English
A further elaboration of the morphism-level map in the functorMap construction, ensuring functorial behavior on arrows via image-mono maps.
Русский
Дальнейшее разъяснение отображения на стрелки в construction functorMap, обеспечивающее факториальность по стрелкам через инъекцию образа.
LaTeX
$$$$ \mathrm{functorMap}(f,i,j) \text{ preserves composition and identity on arrows} $$$$
Lean4
/-- The simplicial thickening defines a functor from the category of linear orders to the category of
simplicial categories
-/
@[simps]
noncomputable def functor {J K : Type u} [LinearOrder J] [LinearOrder K] (f : J →o K) :
EnrichedFunctor SSet (SimplicialThickening J) (SimplicialThickening K)
where
obj := f
map i j := nerveMap ((functorMap f i j))
map_id
i := by
ext
simp only [eId, EnrichedCategory.id]
exact Functor.ext (by cat_disch)
map_comp i j
k := by
ext
simp only [eComp, EnrichedCategory.comp]
exact Functor.ext (by cat_disch)
