📐 Category Theory III: Limits and Colimits

Table of Contents

• #1. Diagrams and Cones|1. Diagrams and Cones
  • #1.1 The Shape Category|1.1 The Shape Category
  • #1.2 Cones and Cocones|1.2 Cones and Cocones
• #2. Limits and Colimits|2. Limits and Colimits
  • #2.1 The Universal Property|2.1 The Universal Property
  • #2.2 Uniqueness Up to Unique Isomorphism|2.2 Uniqueness Up to Unique Isomorphism
  • #2.3 Special Cases|2.3 Special Cases
• #3. Limits in Set|3. Limits in Set
  • #3.1 Explicit Description of the Limit|3.1 Explicit Description of the Limit
  • #3.2 Explicit Description of the Colimit|3.2 Explicit Description of the Colimit
• #4. Products and Exponentials|4. Products and Exponentials
• #5. Preservation, Reflection, and Creation of Limits|5. Preservation, Reflection, and Creation of Limits
  • #5.1 Definitions|5.1 Definitions
  • #5.2 The Creation Theorem|5.2 The Creation Theorem
• #6. Projective and Injective Objects|6. Projective and Injective Objects
  • #6.1 Projective Objects|6.1 Projective Objects
  • #6.2 Injective Objects|6.2 Injective Objects
  • #6.3 Regular Injective Objects|6.3 Regular Injective Objects
• #7. Limits as Colimits: Formal Power Series|7. Limits as Colimits: Formal Power Series
• #8. Regular Monics and Epics|8. Regular Monics and Epics
  • #8.1 Definitions and Hierarchy|8.1 Definitions and Hierarchy
  • #8.2 Characterizations and Examples|8.2 Characterizations and Examples
• #9. Building Limits from Products and Equalizers|9. Building Limits from Products and Equalizers
• #10. Algebraic Categories: Widget and Chad|10. Algebraic Categories: Widget and Chad
• #11. Limits as Right Adjoints to the Diagonal|11. Limits as Right Adjoints to the Diagonal
• #12. Limits in Functor Categories|12. Limits in Functor Categories
• #13. Representable Functors Preserve Limits|13. Representable Functors Preserve Limits
• #14. Adjoints and Limit Preservation|14. Adjoints and Limit Preservation
• #15. The Density Theorem and Cartesian Closed Categories|15. The Density Theorem and Cartesian Closed Categories
  • #15.1 The Density Theorem|15.1 The Density Theorem
  • #15.2 Cartesian Closed Categories|15.2 Cartesian Closed Categories
• #16. Stability of Morphism Classes|16. Stability of Morphism Classes
• #17. Optional: Homological Algebra via Category Theory|17. Optional: Homological Algebra via Category Theory
• #18. The Art of the Diagram Chase|18. The Art of the Diagram Chase 🧩
• #19. Complete and Cocomplete Categories|19. Complete and Cocomplete Categories 🌐
• #20. Functoriality of Limits|20. Functoriality of Limits ⚙️
• #21. Interactions between Limits and Colimits|21. Interactions between Limits and Colimits 🔄
• #References|References

1. Diagrams and Cones 🗺️

1.1 The Shape Category

A diagram in a category \(\mathcal{C}\) is simply a functor \(D : \mathcal{I} \to \mathcal{C}\), where \(\mathcal{I}\) is a small category called the index category or shape. The objects of \(\mathcal{I}\) label the vertices of the diagram, and the morphisms of \(\mathcal{I}\) label the arrows between them.

Definition (Diagram). Let \(\mathcal{C}\) be a category and \(\mathcal{I}\) a small category. A diagram of shape \(\mathcal{I}\) in \(\mathcal{C}\) is a functor \(D : \mathcal{I} \to \mathcal{C}\).

1.2 Cones and Cocones

Definition (Cone). Let \(D : \mathcal{I} \to \mathcal{C}\) be a diagram. A cone over \(D\) is a pair \((N, (\pi_i)_{i \in \mathcal{I}})\) where \(N \in \mathcal{C}\) is an object (the apex) and \(\pi_i : N \to Di\) is a morphism for each object \(i \in \mathcal{I}\), subject to the commutativity condition: for every morphism \(u : i \to j\) in \(\mathcal{I}\), \[D(u) \circ \pi_i = \pi_j.\]

Equivalently, in terms of the constant functor \(\Delta_N : \mathcal{I} \to \mathcal{C}\) (which sends every object to \(N\) and every morphism to \(\mathrm{id}_N\)), a cone \((N, (\pi_i))\) is precisely a natural transformation \(\pi : \Delta_N \Rightarrow D\).

\usepackage{tikz-cd}
\begin{document}
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& N \arrow[dl, "\pi_i"'] \arrow[dr, "\pi_j"] & \\
Di \arrow[rr, "D(u)"'] & & Dj
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Definition (Cocone). A cocone under \(D\) is a pair \((N, (\iota_i)_{i \in \mathcal{I}})\) where \(\iota_i : Di \to N\) for each \(i \in \mathcal{I}\), subject to the cocommutativity condition: for every \(u : i \to j\) in \(\mathcal{I}\), \[\iota_j \circ D(u) = \iota_i.\]

Equivalently, a cocone is a natural transformation \(\iota : D \Rightarrow \Delta_N\).

2. Limits and Colimits 🎯

2.1 The Universal Property

Definition (Limit). A limit of \(D\) is a cone \((L, (\pi_i)_{i \in \mathcal{I}})\) over \(D\) that is terminal in the category of cones over \(D\): for every cone \((N, (\phi_i)_{i \in \mathcal{I}})\) over \(D\), there exists a unique morphism \(! : N \to L\) in \(\mathcal{C}\) such that \[\pi_i \circ \, ! \;= \phi_i \quad \text{for all } i \in \mathcal{I}.\]

We denote the apex \(L\) by \(\varprojlim D\) or \(\lim_{\mathcal{I}} D\) or simply \(\lim D\).

\usepackage{tikz-cd}
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N \arrow[dr, "\varphi_i"'] \arrow[r, "!", dashed] & {L = \lim D} \arrow[d, "\pi_i"] \\
& Di
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Definition (Colimit). A colimit of \(D\) is a cocone \((L, (\iota_i)_{i \in \mathcal{I}})\) under \(D\) that is initial among cocones: for every cocone \((N, (\psi_i)_{i \in \mathcal{I}})\) under \(D\), there exists a unique morphism \(! : L \to N\) such that \[! \circ \iota_i = \psi_i \quad \text{for all } i \in \mathcal{I}.\]

We denote the nadir \(L\) by \(\varinjlim D\) or \(\mathrm{colim}_{\mathcal{I}} D\) or simply \(\mathrm{colim}\, D\).

2.2 Uniqueness Up to Unique Isomorphism

Proposition (Uniqueness of limits). If \((L, (\pi_i))\) and \((L', (\pi'_i))\) are both limits of \(D : \mathcal{I} \to \mathcal{C}\), then there is a unique isomorphism \(\phi : L \xrightarrow{\sim} L'\) compatible with the projections: \(\pi'_i \circ \phi = \pi_i\) for all \(i\).

Proof sketch. By the universal property applied to \((L, (\pi_i))\) as a cone over \(D\), there is a unique \(\phi : L \to L'\) with \(\pi'_i \circ \phi = \pi_i\). Similarly there is a unique \(\psi : L' \to L\) with \(\pi_i \circ \psi = \pi'_i\). Then \(\psi \circ \phi\) and \(\mathrm{id}_L\) are both morphisms \(L \to L\) compatible with \((\pi_i)\); by uniqueness \(\psi \circ \phi = \mathrm{id}_L\). Similarly \(\phi \circ \psi = \mathrm{id}_{L'}\). \(\square\)

This uniqueness justifies speaking of “the” limit of a diagram.

2.3 Special Cases

The following table catalogues the most important special instances of limits and colimits, obtained by varying the shape \(\mathcal{I}\).

[!EXAMPLE]- Pullbacks in concrete categories 💡 The pullback of a cospan \(A \xrightarrow{f} C \xleftarrow{g} B\) is the limit of the diagram of shape \(\bullet \to \bullet \leftarrow \bullet\). It is the object \(A \times_C B\) with projections to \(A\) and \(B\), such that the square commutes and is universal with this property.

In \(\mathbf{Set}\): \(A \times_C B = \{(a, b) \in A \times B \mid f(a) = g(b)\}\) — the “fibered product” over \(C\).

In \(\mathbf{Top}\): the pullback of continuous maps \(f : A \to C\) and \(g : B \to C\) is \(\{(a,b) \in A \times B : f(a) = g(b)\}\) with the subspace topology.

