Products and coproducts as limits and colimits of discrete diagrams, including their existence-and-uniqueness equations

Depends on

• Limits and colimits as terminal cones and initial cocones, with existence and uniqueness in their universal properties
• An indexed family $(A_i)_{i \in I}$ is a function with domain $I$; $\{A_i : i \in I\}$ is its range

Used by

• A limit in a full subcategory need not be the ambient limit Counterexample
• A reflexive coequalizer of sets not preserved by Set(ℕ,-) Counterexample
• Assuming Choice, nonempty sets have all small products but a parallel pair with no equalizer and hence a diagram with no limit Counterexample
• Filtered colimits in Set need not commute with countably infinite products Counterexample
• Canonical morphism from a finite coproduct to a finite product Definition
• Direct sum total complex of a double complex Definition
• Exponential object Definition
• Product total complex of a double complex Definition
• Products of classical algebraic sets and their universal property Definition
• In a poset regarded as a category, products are infima, coproducts are suprema, and equalizers are automatic Example
• Products in Set are Cartesian products and coproducts are tagged disjoint unions Example
• Under the definable-class diagram convention, the empty set is the product of the large family of all sets Example
• FALSE: every functor on CᵒᵖtimesC has an end False statement
• FALSE: every functor preserves the ends that exist in its domain False statement
• Associator, symmetry and unitors of the abelian sheaf tensor product Lemma
• Countable sequence groups and tail filtrations Lemma
• The total differential squares to zero Lemma
• A poset category is complete exactly when every small family has an infimum, and cocomplete exactly when every small family has a supremum Proposition
• Biproducts are associative, commutative, and unital up to canonical isomorphism Proposition
• Products and coproducts of complexes are degreewise when they exist and preserve differentials Proposition
• Sum and product totalisations agree on finite diagonal double complexes Proposition
• A cartesian closed preorder has relative implications Theorem
• A category with finite products is monoidal Theorem
• A power by a set is the product of that many copies and a copower is the coproduct Theorem
• A set-valued coend is the disjoint union of the diagonal values modulo the dinaturality relation Theorem
• An end is the equalizer of two products, and a coend the coequalizer of two coproducts Theorem
• Assuming Choice, a small category with products or coproducts indexed by the cardinality of its morphism set is a preorder Theorem
• Biproduct data characterisation without addition Theorem
• Every small colimit can be constructed as a coequalizer between coproducts over the arrows and objects of the index category Theorem
• Every small limit can be constructed as an equalizer between products over the objects and arrows of the index category Theorem
• Finite, nonempty finite, and connected finite (co)limit criteria in terms of products, equalizers, pullbacks, terminal objects, and their duals Theorem

Dependency tree · two levels

6 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• E. Riehl, Category Theory in Context, Definitions 3.1.9 and 3.1.13 (standard reference, not scraped)