When is the realisation functor conservative?

Let 𝒞 be a cocomplete category and let U : ℐ → 𝒞 be a small diagram. Then there is an induced realisation–nerve adjunction,

RN : 𝒞 → [ℐop, Set]

where N(C) is the presheaf 𝒞(U–, C) and R is defined by left Kan extension. The comonad induced by the realisation–nerve adjunction is precisely the density comonad induced by U, so it is natural to ask, when is R comonadic?

Beck’s monadicity theorem implies R is comonadic if and only if R is conservative and preserves certain equaliser diagrams in [ℐop, Set], and Thomas Athorne described some sufficient conditions in his thesis. We discuss here his condition for conservativity.

Let us say that U satisfies the repeated element condition if the components of the adjunction unit are monomorphisms in [ℐop, Set]. Unravelling the definitions, this amounts to saying R reflects equality between presheaf morphisms with representable domain.

First, suppose R is conservative. Since [ℐop, Set] has coequalisers, R must also be faithful. Thus, R reflects monomorphisms, and in particular, U satisfies the repeated element condition.

Conversely, suppose U satisfies the repeated element condition. By general nonsense about adjunctions, this happens if and only if R is faithful. Since [ℐop, Set] is a balanced category, to verify that R reflects isomorphisms, it is enough to show that R reflects epimorphisms. This is easily done by considering cokernel pairs.

We may therefore conclude that the following are equivalent:

  1. U satisfies the repeated element condition.
  2. R reflects monomorphisms.
  3. R is faithful.
  4. R is conservative.
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