In \(\mathbf{Grp}\): the pullback is the fiber product \(\{(a, b) \in A \times B : f(a) = g(b)\}\) with componentwise group structure.

The dual construction — the pushout of a span \(A \xleftarrow{f} C \xrightarrow{g} B\) — is \(A \sqcup_C B\) (the amalgamated sum). In \(\mathbf{Set}\): the pushout is \((A \sqcup B)/{\sim}\) where \(f(c) \sim g(c)\) for all \(c \in C\). In \(\mathbf{Grp}\): the pushout is the free product with amalgamation \(A *_C B\).

[!EXAMPLE]- Equalizers and kernel pairs The equalizer of \(f, g \colon A \rightrightarrows B\) is the limit of the parallel pair diagram. In \(\mathbf{Set}\): \(\mathrm{eq}(f,g) = \{a \in A \mid f(a) = g(a)\} \hookrightarrow A\).

In \(\mathbf{Grp}\): the equalizer of \(f, g \colon G \rightrightarrows H\) is the subgroup \(\{x \in G \mid f(x) = g(x)\}\) with the inclusion homomorphism.

The kernel pair of a morphism \(f \colon A \to B\) is the pullback of \(f\) along itself: the object \(A \times_B A\) with two projection morphisms \(p_1, p_2 \colon A \times_B A \rightrightarrows A\). In \(\mathbf{Set}\), this is the equivalence relation \(\{(a, a') \mid f(a) = f(a')\}\). Kernel pairs are fundamental to the theory of regular categories.

3. Limits in Set 🔢

3.1 Explicit Description of the Limit

For a small diagram \(D : \mathcal{I} \to \mathbf{Set}\), the limit has an entirely explicit description as a subset of a product.

Theorem (Limits in Set). The limit of \(D : \mathcal{I} \to \mathbf{Set}\) is \[\varprojlim D \;=\; \left\{\; (x_i)_{i \in \mathcal{I}} \in \prod_{i \in \mathcal{I}} D(i) \;\Big|\; D(u)(x_i) = x_j \text{ for all } u : i \to j \text{ in } \mathcal{I} \;\right\},\] with projections \(\pi_j : \varprojlim D \to D(j)\) given by \((x_i)_{i \in \mathcal{I}} \mapsto x_j\).

Proof sketch. One verifies that the given set with projections satisfies the universal property of a limit cone. Given any cone \((N, (\phi_i))\), define \(! : N \to \varprojlim D\) by \(!(n) = (\phi_i(n))_{i \in \mathcal{I}}\). This is the unique map with \(\pi_i \circ \, ! = \phi_i\), and it lands in the equalizing subset by the cone condition \(D(u) \circ \phi_i = \phi_j\). \(\square\)

3.2 Explicit Description of the Colimit

Theorem (Colimits in Set). The colimit of \(D : \mathcal{I} \to \mathbf{Set}\) is \[\varinjlim D \;=\; \left(\bigsqcup_{i \in \mathcal{I}} D(i)\right) \Big/ {\sim},\] where \(\sim\) is the equivalence relation generated by: \(x_i \sim x_j\) whenever there exists \(u : i \to j\) in \(\mathcal{I}\) with \(D(u)(x_i) = x_j\). The structure maps are \(\iota_i : D(i) \to \varinjlim D\), sending \(x\) to its equivalence class \([x]\).

Proof sketch. Given any cocone \((N, (\psi_i))\), define \(! : \varinjlim D \to N\) by \(!([x_i]) = \psi_i(x_i)\). This is well-defined because the cocone condition \(\psi_j \circ D(u) = \psi_i\) ensures that equivalent elements map to the same element of \(N\). \(\square\)

4. Products and Exponentials 🧮

Let \(\mathcal{C}\) be a category with finite products. For fixed \(B \in \mathcal{C}\), the product functor \(- \times B : \mathcal{C} \to \mathcal{C}\) sends \(A \mapsto A \times B\) and \((f : A \to A') \mapsto (f \times \mathrm{id}_B : A \times B \to A' \times B)\).

Proposition. There is a natural isomorphism \[\mathcal{C}(A, B \times C) \;\cong\; \mathcal{C}(A, B) \times \mathcal{C}(A, C)\] naturally in \(A\), \(B\), and \(C\).

Proof sketch. The bijection sends \(f : A \to B \times C\) to the pair \((\pi_1 \circ f, \pi_2 \circ f)\). The inverse sends \((g, h)\) to the unique \(\langle g, h \rangle : A \to B \times C\) given by the universal property of the product. Naturality in each variable follows by functoriality of composition. \(\square\)

5. Preservation, Reflection, and Creation of Limits ⚙️

5.1 Definitions

Let \(F : \mathcal{C} \to \mathcal{D}\) be a functor and \(\mathcal{I}\) a small category.

Definition (Preservation). \(F\) preserves limits of shape \(\mathcal{I}\) if, for every diagram \(D : \mathcal{I} \to \mathcal{C}\) and every limit cone \((L, (\pi_i))\) over \(D\) in \(\mathcal{C}\), the cone \((FL, (F\pi_i))\) is a limit of \(FD : \mathcal{I} \to \mathcal{D}\).

Definition (Reflection). \(F\) reflects limits of shape \(\mathcal{I}\) if, for every diagram \(D : \mathcal{I} \to \mathcal{C}\) and every cone \((L, (\pi_i))\) over \(D\): if \((FL, (F\pi_i))\) is a limit of \(FD\) in \(\mathcal{D}\), then \((L, (\pi_i))\) is already a limit of \(D\) in \(\mathcal{C}\).

Definition (Creation). \(F\) creates limits of shape \(\mathcal{I}\) if \(F\) reflects limits of shape \(\mathcal{I}\), and moreover: for every diagram \(D : \mathcal{I} \to \mathcal{C}\) such that \(FD\) has a limit \((M, (\mu_i))\) in \(\mathcal{D}\), there exists a unique cone \((L, (\pi_i))\) over \(D\) in \(\mathcal{C}\) such that \((FL, (F\pi_i)) = (M, (\mu_i))\), and this unique cone is a limit cone.

5.2 The Creation Theorem

Theorem. If \(F : \mathcal{C} \to \mathcal{D}\) creates limits of shape \(\mathcal{I}\), and \(D : \mathcal{I} \to \mathcal{C}\) is a diagram such that \(FD\) has a limit in \(\mathcal{D}\), then: 1. \(D\) has a limit in \(\mathcal{C}\). 2. \(F\) preserves this limit.

Proof. Since \(F\) creates limits, the existence of a limit of \(FD\) implies the existence of a unique lift \((L, (\pi_i))\) over \(D\) that maps to it, and this lift is a limit cone. This gives (1). For (2): since \(F\) maps the limit cone \((L, (\pi_i))\) to the limit cone \((FL, (F\pi_i)) = (M, (\mu_i))\) of \(FD\), \(F\) preserves this limit. \(\square\)

6. Projective and Injective Objects 🔩

6.1 Projective Objects

Definition (Projective object). An object \(P \in \mathcal{C}\) is projective if the representable functor \(\mathcal{C}(P, -) : \mathcal{C} \to \mathbf{Set}\) preserves epimorphisms: for every epimorphism \(e : A \twoheadrightarrow B\) and every morphism \(f : P \to B\), there exists a morphism \(g : P \to A\) with \(e \circ g = f\).

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& P \arrow[d, "f"] \arrow[dl, dashed, "\exists g"'] \\
A \arrow[r, "e"', two heads] & B
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6.2 Injective Objects

Definition (Injective object). An object \(I \in \mathcal{C}\) is injective if the contravariant representable \(\mathcal{C}(-, I) : \mathcal{C}^{\mathrm{op}} \to \mathbf{Set}\) sends monomorphisms to surjections: for every monomorphism \(m : A \hookrightarrow B\) and every morphism \(f : A \to I\), there exists \(g : B \to I\) with \(g \circ m = f\).

6.3 Regular Injective Objects

Definition (Regular injective). An object \(I \in \mathcal{C}\) is regular injective if it has the extension property only with respect to regular monomorphisms (i.e., equalizer embeddings) rather than all monomorphisms.

7. Limits as Colimits: Formal Power Series 🌀

This section illustrates how limits arise in classical algebra through a categorical lens.

Let \(k\) be a field. For each \(n \geq 0\), consider the truncated polynomial ring \(k[X]/(X^n)\). For \(m \geq n\), there is a canonical surjective ring homomorphism \[\rho_{m,n} : k[X]/(X^m) \twoheadrightarrow k[X]/(X^n), \qquad [f(X)] \mapsto [f(X) \mod X^n],\] and these satisfy the coherence condition \(\rho_{n,p} \circ \rho_{m,n} = \rho_{m,p}\) for \(m \geq n \geq p \geq 0\). Thus the rings \(\{k[X]/(X^n)\}_{n \geq 0}\) together with the maps \(\rho_{m,n}\) form an inverse system (a diagram indexed by the poset \((\mathbb{N}, \leq)^{\mathrm{op}}\)).

