CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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Every category monadic over Set is complete

Statement

If $U : D \to S e t$ is monadic, then $D$ has every small limit.

Facts & Assumptions

Given: A monadic functor $U : D \to S e t$ in the sense of Monadic and strictly monadic functors.

[L1]

Every small diagram in $S e t$ has a limit (Set has all small limits, realized as compatible tuples in a set-indexed product).

[L2]

An Eilenberg–Moore forgetful functor strictly creates every base limit (The Eilenberg–Moore forgetful functor strictly creates every limit that exists in the base).

[L3]

Equivalences preserve, reflect, and create all existing limits in the ordinary isomorphism-invariant sense (Equivalences preserve, reflect, and create limits and colimits in the isomorphism-invariant sense).

Proof

technique · direct

1.1L1L2

For any small diagram in the Eilenberg–Moore category of the induced monad on $S e t$ , [L1] supplies its underlying limit and [L2] lifts that limit; hence this Eilenberg–Moore category is complete.

2.1step 1.1L3∎

Monadicity says that the comparison $K : D \to {S e t}^{T}$ is an equivalence. By [L3], each small limit from step 1.1 transports across $K$ , so $D$ is complete.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

15 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, 2nd ed., Corollary 5.6.7 (standard reference, not scraped)