faithful — Mathlib

English

The standard nerve functor nerveFunctor is faithful; it injectively maps morphisms on objects of the domain to morphisms between nerves.

Русский

Функтор nerveFunctor верен; он инъективно отображает морфизмы на основе области определения в морфизмы между нервами.

LaTeX

$$$\text{nerveFunctor}$ is faithful: for all f,g,\; \text{nerveFunctor.map } f = \text{nerveFunctor.map } g \Rightarrow f=g$$$

Lean4

/-- Repleteness exists for full and faithful functors but not fully faithful functors, which is
why we do this inefficiently. -/
instance faithful : nerveFunctor.{u, u}.Faithful :=
  have : (Nerve.nerveFunctor₂ ⋙ SSet.Truncated.cosk 2).Faithful := Faithful.comp nerveFunctor₂ (SSet.Truncated.cosk 2)
  Functor.Faithful.of_iso Nerve.cosk₂Iso.symm

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