Proposition. The ring of formal power series \(k\llbracket X \rrbracket\) is (isomorphic to) the limit of this inverse system in the category \(\mathbf{Ring}\): \[k\llbracket X \rrbracket \;\cong\; \varprojlim_{n \in \mathbb{N}} k[X]/(X^n).\]

Proof sketch. The limit is computed by the explicit formula from §3.1 (applied in the enriched sense for rings, or equivalently using the underlying sets): an element of \(\varprojlim k[X]/(X^n)\) is a compatible sequence \((a_n)_{n \geq 0}\) with \(a_n \in k[X]/(X^n)\) and \(\rho_{m,n}(a_m) = a_n\). Such a sequence is precisely determined by its coefficients \(c_0, c_1, c_2, \ldots \in k\) (the coefficients of a formal power series), via \(a_n = c_0 + c_1 X + \cdots + c_{n-1}X^{n-1}\) in \(k[X]/(X^n)\). The ring operations on the limit match those of \(k\llbracket X \rrbracket\) because truncation is a ring homomorphism. \(\square\)

The \(k\)-linear dual \(k[X]^* = \mathrm{Hom}_k(k[X], k)\) also has a canonical identification with \(k\llbracket X \rrbracket\): a \(k\)-linear functional \(\phi : k[X] \to k\) is determined by its values on the basis \(\{1, X, X^2, \ldots\}\), i.e., by a sequence \((c_n)_{n \geq 0} = (\phi(X^n))\), and this is exactly a formal power series \(\sum c_n X^n\).

Key categorical point. \(k\llbracket X \rrbracket = \varprojlim_n k[X]/(X^n)\) is the completion of \(k[X]\) at the maximal ideal \((X)\), i.e., the limit (= colimit of the inverse system = limit in the category-theoretic sense) of the diagram of finite approximations. The dual identification \(k[X]^* \cong k\llbracket X \rrbracket\) then follows from the fact that linear functionals on a filtered colimit correspond to compatible sequences of functionals on the finite stages.

8. Regular Monics and Epics 🔱

8.1 Definitions and Hierarchy

Definition (Regular monic). A morphism \(m : A \to B\) in \(\mathcal{C}\) is a regular monomorphism if it arises as an equalizer: there exist morphisms \(f, g : B \rightrightarrows C\) such that \((A, m)\) is the equalizer of \(f\) and \(g\).

Definition (Split monic). A morphism \(m : A \to B\) is a split monomorphism (or section) if there exists \(r : B \to A\) (a retraction) with \(r \circ m = \mathrm{id}_A\).

The following chain of implications holds in any category:

\[\text{split monic} \implies \text{regular monic} \implies \text{monic}\]

Proof of first implication. If \(r \circ m = \mathrm{id}_A\), then \(m\) is the equalizer of \(m \circ r : B \to B\) and \(\mathrm{id}_B\): indeed, \(m \circ r \circ m = m = \mathrm{id}_B \circ m\), so \(m\) factors through this equalizer; conversely, if \(f : X \to B\) satisfies \((m \circ r) \circ f = f\), then \(r \circ f : X \to A\) is the unique factoring since \(m \circ (r \circ f) = f\). \(\square\)

Proof of second implication. If \(m\) is the equalizer of \(f, g : B \rightrightarrows C\), suppose \(m \circ h = m \circ k\). Then \(f \circ m \circ h = g \circ m \circ h\), i.e., \(f \circ m \circ h = g \circ m \circ k\). Since \(m\) is an equalizer, \(h = k\) by the universal property. \(\square\)

The dual hierarchy for epimorphisms: \[\text{split epic} \implies \text{regular epic} \implies \text{epic}\]

Definition (Regular epic). A morphism \(e : A \to B\) is a regular epimorphism if it arises as a coequalizer of some parallel pair \(f, g : C \rightrightarrows A\).

Definition (Split epic). A morphism \(e : A \to B\) is a split epimorphism if there exists \(s : B \to A\) with \(e \circ s = \mathrm{id}_B\).

8.2 Characterizations and Examples

Proposition (Isomorphisms from epic + regular monic). A morphism \(f : A \to B\) is an isomorphism if and only if it is both an epimorphism and a regular monomorphism.

Proof sketch. (\(\Rightarrow\)) Isomorphisms are both epic and (split, hence regular) monic. (\(\Leftarrow\)) Suppose \(f\) is the equalizer of \(g, h : B \rightrightarrows C\) and is also epic. From \(g \circ f = h \circ f\) and \(f\) being epic, \(g = h\). Then the equalizer of \(g = h\) is the identity \(\mathrm{id}_B\), so \(f\) is isomorphic to the identity, hence an isomorphism. \(\square\)

9. Building Limits from Products and Equalizers 🏗️

The following theorem shows that one need not check all limit shapes separately: a small number of “primitive” limit types suffice to generate all finite limits.

Theorem (Finite limits from products and equalizers). A category \(\mathcal{C}\) has all finite limits if and only if it has: 1. A terminal object (limit of the empty diagram), 2. Binary products, 3. Equalizers of parallel pairs.

Proof sketch. The “only if” direction is trivial. For the “if” direction, let \(D : \mathcal{I} \to \mathcal{C}\) be a finite diagram. Construct the limit as an equalizer of two maps between products, as follows. Form the products \[P = \prod_{i \in \mathrm{ob}(\mathcal{I})} Di, \qquad Q = \prod_{u : i \to j \text{ in } \mathcal{I}} Dj.\] Define two maps \(s, t : P \to Q\): the \(u\)-component of \(s\) is \(\pi_j : P \to Dj\), and the \(u\)-component of \(t\) is \(D(u) \circ \pi_i : P \to Dj\). Then \(\mathrm{eq}(s, t)\) is (canonically isomorphic to) \(\lim D\). \(\square\)

Corollary. A category with a terminal object and all pullbacks has all finite limits.

Proof. Equalizers can be built from pullbacks: the equalizer of \(f, g : A \rightrightarrows B\) is the pullback of \(\langle f, g \rangle : A \to B \times B\) along the diagonal \(\Delta : B \to B \times B\). \(\square\)

\usepackage{tikz-cd}
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\begin{tikzcd}
A \arrow[dr, "\langle f{,}g\rangle"] & \\
B \arrow[r, "\Delta"'] & B \times B
\end{tikzcd}
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The equalizer of \(f, g\) is the pullback of this cospan. In particular, categories with terminal objects and pullbacks have all finite limits.

Dual statement. A category has all finite colimits if and only if it has an initial object, binary coproducts, and coequalizers.

Proposition (Pushouts preserve epics). In a category \(\mathcal{C}\), if \(e : A \twoheadrightarrow B\) is a regular epimorphism and \[\begin{array}{ccc} A & \xrightarrow{f} & C \\ {\scriptstyle e}\downarrow & & \downarrow{\scriptstyle e'} \\ B & \xrightarrow{g} & D \end{array}\] is a pushout square, then \(e' : C \twoheadrightarrow D\) is also a regular epimorphism.

10. Algebraic Categories: Widget and Chad 🔧

A widget is a set \(W\) equipped with: - Distinguished elements \(0, 1 \in W\), - A ternary operation \([-,-,-] : W^3 \to W\), - Unary operations \(\tilde{r} : W \to W\) for each \(r \in \mathbb{Q}\),

satisfying the axioms, for all \(a, b, c, d \in W\): 1. \([a, b, b] = a\), 2. \([a, a, b] = b\), 3. \([a, [a, b, c], d] = [a, b, d]\), 4. \(\tilde{r}([a, b, c]) = [\tilde{r}(a), \tilde{r}(b), \tilde{r}(c)]\).

The category \(\mathbf{Widget}\) has widgets as objects and functions preserving \(0\), \(1\), \([-,-,-]\), and all \(\tilde{r}\) as morphisms. There is a forgetful functor \(U : \mathbf{Widget} \to \mathbf{Set}\).

Theorem. The forgetful functor \(U : \mathbf{Widget} \to \mathbf{Set}\) creates all limits.

