）
	bed	床
	beer	啤酒
	bell	铃铛
	bell-slash	铃铛（静音）
	bicycle	自行车
	binoculars	双筒望远镜
	biohazard	生物危害
	bird	鸟
	birthday-cake	生日蛋糕
	bluetooth	蓝牙
	boat	船
	bone	骨头
	book	书
	book-open	打开的书
	bookmark	书签
	border-all	全部边框
	border-none	无边框
	bowling-ball	保龄球
	box	盒子
	box-open	打开的盒子
	braille	盲文
	briefcase	公文包
	broadcast-tower	广播塔
	brush	刷子
	bug	虫子
	building	建筑
	building-columns	建筑（带柱子）
	bullhorn	扩音器
	bullseye	靶心
	bus	公交车
	cab	出租车
	calculator	计算器
	calendar	日历
	calendar-alt	日历（简化）
	calendar-check	日历（带勾选）
	calendar-day	日历（按天）
	calendar-minus	日历（减号）
	calendar-plus	日历（加号）
	calendar-week	日历（按周）
	camera	相机
	camera-retro	复古相机
	campground	露营地
	candle	蜡烛
	candy-cane	拐杖糖
	canister	罐
	caps-lock	大小写锁定
	car	汽车
	car-alt	汽车（简化）
	car-battery	汽车电池
	caret-down	插入符号（下）
	caret-left	插入符号（左）
	caret-right	插入符号（右）
	caret-up	插入符号（上）
	cart-arrow-down	购物车（向下箭头）
	cart-plus	购物车（加号）
	cat	猫
	certificate	证书
	chair	椅子
	chalkboard	黑板
	cheese	奶酪
	chess	国际象棋
	chess-bishop	国际象棋（主教）
	chess-board	国际象棋棋盘
	chess-king	国际象棋（国王）
	chess-knight	国际象棋（骑士）
	chess-pawn	国际象棋（兵）
	chess-queen	国际象棋（皇后）
	chess-rook	国际象棋（车）
	child	小孩
	circle	圆形
	circle-check	圆形（勾选）
	circle-notch	圆形（缺口）
	circle-o	圆形（空心）
	circle-of-fifths	五度圈
	circle-play	圆形（播放）
	circle-plus	圆形（加号）
	circle-question	圆形（问号）
	circle-xmark	圆形（叉号）
	clipboard	剪贴板
	clipboard-check	剪贴板（勾选）
	clipboard-list	剪贴板（列表）
	clock	时钟
	clone	克隆
	closed-captioning	隐藏字幕
	cloud	云
	cloud-download-alt	云下载
	cloud-upload-alt	云上传
	code	代码
	code-branch	代码分支
	coffee	咖啡
	cog	齿轮
	cogs	齿轮组
	coins	硬币
	column	列
	comment	评论
	comment-alt	评论（简化）
	comment-dots	评论（点）
	comments	评论组
	compass	指南针
	compress	压缩
	computer	电脑
	concrete	混凝土
	cone	圆锥体
	connectdevelop	ConnectDevelop
	cookie	饼干
	copy	复制
	copyright	版权
	couch	沙发
	cow	牛
	credit-card	信用卡
	crop	裁剪
	cross	十字架
	crosshairs	十字准星
	crown	皇冠
	cube	立方体
	cubes	立方体组
	cut	剪切
	cutlery	餐具
	dashboard	仪表盘
	database	数据库
	deaf	聋人
	desktop	桌面
	dev	开发
	diamond	钻石
	dice	骰子
	dice-d20	骰子（D20）
	dice-d6	骰子（D6）
	direction	方向
	disabled	禁用
	docker	Docker
	dog	狗
	dollar-sign	美元符号
	dot-circle	点圆形
	download	下载
	draggable	可拖拽
	dragon	龙
	draw-polygon	绘制多边形
	dribbble	Dribbble（设计平台）
	drum	鼓
	drumstick-bite	鸡腿
	dropbox	Dropbox（云存储）
	drown	溺水
	dvd	DVD
	edit	编辑
	eject	弹出
	egg	鸡蛋
	electricity	电
	elevator	电梯
	ellipsis-h	水平省略号
	ellipsis-v	垂直省略号
	envelope	信封
	envelope-open	打开的信封
	envelope-square	方形信封
	equalizer	均衡器
	eraser	橡皮擦
	euro-sign	欧元符号
	exchange-alt	交换
	exclamation	感叹号
	exclamation-circle	圆形感叹号
	exclamation-triangle	三角形感叹号
	expand	展开
	external-link	外部链接
	external-link-alt	外部链接（简化）
	eye	眼睛
	eye-slash	眼睛（斜杠）
	face-grin-stars	笑脸（星星眼）
	face-laugh-beam	笑脸（大笑）
	face-meh	笑脸（平淡）
	face-smile	笑脸（微笑）
	facebook	Facebook（社交平台）
