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Under dependent choice, categories of models for algebraic theories are complete and cocomplete
Statement
Assume dependent choice. Every category of models for an algebraic theory is complete and cocomplete.
Facts & Assumptions
Given: Dependent choice and a category of models for an algebraic theory.
A category of models for an algebraic theory is equipped with a finitary monadic functor to (Categories of models for algebraic theories).
Every small diagram in has a limit (Set has all small limits, realized as compatible tuples in a set-indexed product).
Every small diagram in has a colimit (Set has all small colimits, realized as a quotient of a set-indexed disjoint union).
Equivalences preserve and reflect existing limits and colimits (Equivalences preserve, reflect, and create limits and colimits in the isomorphism-invariant sense).
Under dependent choice, a finitary monad on a complete cocomplete locally small category has a complete and cocomplete Eilenberg–Moore category (Under dependent choice, a finitary monad on a complete cocomplete locally small category has complete and cocomplete algebras).
Proof
By [L2] and [L3], is complete and cocomplete; it is locally small because each function collection between two sets is a set. This includes empty diagrams.
The finitary monad induced by [L1] satisfies the hypotheses of [L5], so under dependent choice its Eilenberg–Moore category is complete and cocomplete.
The comparison supplied by [L1] is an equivalence over . Transporting the limits and colimits of step 2.1 across it by [L4] proves that is complete and cocomplete.
Depends on
- Categories of models for algebraic theories
- Under dependent choice, a finitary monad on a complete cocomplete locally small category has complete and cocomplete algebras
- Set has all small limits, realized as compatible tuples in a set-indexed product
- Set has all small colimits, realized as a quotient of a set-indexed disjoint union
- Equivalences preserve, reflect, and create limits and colimits in the isomorphism-invariant sense
- The axiom of dependent choice: a relation in which every element is related to something admits an $\mathbb{N}$-indexed chain
Used by
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Dependency tree · two levels
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Sources
- E. Riehl, Category Theory in Context, 2nd ed., Corollary 5.6.14 (standard reference, not scraped)