Proof sketch. Given a diagram \(D : \mathcal{I} \to \mathbf{Widget}\), the limit of \(UD : \mathcal{I} \to \mathbf{Set}\) is the set \[L = \left\{(w_i)_{i \in \mathcal{I}} \in \prod_{i} W_i \;\middle|\; D(u)(w_i) = w_j \text{ for all } u : i \to j \right\}.\] Define widget structure on \(L\) pointwise: \(0_L = (0_{W_i})_i\), \(1_L = (1_{W_i})_i\), \([(a_i), (b_i), (c_i)] = ([a_i, b_i, c_i])_i\), and \(\tilde{r}((a_i)_i) = (\tilde{r}(a_i))_i\). The closure of \(L\) under these operations follows from the fact that all structure maps \(D(u)\) are widget homomorphisms (they commute with the operations). The resulting cone is the unique lift of the limit cone of \(UD\) to \(\mathbf{Widget}\), and it satisfies the universal property. \(\square\)

Corollary. \(\mathbf{Widget}\) has all limits, and \(U\) preserves them (by the creation theorem from §5.2).

A chad is a simplified version of a widget with fewer operations (e.g., omitting some of the \(\tilde{r}\) or the distinguished elements). The forgetful functor \(\mathbf{Widget} \to \mathbf{Chad}\) similarly creates all limits, by an entirely analogous pointwise argument.

11. Limits as Right Adjoints to the Diagonal 🔄

Definition (Diagonal functor). For a small category \(\mathcal{I}\) and a category \(\mathcal{C}\), the diagonal functor \(\Delta : \mathcal{C} \to [\mathcal{I}, \mathcal{C}]\) sends each object \(c \in \mathcal{C}\) to the constant diagram \(\Delta_c : \mathcal{I} \to \mathcal{C}\) (with \(\Delta_c(i) = c\) for all \(i\) and \(\Delta_c(u) = \mathrm{id}_c\) for all \(u\)), and each morphism \(f : c \to c'\) to the natural transformation \(\Delta_f : \Delta_c \Rightarrow \Delta_{c'}\) with all components equal to \(f\).

Theorem. \(\mathcal{C}\) has all limits of shape \(\mathcal{I}\) if and only if \(\Delta : \mathcal{C} \to [\mathcal{I}, \mathcal{C}]\) has a right adjoint \(\lim_{\mathcal{I}} : [\mathcal{I}, \mathcal{C}] \to \mathcal{C}\).

Proof. The adjunction \(\Delta \dashv \lim_{\mathcal{I}}\) means there is a natural bijection \[[\mathcal{I}, \mathcal{C}]({\Delta_c, D}) \;\cong\; \mathcal{C}(c, \lim_{\mathcal{I}} D).\] The left side is exactly \(\mathrm{Cone}(c, D)\) (natural transformations from the constant functor at \(c\) to \(D\), i.e., cones over \(D\) with apex \(c\)). A right adjoint to \(\Delta\) thus assigns to each diagram \(D\) an object \(\lim D\) representing the functor \(c \mapsto \mathrm{Cone}(c, D)\), which is precisely the universal property of the limit. \(\square\)

Dually. \(\mathcal{C}\) has all colimits of shape \(\mathcal{I}\) if and only if \(\Delta : \mathcal{C} \to [\mathcal{I}, \mathcal{C}]\) has a left adjoint \(\mathrm{colim}_{\mathcal{I}} : [\mathcal{I}, \mathcal{C}] \to \mathcal{C}\).

12. Limits in Functor Categories 🗂️

Let \(\mathcal{C}\) be a small category and \(\mathcal{D}\) a category with all limits of shape \(\mathcal{I}\).

Theorem (Pointwise limits in functor categories). The functor category \([\mathcal{C}, \mathcal{D}]\) has all limits of shape \(\mathcal{I}\), computed pointwise: for a diagram \(\mathbf{F} : \mathcal{I} \to [\mathcal{C}, \mathcal{D}]\) (i.e., a family of functors \(F_i : \mathcal{C} \to \mathcal{D}\) with natural transformations \(\mathbf{F}(u) : F_i \Rightarrow F_j\) for \(u : i \to j\)), the limit functor \(\lim_{\mathcal{I}} \mathbf{F} : \mathcal{C} \to \mathcal{D}\) is defined by \[(\lim_{\mathcal{I}} \mathbf{F})(c) \;=\; \lim_{\mathcal{I}} (F_-(c)) \;=\; \lim_{\mathcal{I}}(i \mapsto F_i(c)).\]

Proof sketch. For each \(c \in \mathcal{C}\), the evaluation functor \(\mathrm{ev}_c : [\mathcal{C}, \mathcal{D}] \to \mathcal{D}\) sends \(\mathbf{F}\) to \(F_i(c)\) and the maps to the corresponding natural transformation components. Since \(\mathcal{D}\) has limits, \(\lim_{\mathcal{I}}(F_-(c))\) exists for each \(c\). One checks that this assignment is functorial in \(c\) (using the fact that limit cones are natural) and satisfies the universal property in \([\mathcal{C}, \mathcal{D}]\). \(\square\)

Monics in presheaf categories. A natural transformation \(\alpha : F \Rightarrow G\) in \([\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]\) is a monomorphism if and only if each component \(\alpha_c : F(c) \to G(c)\) is injective (a monomorphism in \(\mathbf{Set}\)).

Proof. If all \(\alpha_c\) are injective, then for any natural transformations \(\beta, \gamma : H \Rightarrow F\) with \(\alpha \circ \beta = \alpha \circ \gamma\), we get \(\alpha_c \circ \beta_c = \alpha_c \circ \gamma_c\) for all \(c\), and injectivity gives \(\beta_c = \gamma_c\) for all \(c\), so \(\beta = \gamma\). Conversely, if some \(\alpha_c\) is not injective, one can construct \(\beta \neq \gamma\) with \(\alpha \circ \beta = \alpha \circ \gamma\) by precomposing \(\alpha_c\) with two distinct maps into \(F(c)\). \(\square\)

Epics are subtler. Epimorphisms in \([\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]\) need not be pointwise surjective. Surprisingly, a natural transformation can be epic without all its components being surjective — see e.g. the inclusion of a non-surjective dense subfunctor.

13. Representable Functors Preserve Limits 🔍

Theorem. For any object \(A \in \mathcal{C}\), the representable functor \(H_A = \mathcal{C}(A, -) : \mathcal{C} \to \mathbf{Set}\) preserves all limits that exist in \(\mathcal{C}\).

Proof. Let \((L, (\pi_i))\) be a limit of \(D : \mathcal{I} \to \mathcal{C}\). We must show \((H_A(L), (H_A(\pi_i)))\) is a limit of \(H_A \circ D\) in \(\mathbf{Set}\), i.e., that the canonical map \[\mathcal{C}(A, L) \xrightarrow{\sim} \varprojlim_{i \in \mathcal{I}} \mathcal{C}(A, Di)\] is a bijection. But this is immediate: a morphism \(f : A \to L\) determines a compatible family \(({\pi_i \circ f : A \to Di})_{i \in \mathcal{I}}\) (a cone over \(D\) from \(A\)), and by the universal property of \(L\), every such family arises from a unique \(f\). \(\square\)

Theorem (Yoneda embedding preserves limits). The Yoneda embedding \(H^- : \mathcal{C} \to [\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]\), defined by \(A \mapsto H^A = \mathcal{C}(-, A)\), preserves all limits.

Proof. By the pointwise description of limits in presheaf categories (§12), a limit of a diagram in \([\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]\) is computed pointwise. For a limit \((L, (\pi_i))\) in \(\mathcal{C}\), the cone \((H^L, (H^{\pi_i}))\) has components \[(H^L)(B) = \mathcal{C}(B, L), \quad (H^{\pi_i})(B) = \mathcal{C}(B, \pi_i).\] At each \(B\), this is a limit cone in \(\mathbf{Set}\) by the previous theorem (applied to \(\mathcal{C}(B, -)\)). Since pointwise limit cones are limit cones in the functor category, \(H^-\) preserves limits. \(\square\)

14. Adjoints and Limit Preservation 🔗

This is one of the most important theorems in category theory.

Theorem (Right adjoints preserve limits; left adjoints preserve colimits). Let \(F \dashv G : \mathcal{C} \rightleftharpoons \mathcal{D}\) be an adjunction. - \(G\) (the right adjoint) preserves all limits that exist in \(\mathcal{D}\). - \(F\) (the left adjoint) preserves all colimits that exist in \(\mathcal{C}\).

Proof (right adjoints preserve limits). Let \((L, (\pi_i)_{i \in \mathcal{I}})\) be a limit of \(D : \mathcal{I} \to \mathcal{D}\). We must show that \((GL, (G\pi_i)_{i \in \mathcal{I}})\) is a limit of \(GD : \mathcal{I} \to \mathcal{C}\).