	facebook-f	Facebook（简化）
	facebook-messenger	Facebook Messenger（聊天工具）
	factory	工厂
	fan	风扇
	fast-backward	快退
	fast-forward	快进
	fax	传真
	feather	羽毛
	feed	饲料
	female	女性
	ferry	渡轮
	file	文件
	file-alt	文件（简化）
	file-archive	归档文件
	file-audio	音频文件
	file-code	代码文件
	file-excel	Excel文件
	file-image	图片文件
	file-pdf	PDF文件
	file-powerpoint	PowerPoint文件
	file-signature	签名文件
	file-text	文本文件
	file-video	视频文件
	file-word	Word文件
	fill-drip	填充（滴）
	film	电影
	filter	过滤器
	fire	火
	fire-alt	火（简化）
	fire-extinguisher	灭火器
	first-aid	急救
	fish	鱼
	flag	旗帜
	flag-checkered	格子旗
	flag-usa	美国国旗
	flask	烧瓶
	flight	飞行
	flipboard	Flipboard（新闻应用）
	float-left	左浮动
	float-right	右浮动
	floor-lamp	落地灯
	flower	花
	folder	文件夹
	folder-open	打开的文件夹
	font	字体
	football-ball	足球
	forklift	叉车
	form	表单
	forumbee	Forumbee（论坛平台）
	forward	前进
	frown	皱眉
	frown-open	皱眉（开口）
	fuel-pump	加油站
	gamepad	游戏手柄
	gavel	法槌
	gem	宝石
	genderless	无性别
	get-pocket	Pocket（稍后阅读工具）
	gift	礼物
	glass-cheers	干杯
	glass-water	水杯
	globe	地球
	globe-americas	地球（美洲）
	globe-asia	地球（亚洲）
	globe-europe	地球（欧洲）
	goat	山羊
	gopuram	寺庙塔楼
	graduation-cap	毕业帽
	grave	坟墓
	guitar	吉他
	hammer	锤子
	hand-holding-heart	手捧心
	hand-holding-usd	手持美元
	handshake	握手
	handshake-alt	握手（简化）
	hard-hat	安全帽
	hash	#符号
	headphones	耳机
	heart	心
	heart-broken	破碎的心
	heat	热
	helicopter	直升机
	highlighter	荧光笔
	hill-rockslide	山崩
	history	历史
	home	家
	hospital	医院
	hospital-user	医院用户
	hot-tub	热水浴缸
	hourglass	沙漏
	hourglass-end	沙漏（结束）
	hourglass-half	沙漏（一半）
	hourglass-start	沙漏（开始）
	house-chimney	房子（烟囱）
	house-chimney-user	房子（烟囱用户）
	house-user	房子（用户）
	html5	HTML5
	hubspot	HubSpot（营销平台）
	ice-cream	冰淇淋
	id-card	身份证
	id-card-alt	身份证（简化）
	image	图片
	images	图片组
	inbox	收件箱
	indent	缩进
	industry	工业
	infinity	无限
	info	信息
	info-circle	圆形信息
	instagram	Instagram（社交平台）
	internet-explorer	Internet Explorer（浏览器）
	italic	斜体
	itunes	iTunes
	itunes-note	iTunes（音符）
	jaguar	捷豹
	jawbone	Jawbone（智能硬件品牌）
	jeddah	吉达
	jenkins	Jenkins（自动化工具）
	jersey	运动衫
	joomla	Joomla（CMS平台）
	journal-whills	期刊
	jupyter	Jupyter（数据分析工具）
	key	钥匙
	keyboard	键盘
	kickstarter	Kickstarter（众筹平台）
	kickstarter-k	Kickstarter（简化）
	ladder	梯子
	laptop	笔记本电脑
	laptop-code	笔记本电脑（代码）
	laptop-house	笔记本电脑（家）
	laptop-medical	笔记本电脑（医疗）
	lastfm	Last.fm（音乐平台）
	lastfm-square	Last.fm（方形）
	laugh	笑
	layer-group	图层组
	leaf	叶子
	leanpub	Leanpub（电子书平台）
	left-right	左右
	lemon	柠檬
	less-than	小于号
	level-down-alt	向下水平
	level-up-alt	向上水平
	life-ring	救生圈
	lightbulb	灯泡
	link	链接
	link-slash	链接（斜杠）
	linux	Linux
	list	列表
	list-alt	列表（简化）
	list-check	列表（勾选）
	list-ol	有序列表
	list-ul	无序列表
	location-arrow	定位箭头
	lock	锁
	lock-open	打开的锁
	log	日志
	login	登录
	logout	登出
	long-arrow-alt-down	长向下箭头
	long-arrow-alt-left	长向左箭头
