7.2 Definition and examples of limits

Limits in category theory are universal constructions that capture common patterns across mathematical structures. They formalize ideas like products, equalizers, and pullbacks, providing a unified framework for understanding these concepts.

From sets to groups to topological spaces, limits show up in various forms. By studying their properties and calculations, we gain insights into the underlying structures and relationships between different mathematical objects.

Definition and Examples of Limits

Formal definition of limits

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• A limit of a diagram $D: J \to C$ consists of an object $L$ in $C$ together with a family of morphisms $\{\pi_j: L \to D(j)\}_{j \in J}$ satisfying the following conditions:
  • For every morphism $f: j \to j'$ in $J$, the diagram commutes $D(f) \circ \pi_j = \pi_{j'}$ ensures consistency among the morphisms
  • For any other object $X$ with a family of morphisms $\{\psi_j: X \to D(j)\}_{j \in J}$ making the corresponding diagrams commute, there exists a unique morphism $u: X \to L$ such that $\pi_j \circ u = \psi_j$ for all $j \in J$ guarantees the universality of the limit

Examples across categories

• In the category of sets:
  • Products form limits where the product of sets $A$ and $B$ is the set $A \times B = \{(a, b) | a \in A, b \in B\}$ with projection morphisms $\pi_1: A \times B \to A$ and $\pi_2: A \times B \to B$ (Cartesian product)
  • Equalizers form limits where given functions $f, g: A \to B$, the equalizer is the subset $E = \{a \in A | f(a) = g(a)\}$ with the inclusion morphism $i: E \to A$ (kernel of a homomorphism)
• In the category of groups:
  • Products form limits where the product of groups $G$ and $H$ is the direct product $G \times H$ with componentwise multiplication and projection homomorphisms (direct product)
  • Pullbacks form limits where given group homomorphisms $f: A \to C$ and $g: B \to C$, the pullback is the subgroup of $A \times B$ consisting of pairs $(a, b)$ such that $f(a) = g(b)$ (fiber product)
• In the category of topological spaces:
  • Products form limits where the product of spaces $X$ and $Y$ is the space $X \times Y$ with the product topology and projection continuous maps (product topology)
  • Inverse limits form limits where given a directed system of spaces $\{X_i, f_{ij}: X_j \to X_i\}$, the inverse limit is the subspace of $\prod X_i$ consisting of points $(x_i)$ such that $f_{ij}(x_j) = x_i$ for all $i \leq j$ (projective limit)

Limits and universal properties

• Limits are characterized by a universal property where:
  • The morphisms $\{\pi_j: L \to D(j)\}_{j \in J}$ are "universal" among all families of morphisms from an object to the diagram $D$ that make the corresponding diagrams commute
  • This universal property uniquely determines the limit up to isomorphism guarantees uniqueness
• Examples of universal properties related to limits include:
  • Products have morphisms $\pi_1: A \times B \to A$ and $\pi_2: A \times B \to B$ that are universal among all pairs of morphisms from an object to $A$ and $B$ separately (universal property of the product)
  • Equalizers have the inclusion morphism $i: E \to A$ that is universal among all morphisms from an object to $A$ that equalize $f$ and $g$ (universal property of the equalizer)

Calculation in specific categories

• In the category of sets:
  • Product of sets $A$ and $B$ is calculated as $A \times B = \{(a, b) | a \in A, b \in B\}$ (Cartesian product)
  • Equalizer of functions $f, g: A \to B$ is calculated as $E = \{a \in A | f(a) = g(a)\}$ (kernel)
• In the category of groups:
  • Product of groups $G$ and $H$ is calculated as $G \times H$ with componentwise multiplication (direct product)
  • Pullback of group homomorphisms $f: A \to C$ and $g: B \to C$ is calculated as the subgroup of $A \times B$ consisting of pairs $(a, b)$ such that $f(a) = g(b)$ (fiber product)
• In the category of topological spaces:
  • Product of spaces $X$ and $Y$ is calculated as $X \times Y$ with the product topology (product space)
  • Inverse limit of a directed system $\{X_i, f_{ij}: X_j \to X_i\}$ is calculated as the subspace of $\prod X_i$ consisting of points $(x_i)$ such that $f_{ij}(x_j) = x_i$ for all $i \leq j$ (projective limit)