By the representability criterion (§13 applied to \(\mathcal{C}(A, -)\)), it suffices to show that for every \(A \in \mathcal{C}\), the canonical map \[\mathcal{C}(A, GL) \xrightarrow{\;(G\pi_i \circ -)_i\;} \varprojlim_{i \in \mathcal{I}} \mathcal{C}(A, GDi)\] is a bijection. Using the adjunction bijection \(\mathcal{C}(A, GX) \cong \mathcal{D}(FA, X)\) (natural in \(X \in \mathcal{D}\)), this map is naturally identified with: \[\mathcal{D}(FA, L) \xrightarrow{\;(\pi_i \circ -)_i\;} \varprojlim_{i \in \mathcal{I}} \mathcal{D}(FA, Di).\] This is a bijection because \(\mathcal{D}(FA, -)\) is a representable functor, hence preserves all limits by §13. Since the bijection is natural in \(A\), we conclude that \((GL, (G\pi_i))\) is a limit of \(GD\). \(\square\)

Proof (left adjoints preserve colimits). By duality: a left adjoint \(F : \mathcal{C} \to \mathcal{D}\) is a right adjoint in the opposite categories, \(F^{\mathrm{op}} : \mathcal{C}^{\mathrm{op}} \to \mathcal{D}^{\mathrm{op}}\) is right adjoint to \(G^{\mathrm{op}}\). Limits in \(\mathcal{C}^{\mathrm{op}}\) are colimits in \(\mathcal{C}\), so the result follows by applying the previous proof to the opposite adjunction. \(\square\)

15. The Density Theorem and Cartesian Closed Categories 🌐

15.1 The Density Theorem

The category of elements of a presheaf \(X \in [\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]\) is the category \(\int X\) (also written \(\mathbf{el}(X)\)) defined as follows: - Objects: pairs \((A, x)\) with \(A \in \mathcal{C}\) and \(x \in X(A)\). - Morphisms \((A, x) \to (B, y)\): morphisms \(f : A \to B\) in \(\mathcal{C}\) such that \(X(f)(y) = x\) (i.e., \(f^*(y) = x\)). - There is a forgetful functor \(\pi : \int X \to \mathcal{C}\) sending \((A, x) \mapsto A\).

Theorem (Density / co-Yoneda Lemma). Every presheaf \(X \in [\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]\) is a canonical colimit of representable presheaves. Specifically, \[X \;\cong\; \mathrm{colim}_{(A, x) \in \int X} \mathcal{C}(-, A).\] The colimit is indexed by \((\int X)^{\mathrm{op}}\), with the cocone component at \((A, x)\) being the natural transformation \(\mathcal{C}(-, A) \Rightarrow X\) corresponding (via Yoneda) to \(x \in X(A)\).

Proof. We must show that \(X\) represents the functor \([\mathcal{C}^{\mathrm{op}}, \mathbf{Set}](\mathrm{colim}\, \mathcal{C}(-, A), -) \cong \varprojlim_{(A,x) \in \int X} [\mathcal{C}^{\mathrm{op}}, \mathbf{Set}](\mathcal{C}(-, A), -)\). By the Yoneda lemma, \([\mathcal{C}^{\mathrm{op}}, \mathbf{Set}](\mathcal{C}(-, A), Y) \cong Y(A)\) naturally. So the limit becomes \(\varprojlim_{(A, x) \in \int X} Y(A)\). One verifies that this limit is naturally isomorphic to \([\mathcal{C}^{\mathrm{op}}, \mathbf{Set}](X, Y)\) via the Yoneda correspondence — a natural transformation \(X \Rightarrow Y\) assigns to each \((A, x)\) an element \(\alpha_A(x) \in Y(A)\), which is precisely a compatible family indexed by \(\int X\). \(\square\)

15.2 Cartesian Closed Categories

Definition (Cartesian closed category). A category \(\mathcal{C}\) is cartesian closed (a CCC) if: 1. \(\mathcal{C}\) has finite products (including a terminal object \(1\)), 2. For each pair of objects \(A, B \in \mathcal{C}\), there exists an exponential object \(B^A\) (also written \([A, B]\)) together with an evaluation morphism \(\mathrm{ev} : B^A \times A \to B\), such that for every \(C \in \mathcal{C}\), the function \[\mathcal{C}(C \times A, B) \xrightarrow{\sim} \mathcal{C}(C, B^A), \qquad f \mapsto \lambda f,\] is a bijection, natural in \(C\). Here \(\lambda f\) is the unique morphism with \(\mathrm{ev} \circ (\lambda f \times \mathrm{id}_A) = f\).

Equivalently: for each \(A\), the functor \(- \times A : \mathcal{C} \to \mathcal{C}\) has a right adjoint \((-)^A : \mathcal{C} \to \mathcal{C}\): \[- \times A \;\dashv\; (-)^A.\]

Presheaf categories are cartesian closed. For \(F, G \in [\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]\), the exponential \(G^F\) is defined by \[G^F(A) = [\mathcal{C}^{\mathrm{op}}, \mathbf{Set}](\mathcal{C}(-, A) \times F, G) \;\cong\; [\mathcal{C}^{\mathrm{op}}, \mathbf{Set}](F, G^{\mathcal{C}(-, A)}),\] or equivalently, by the Yoneda lemma: \[G^F(A) = \mathrm{Nat}(\mathcal{C}(-, A) \times F, G).\] When \(F = \mathcal{C}(-, B)\) is representable, the Yoneda lemma gives \(G^F(A) \cong G(A)^{F(A)}\)… wait, more precisely: \(G^{\mathcal{C}(-, B)}(A) \cong \mathrm{Nat}(\mathcal{C}(-, A) \times \mathcal{C}(-, B), G) \cong G(A \times B)\) (when \(\mathcal{C}\) has products).

Proposition. If \(\mathcal{C}\) is a cartesian closed category, then the Yoneda embedding \(H^- : \mathcal{C} \to [\mathcal{C}^{\mathrm{op}}, \mathbf{Set}]\) preserves finite products and exponentials.

Proof sketch. Preservation of products: \(H^{A \times B}(C) = \mathcal{C}(C, A \times B) \cong \mathcal{C}(C, A) \times \mathcal{C}(C, B) = (H^A \times H^B)(C)\) naturally in \(C\). Preservation of exponentials: \(H^{B^A}(C) = \mathcal{C}(C, B^A) \cong \mathcal{C}(C \times A, B)\), and \((H^B)^{H^A}(C) = \mathrm{Nat}(\mathcal{C}(-, C) \times \mathcal{C}(-, A), \mathcal{C}(-, B)) \cong \mathcal{C}(C \times A, B)\) by Yoneda. \(\square\)

16. Stability of Morphism Classes 📊

For the six classes of morphisms — monics, regular monics, split monics, epics, regular epics, split epics — we ask two questions: - (C) Is the class closed under composition? - (P) Is the class stable under pullback (i.e., if \(f\) is in the class, is its pullback along any map also in the class)?

The following table summarizes the answers:

Justifications (selected):

Monics are closed under composition. If \(f, g\) are monic and \(f \circ g \circ h = f \circ g \circ k\), then \(g \circ h = g \circ k\) (by \(f\) monic), then \(h = k\) (by \(g\) monic).

Monics are stable under pullback. If \(m : A \to B\) is monic and the square \[\begin{array}{ccc} A' & \xrightarrow{m'} & B' \\ \downarrow & & \downarrow \\ A & \xrightarrow{m} & B \end{array}\] is a pullback, then \(m'\) is monic: if \(m' \circ h = m' \circ k\), then both \((h, k)\) induce the same morphism into the pullback corner, so they agree.

Epics are not pullback stable, even in Set. Let \(e : \{a, b\} \twoheadrightarrow \{*\}\) be the unique surjection (an epic/split epic in \(\mathbf{Set}\)). Pull it back along \(s : \{*\} \to \{*\}\) (say the identity): the pullback is \(\{a, b\} \to \{*\}\), which is still surjective. However, consider a non-surjective map \(i : \emptyset \to \{*\}\): the pullback of \(e\) along \(i\) is \(\emptyset \to \emptyset\), which is vacuously surjective. More interesting: in \(\mathbf{Top}\), the quotient map \([0,1] \twoheadrightarrow S^1\) is a regular epic; its pullback along the inclusion of a point \(\{p\} \hookrightarrow S^1\) is a single fiber, which need not be surjective.

Split monics are closed under composition. If \(r \circ m = \mathrm{id}\) and \(s \circ n = \mathrm{id}\), then \((m \circ s) \circ (n \circ r) = m \circ (s \circ n) \circ r = m \circ r = \mathrm{id}\)… wait: \((r \circ s) \circ (n \circ m) = r \circ (s \circ n) \circ m = r \circ m = \mathrm{id}\). So \(n \circ m\) is split monic with retraction \(r \circ s\).