	long-arrow-alt-right	长向右箭头
	long-arrow-alt-up	长向上箭头
	low-vision	低视力
	magnet	磁铁
	mail-bulk	批量邮件
	mailchimp	Mailchimp（邮件营销平台）
	male	男性
	map	地图
	map-location-dot	地图（定位点）
	map-marked-alt	地图（标记）
	map-marker-alt	地图标记
	map-pin	地图钉
	map-signs	地图（标志）
	mars	火星
	mars-and-venus	火星和金星
	mars-double	双火星
	mask	面具
	maxcdn	MaxCDN
	meanpath	Meanpath
	medium	Medium（博客平台）
	medkit	医疗箱
	meetup	Meetup（线下活动平台）
	meh	平淡
	meh-blank	平淡（空白）
	memory	内存
	menu	菜单
	mercury	水星
	meter-high	高度计
	meter-low	低高度计
	meter-square	平方计
	microchip	微芯片
	microphone	麦克风
	microphone-lines	麦克风（线）
	microphone-slash	麦克风（斜杠）
	microsoft	Microsoft
	minimize	最小化
	minus	减号
	minus-circle	圆形减号
	minus-square	方形减号
	mobile	手机
	mobile-alt	手机（简化）
	mobile-screen	手机屏幕
	mobile-screen-button	手机屏幕（按钮）
	modulo	模
	moon	月亮
	moon-stars	月亮星星
	mortar-pestle	研钵和杵
	motorcycle	摩托车
	mountain	山
	mountain-city	山城市
	mouse	鼠标
	mug-hot	热 mug
	music	音乐
	nasa	NASA
	navicon	导航图标
	neuter	中性
	new	新
	newspaper	报纸
	node	节点
	node-js	Node.js
	noise	噪音
	not-equal	不等于
	notebook	笔记本
	notebook-alt	笔记本（简化）
	notebook-medical	医疗笔记本
	nuclear	核能
	object-group	对象组
	object-ungroup	对象取消组
	odnoklassniki	Odnoklassniki（社交平台）
	odnoklassniki-square	Odnoklassniki（方形）
	oil-can	油罐
	old-republic	旧共和国
	olympics	奥运会
	om	欧姆
	opencart	OpenCart（电商平台）
	openid	OpenID
	opera	Opera（浏览器）
	option	选项
	outdent	减少缩进
	page-break	分页符
	paint-brush	画笔
	paint-roller	油漆滚筒
	palette	调色板
	paper-plane	纸飞机
	paper-plane-alt	纸飞机（简化）
	paragraph	段落
	park	公园
	pause	暂停
	pause-circle	圆形暂停
	pencil	铅笔
	pencil-alt	铅笔（简化）
	pencil-ruler	铅笔尺子
	people-arrows	人们箭头
	people-carry	人们携带
	percent	百分比
	perfect-square	完美平方
	phone	电话
	phone-alt	电话（简化）
	phone-laptop	电话笔记本电脑
	phone-slash	电话（斜杠）
	photo-film	照片胶片
	picture-o	图片（空心）
	piggy-bank	存钱罐
	plane-departure	飞机起飞
	plane-arrival	飞机到达
	play	播放
	play-circle	圆形播放
	playstation	PlayStation
	plus	加号
	plus-circle	圆形加号
	plus-square	方形加号
	podcast	播客
	poo	便便
	poo-storm	便便风暴
	power-off	电源关闭
	print	打印
	prison	监狱
	problem	问题
	project-diagram	项目图表
	pulse	脉冲
	qrcode	二维码
	question	问号
	question-circle	圆形问号
	quote-left	左引号
	quote-right	右引号
	radio	收音机
	random	随机
	raspberry-pi	树莓派
	react	React
	readme	README
	receipt	收据
	record-vinyl	黑胶唱片
	recycle	回收
	red-river	红河
	registered	注册
	remove	移除
	repeat	重复
	reply	回复
	reply-all	全部回复
	repository	仓库
	repository-alt	仓库（简化）
	request	请求
	resize	调整大小
	retweet	转发
	road	道路
	road-barrier	路障
	robot	机器人
	rocket	火箭
	rotate	旋转
	rotate-left	向左旋转
	rotate-right	向右旋转
	rss	RSS
	rss-square	RSS（方形）
	rub	卢布

References

Lane, S. M. (1998), Categories for the working mathematician, Graduate texts in mathematics, New York, NY: Springer-Verlag. https://doi.org/10.1007/978-1-4757-4721-8.