17. Optional: Homological Algebra via Category Theory 🧬

This section sketches how the categorical machinery of limits, colimits, projective and injective objects organizes the foundations of homological algebra.

An abelian category is an additive category in which every morphism has a kernel and cokernel, every monic is a kernel, and every epic is a cokernel. Examples: \(\mathbf{Ab}\), \(R\text{-}\mathbf{Mod}\), \(\mathrm{QCoh}(X)\).

In an abelian category \(\mathcal{A}\), a chain complex \((C_\bullet, \partial)\) is a sequence \[\cdots \to C_{n+1} \xrightarrow{\partial_{n+1}} C_n \xrightarrow{\partial_n} C_{n-1} \to \cdots\] with \(\partial_n \circ \partial_{n+1} = 0\). The \(n\)-th homology is \(H_n(C_\bullet) = \ker \partial_n / \mathrm{im}\, \partial_{n+1}\), which is well-defined in any abelian category.

Projective resolutions. A projective resolution of \(A \in \mathcal{A}\) is an exact sequence \[\cdots \to P_2 \to P_1 \to P_0 \to A \to 0\] with each \(P_n\) projective. Such resolutions exist whenever \(\mathcal{A}\) has enough projectives (every object admits an epic from a projective).

Derived functors. Let \(T : \mathcal{A} \to \mathcal{B}\) be a right-exact additive functor. The left derived functors \(L_n T\) are defined by: choose a projective resolution \(P_\bullet \to A\), apply \(T\) to get a complex \(T(P_\bullet)\), then \(L_n T(A) = H_n(T(P_\bullet))\). The key fact is that this is independent of the choice of projective resolution.

Ext and Tor. For \(A \in \mathcal{A}\), the functors \(\mathrm{Ext}^n(A, -)\) are the right derived functors of \(\mathcal{A}(A, -) : \mathcal{A} \to \mathbf{Ab}\): \[\mathrm{Ext}^n(A, B) = R^n \mathcal{A}(A, -)(B) = H^n(\mathcal{A}(P_\bullet, B)),\] where \(P_\bullet \to A\) is a projective resolution of \(A\). For \(R\)-modules, \(\mathrm{Tor}_n(A, -)\) is the \(n\)-th left derived functor of \(A \otimes_R -\).

Computation. In \(\mathbf{Ab}\), the projective resolution of \(\mathbb{Z}/n\mathbb{Z}\) is \[0 \to \mathbb{Z} \xrightarrow{\times n} \mathbb{Z} \to \mathbb{Z}/n\mathbb{Z} \to 0.\] Applying \(\mathrm{Hom}(-, \mathbb{Z})\) gives \[0 \to \mathbb{Z} \xrightarrow{\times n} \mathbb{Z} \to 0,\] from which \(\mathrm{Ext}^0(\mathbb{Z}/n\mathbb{Z}, \mathbb{Z}) = 0\) (since \(\mathbb{Z}/n\mathbb{Z}\) has no maps to \(\mathbb{Z}\)) and \(\mathrm{Ext}^1(\mathbb{Z}/n\mathbb{Z}, \mathbb{Z}) \cong \mathbb{Z}/n\mathbb{Z}\) (the cokernel of \(\times n : \mathbb{Z} \to \mathbb{Z}\)).

18. The Art of the Diagram Chase 🧩

A diagram chase is a proof technique in which one establishes a result by following elements through a commutative diagram, using commutativity and exactness conditions to derive the conclusion. The technique is most powerful in abelian categories, where kernels, cokernels, and exact sequences are available, but it generalizes to any regular category via generalized elements.

The Five Lemma

Consider a commutative diagram of abelian groups with exact rows:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
A_1 \arrow[r, "\alpha_1"] \arrow[d, "f_1"'] & A_2 \arrow[r, "\alpha_2"] \arrow[d, "f_2"'] & A_3 \arrow[r, "\alpha_3"] \arrow[d, "f_3"'] & A_4 \arrow[r, "\alpha_4"] \arrow[d, "f_4"'] & A_5 \arrow[d, "f_5"'] \\
B_1 \arrow[r, "\beta_1"'] & B_2 \arrow[r, "\beta_2"'] & B_3 \arrow[r, "\beta_3"'] & B_4 \arrow[r, "\beta_4"'] & B_5
\end{tikzcd}
\end{document}

Theorem (Five Lemma). In the diagram above with exact rows: - If \(f_1\) is epic and \(f_2, f_4\) are isomorphisms, then \(f_3\) is monic. - If \(f_5\) is monic and \(f_2, f_4\) are isomorphisms, then \(f_3\) is epic. - Hence if \(f_1, f_2, f_4, f_5\) are all isomorphisms, then \(f_3\) is an isomorphism.

Proof (injectivity of \(f_3\)). Suppose \(a_3 \in A_3\) with \(f_3(a_3) = 0\). Apply \(\beta_3 \circ f_3 = f_4 \circ \alpha_3\) to get \(f_4(\alpha_3(a_3)) = 0\). Since \(f_4\) is injective, \(\alpha_3(a_3) = 0\). By exactness of the top row at \(A_3\), there exists \(a_2 \in A_2\) with \(\alpha_2(a_2) = a_3\). Now \(\beta_2(f_2(a_2)) = f_3(\alpha_2(a_2)) = f_3(a_3) = 0\). By exactness of the bottom row at \(B_2\), there exists \(b_1 \in B_1\) with \(\beta_1(b_1) = f_2(a_2)\). Since \(f_1\) is surjective, there exists \(a_1 \in A_1\) with \(f_1(a_1) = b_1\). Then \(f_2(\alpha_1(a_1)) = \beta_1(f_1(a_1)) = \beta_1(b_1) = f_2(a_2)\). Since \(f_2\) is injective, \(\alpha_1(a_1) = a_2\). By exactness at \(A_2\), \(a_3 = \alpha_2(a_2) = \alpha_2(\alpha_1(a_1)) = 0\). \(\square\)

Proof (surjectivity of \(f_3\)). Let \(b_3 \in B_3\). Apply \(\beta_3\) to get \(\beta_3(b_3) \in B_4\). Since \(f_4\) is surjective, write \(\beta_3(b_3) = f_4(a_4)\) for some \(a_4 \in A_4\). Then \(f_5(\alpha_4(a_4)) = \beta_4(f_4(a_4)) = \beta_4(\beta_3(b_3)) = 0\) (by exactness at \(B_4\)). Since \(f_5\) is injective, \(\alpha_4(a_4) = 0\), so \(a_4 \in \ker \alpha_4 = \mathrm{im}\, \alpha_3\). Write \(a_4 = \alpha_3(a_3)\). Then \(\beta_3(f_3(a_3)) = f_4(\alpha_3(a_3)) = f_4(a_4) = \beta_3(b_3)\), so \(b_3 - f_3(a_3) \in \ker \beta_3 = \mathrm{im}\, \beta_2\). Write \(b_3 - f_3(a_3) = \beta_2(b_2)\). Since \(f_2\) is surjective, \(b_2 = f_2(a_2)\). Then \(f_3(a_3 + \alpha_2(a_2)) = f_3(a_3) + f_3(\alpha_2(a_2)) = f_3(a_3) + \beta_2(f_2(a_2)) = f_3(a_3) + (b_3 - f_3(a_3)) = b_3\). \(\square\)

The Short Five Lemma

Theorem (Short Five Lemma). Consider a commutative diagram with exact rows:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
0 \arrow[r] & A \arrow[r, "i"] \arrow[d, "f"'] & B \arrow[r, "p"] \arrow[d, "g"'] & C \arrow[r] \arrow[d, "h"'] & 0 \\
0 \arrow[r] & A' \arrow[r, "i'"'] & B' \arrow[r, "p'"'] & C' \arrow[r] & 0
\end{tikzcd}
\end{document}

If \(f\) and \(h\) are isomorphisms, then \(g\) is an isomorphism.

Proof sketch. This is a special case of the Five Lemma with \(A_1 = B_1 = 0\) and \(A_5 = B_5 = 0\). Alternatively: for injectivity of \(g\), suppose \(g(b) = 0\). Then \(h(p(b)) = p'(g(b)) = 0\), so \(p(b) = 0\) (since \(h\) is monic), thus \(b = i(a)\) for some \(a\). Then \(i'(f(a)) = g(i(a)) = g(b) = 0\), so \(f(a) = 0\) (since \(i'\) is monic), so \(a = 0\) and \(b = 0\). For surjectivity: given \(b' \in B'\), map \(p'(b') \in C'\); since \(h\) is surjective, \(p'(b') = h(c)\) for some \(c \in C\); choose a preimage \(b \in B\) with \(p(b) = c\) (using surjectivity of \(p\)); then \(p'(b' - g(b)) = 0\), so \(b' - g(b) = i'(a')\) for some \(a'\); since \(f\) is surjective, \(a' = f(a)\), giving \(b' = g(b) + i'(f(a)) = g(b + i(a)) = g(b + i(a))\). \(\square\)

The Snake Lemma

The Snake Lemma is the engine of connecting homomorphisms in long exact sequences of homology.

Theorem (Snake Lemma). Given a commutative diagram of abelian groups with exact rows:

\usepackage{tikz-cd}
\begin{document}
\begin{tikzcd}
& A \arrow[r, "f"] \arrow[d, "\alpha"'] & B \arrow[r, "g"] \arrow[d, "\beta"'] & C \arrow[r] \arrow[d, "\gamma"'] & 0 \\
0 \arrow[r] & A' \arrow[r, "f'"'] & B' \arrow[r, "g'"'] & C'
\end{tikzcd}
\end{document}

there is an exact sequence \[\ker \alpha \xrightarrow{\bar f} \ker \beta \xrightarrow{\bar g} \ker \gamma \xrightarrow{\;\delta\;} \mathrm{coker}\, \alpha \xrightarrow{\bar f'} \mathrm{coker}\, \beta \xrightarrow{\bar g'} \mathrm{coker}\, \gamma,\] where \(\delta\) is the connecting homomorphism defined by the diagram chase: given \(c \in \ker \gamma\), choose \(b \in B\) with \(g(b) = c\); then \(g'(\beta(b)) = \gamma(g(b)) = \gamma(c) = 0\), so \(\beta(b) \in \ker g' = \mathrm{im}\, f'\); write \(\beta(b) = f'(a')\) and set \(\delta(c) = [a'] \in A'/\mathrm{im}\, \alpha\).

[!EXAMPLE]- Snake Lemma applied: long exact sequence of homology The Snake Lemma is the key tool in proving that a short exact sequence of chain complexes \(0 \to A_\bullet \to B_\bullet \to C_\bullet \to 0\) gives a long exact sequence in homology: \[\cdots \to H_n(A) \to H_n(B) \to H_n(C) \xrightarrow{\delta} H_{n-1}(A) \to \cdots\] The connecting homomorphism \(\delta\) is defined by applying the Snake Lemma to the short exact sequence \(0 \to Z_n(A) \to Z_n(B) \to Z_n(C)\) and \(B_{n-1}(A) \to B_{n-1}(B) \to B_{n-1}(C) \to 0\), where \(Z_n, B_n\) denote cycles and boundaries.

Diagram Chasing in General Categories

In non-abelian settings, one cannot refer to elements of objects. Instead, one works with generalized elements: a morphism \(x : T \to A\) is a “\(T\)-shaped element of \(A\).” Commutativity \(f \circ x = g \circ x\) for all \(T\)-elements (all \(T\) and all \(x : T \to A\)) implies \(f = g\) — this is precisely the Yoneda lemma applied to the natural bijection \(\mathcal{C}(T, A) \cong \mathcal{C}(T, A)\).

Definition (Generalized element). A generalized element of \(A \in \mathcal{C}\) of stage \(T\) is a morphism \(x : T \to A\).

💡 In this language, a cone \((L, (\pi_i))\) is a compatible family of generalized elements \(\pi_i : L \to Di\), and the universal property of the limit says that any other such compatible family factors uniquely through \(L\).

This formalism allows diagram chasing in any category, though the arguments become more subtle without exact sequences.

The Salamander Lemma (Brief Mention)

The Salamander Lemma (due to Bergman) is a powerful generalization: for any commutative diagram of abelian groups, the cohomology at any spot (the quotient of the kernel by the image of incoming maps) fits into exact sequences relating it to the cohomology of adjacent spots. Riehl (§1.6) gives a clean treatment. The Snake and Five Lemmas become immediate corollaries.

19. Complete and Cocomplete Categories 🌐

Definition (Complete and cocomplete). A category \(\mathcal{C}\) is complete if it has all small limits — limits of diagrams \(D : \mathcal{I} \to \mathcal{C}\) where \(\mathcal{I}\) is a small category. \(\mathcal{C}\) is cocomplete if it has all small colimits.

The Completeness Theorem

Theorem (Completeness from products and equalizers). A category \(\mathcal{C}\) is complete if and only if it has all small products and all equalizers. Dually, \(\mathcal{C}\) is cocomplete if and only if it has all small coproducts and all coequalizers.

Proof sketch. The “only if” direction is trivial. For “if”: given any small diagram \(D : \mathcal{I} \to \mathcal{C}\), the limit is constructed as an equalizer of two maps between products, exactly as in the finite case (§9), but now using small (possibly infinite) products. Specifically, form \[P = \prod_{i \in \mathrm{ob}(\mathcal{I})} Di, \qquad Q = \prod_{u : i \to j \text{ in } \mathcal{I}} Dj,\] define \(s, t : P \to Q\) by \(s_u = \pi_j\) and \(t_u = D(u) \circ \pi_i\), then \(\lim D = \mathrm{eq}(s, t)\). The small product and equalizer exist by hypothesis. \(\square\)

This shows that the “generating” limit shapes are products (discrete diagrams) and equalizers (parallel pairs).

Examples

Completeness via Generating Limit Shapes

The theorem above shows that one can reduce all small limits to products and equalizers. A refinement: a category has all finite limits iff it has a terminal object, binary products, and equalizers (§9). The “small” version replaces binary products with small products.

Proposition. The following are equivalent for a category \(\mathcal{C}\): 1. \(\mathcal{C}\) is complete. 2. \(\mathcal{C}\) has all small products and all equalizers. 3. \(\mathcal{C}\) has all small products, a terminal object, and all pullbacks.

Preservation of Completeness

Proposition. Let \(F : \mathcal{C} \to \mathcal{D}\). - If \(F\) creates limits and \(\mathcal{C}\) is complete, then \(\mathcal{D}\) is complete. - If \(\mathcal{D}\) is a reflective subcategory of \(\mathcal{C}\) (inclusion has a left adjoint) and \(\mathcal{C}\) is complete, then \(\mathcal{D}\) is complete. - If \(\mathcal{C}\) is complete, then \([\mathcal{J}, \mathcal{C}]\) is complete for any small \(\mathcal{J}\) (by pointwise limits, §12).

20. Functoriality of Limits ⚙️

Limits as Functors

When \(\mathcal{C}\) has all limits of shape \(\mathcal{I}\), the assignment \(D \mapsto \lim D\) defines a functor \[\lim_{\mathcal{I}} : [\mathcal{I}, \mathcal{C}] \to \mathcal{C}.\] This is the right adjoint to the diagonal functor \(\Delta : \mathcal{C} \to [\mathcal{I}, \mathcal{C}]\) (established in §11). Functoriality means: given a natural transformation \(\alpha : D \Rightarrow D'\) in \([\mathcal{I}, \mathcal{C}]\), there is an induced map \[\lim \alpha : \lim D \to \lim D',\] the unique morphism such that \(\pi'_i \circ (\lim \alpha) = \alpha_i \circ \pi_i\) for each \(i \in \mathcal{I}\), where \(\pi, \pi'\) are the respective limit projections. This is obtained by applying the universal property of \(\lim D'\) to the cone \((\lim D,\; (\alpha_i \circ \pi_i)_{i \in \mathcal{I}})\) over \(D'\).

flowchart LR
    subgraph "Functor category [I, C]"
        D["D (diagram)"]
        D2["D' (diagram)"]
        D -- "α (nat. trans.)" --> D2
    end
    subgraph "Base category C"
        limD["lim D"]
        limD2["lim D'"]
        limD -- "lim α" --> limD2
    end
    D -- "lim_I (right adjoint to Δ)" --> limD
    D2 --> limD2

Morphisms of Diagrams and Naturality

Proposition. The induced map \(\lim \alpha : \lim D \to \lim D'\) satisfies \[\pi'_i \circ (\lim \alpha) = \alpha_i \circ \pi_i \quad \text{for each } i \in \mathcal{I}.\] Moreover, \(\lim(-)\) is functorial: \(\lim(\beta \circ \alpha) = (\lim \beta) \circ (\lim \alpha)\) and \(\lim(\mathrm{id}_D) = \mathrm{id}_{\lim D}\).

Proof. By the universal property of \(\lim D'\) applied to the cone over \(D'\) with apex \(\lim D\) and legs \(\alpha_i \circ \pi_i\). The cone condition holds: for \(u : i \to j\) in \(\mathcal{I}\), \(D'(u) \circ (\alpha_i \circ \pi_i) = \alpha_j \circ D(u) \circ \pi_i = \alpha_j \circ \pi_j\) (using naturality of \(\alpha\) and the cone condition for \(\pi\)). Uniqueness gives functoriality. \(\square\)

The Limit Functor Preserves Limits

Theorem. The limit functor \(\lim_{\mathcal{I}} : [\mathcal{I}, \mathcal{C}] \to \mathcal{C}\) is a right adjoint (to \(\Delta\)), hence by RAPL (§14) it preserves all limits that exist in \([\mathcal{I}, \mathcal{C}]\).

Corollary (“Fubini for limits”). A limit of limits is a limit. More precisely, if \(D : \mathcal{I} \times \mathcal{J} \to \mathcal{C}\) is a bifunctor and \(\mathcal{C}\) has all limits of shapes \(\mathcal{I}\), \(\mathcal{J}\), and \(\mathcal{I} \times \mathcal{J}\), then there is a natural isomorphism \[\lim_{\mathcal{I}} \lim_{\mathcal{J}} D(-,-) \;\cong\; \lim_{\mathcal{I} \times \mathcal{J}} D \;\cong\; \lim_{\mathcal{J}} \lim_{\mathcal{I}} D(-,-).\]

[!EXAMPLE]- Fubini for products The Fubini isomorphism for products says \(\prod_{i \in \mathcal{I}} \prod_{j \in \mathcal{J}} D_{ij} \cong \prod_{(i,j) \in \mathcal{I} \times \mathcal{J}} D_{ij} \cong \prod_{j \in \mathcal{J}} \prod_{i \in \mathcal{I}} D_{ij}\), which is the familiar commutativity and associativity of Cartesian products of sets.

Continuity of Hom-Functors

Theorem. For any \(A \in \mathcal{C}\), the hom-functor \(\mathcal{C}(A, -) : \mathcal{C} \to \mathbf{Set}\) preserves all limits: \[\mathcal{C}(A, \lim_{\mathcal{I}} D) \;\cong\; \lim_{\mathcal{I}} \mathcal{C}(A, D(-)).\]

This was already proved in §13. The functoriality of limits strengthens it: the isomorphism is natural in both \(A\) and \(D\), making \(\mathcal{C}(-, -)\) a continuous bifunctor (contravariant in the first argument, covariant in the second).

Definition (Continuous functor). A functor \(F : \mathcal{C} \to \mathcal{D}\) is continuous if it preserves all small limits. Equivalently: for every small diagram \(D : \mathcal{I} \to \mathcal{C}\) and limit \((L, (\pi_i))\) in \(\mathcal{C}\), the cone \((F(L), (F\pi_i))\) is a limit of \(F \circ D\) in \(\mathcal{D}\).

📐 Every right adjoint is continuous (RAPL). Every representable functor is continuous. The forgetful functor \(\mathbf{Grp} \to \mathbf{Set}\) is continuous (it is a right adjoint to the free group functor).

21. Interactions between Limits and Colimits 🔄

Limits and Colimits Do Not Generally Commute

The canonical comparison map \[\mathrm{colim}_{j \in \mathcal{J}}\, \lim_{i \in \mathcal{I}} D(i, j) \;\longrightarrow\; \lim_{i \in \mathcal{I}}\, \mathrm{colim}_{j \in \mathcal{J}} D(i, j)\] exists for any bifunctor \(D : \mathcal{I} \times \mathcal{J} \to \mathcal{C}\), but it is rarely an isomorphism in general. The direction of this map (colim of lim maps to lim of colim) reflects the fact that limits “commute past” colimits in this canonical direction.

[!EXAMPLE]- Counterexample in Set Let \(\mathcal{J} = \{0, 1\}\) (discrete, two objects) and \(\mathcal{I} = \{0 \to 1\}\) (the walking arrow). Consider the diagram \(D : \mathcal{I} \times \mathcal{J} \to \mathbf{Set}\) given by: - \(D(0, 0) = \{a\}\), \(D(0, 1) = \{b\}\) (disjoint singletons) - \(D(1, 0) = D(1, 1) = \{*\}\) (one-point set) - Maps \(D(0,j) \to D(1,j)\) are the unique maps.

Then \(\lim_{\mathcal{I}} D(-,0) = \{a\}\) and \(\lim_{\mathcal{I}} D(-,1) = \{b\}\), so \(\mathrm{colim}_{\mathcal{J}} \lim_{\mathcal{I}} D = \{a, b\}\) (a two-element set).

On the other hand, \(\mathrm{colim}_{\mathcal{J}} D(0,-) = \{a, b\}\) and \(\mathrm{colim}_{\mathcal{J}} D(1,-) = \{*\}\), so \(\lim_{\mathcal{I}} \mathrm{colim}_{\mathcal{J}} D = \{*\}\) (the limit of the map \(\{a,b\} \to \{*\}\), which is \(\{*\}\)).

Thus \(\mathrm{colim}\, \lim = \{a,b\} \not\cong \{*\} = \lim\, \mathrm{colim}\), and the canonical map \(\{a,b\} \to \{*\}\) is not an isomorphism.

Filtered Colimits

Definition (Filtered category). A category \(\mathcal{J}\) is filtered if: 1. \(\mathcal{J}\) is non-empty, 2. Any two objects \(i, j \in \mathcal{J}\) have a common upper bound: there exists \(k\) with morphisms \(i \to k\) and \(j \to k\), 3. Any two parallel morphisms \(f, g : i \rightrightarrows j\) have a common coequalization: there exists \(k\) and \(h : j \to k\) with \(h \circ f = h \circ g\).

Filtered colimits generalize directed colimits (colimits over directed posets, which satisfy conditions 1 and 2 but not necessarily 3).

Filtered Colimits Commute with Finite Limits in Set

Theorem (Filtered colimits commute with finite limits). In \(\mathbf{Set}\), if \(\mathcal{J}\) is filtered and \(\mathcal{I}\) is finite, then the canonical comparison map \[\mathrm{colim}_{j \in \mathcal{J}}\, \lim_{i \in \mathcal{I}} D(i,j) \;\xrightarrow{\;\sim\;}\; \lim_{i \in \mathcal{I}}\, \mathrm{colim}_{j \in \mathcal{J}} D(i,j)\] is an isomorphism for any \(D : \mathcal{I} \times \mathcal{J} \to \mathbf{Set}\).

Proof sketch (for binary products, \(\mathcal{I} = \{0, 1\}\)). An element of \(\mathrm{colim}_{j} (D(0,j) \times D(1,j))\) is an equivalence class \([(j, x, y)]\) with \(x \in D(0,j)\) and \(y \in D(1,j)\). An element of \(\mathrm{colim}_{j} D(0,j) \times \mathrm{colim}_{j} D(1,j)\) is a pair \(([(j_1, x)], [(j_2, y)])\) with \(x \in D(0,j_1)\) and \(y \in D(1,j_2)\). Since \(\mathcal{J}\) is filtered, there exists \(k\) with morphisms \(j_1 \to k\) and \(j_2 \to k\); pushing forward, we can represent both \(x\) and \(y\) at stage \(k\), giving \([(k, x', y')]\). This defines the inverse map. \(\square\)

Proof sketch (for equalizers). Given \([(j, b)]\) in \(\mathrm{colim}_j \mathrm{eq}(f_j, g_j)\), one shows the image in \(\mathrm{colim}_j D(j)\) lies in the equalizer of the colimit maps, using the filteredness to coequalize the witnesses.

🔑 This theorem is the key to understanding the structure of algebraic categories.

Algebraic Importance

Proposition (Algebras are filtered colimits of finitely presented ones). Every group (ring, module, algebra over a theory \(\mathbb{T}\), …) is a filtered colimit of finitely presented ones. Specifically, given a group \(G\), its finitely generated subgroups form a directed system (ordered by inclusion), and \(G = \varinjlim_{H \subseteq G,\, H \text{ f.g.}} H\).

A finitely presented algebra is one with finitely many generators and finitely many relations. The class of all \(\mathbb{T}\)-algebras that are finitely presented is essentially the “compact objects” of the category.

Sifted Colimits

Definition (Sifted colimit). A colimit is sifted if the index category \(\mathcal{J}\) has the property that the diagonal \(\mathcal{J} \to \mathcal{J} \times \mathcal{J}\) is final (cofinal). Equivalently, \(\mathcal{J}\) is sifted iff \(\mathrm{colim}_{\mathcal{J}}\) commutes with finite products in \(\mathbf{Set}\).

Filtered colimits and reflexive coequalizers (coequalizers of parallel pairs that have a common section) are both sifted. The class of sifted colimits characterizes algebraic categories: a functor \(\mathbf{Set}^{\mathrm{op}} \to \mathbf{Set}\) is representable by an algebraic theory iff it preserves sifted colimits.

Summary: When Limits and Colimits Commute

References