mathlib documentation

category_theory.connected_components

Connected components of a category #

Defines a type connected_components J indexing the connected components of a category, and the full subcategories giving each connected component: component j : Type u₁. We show that each component j is in fact connected.

We show every category can be expressed as a disjoint union of its connected components, in particular decomposed J is the category (definitionally) given by the sigma-type of the connected components of J, and it is shown that this is equivalent to J.

def category_theory.connected_components (J : Type u₁) [category_theory.category J] :
Type u₁

This type indexes the connected components of the category J.

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Given an index for a connected component, produce the actual component as a full subcategory.

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The inclusion functor from a connected component to the whole category.

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@[simp]
theorem category_theory.component.ι_obj {J : Type u₁} [category_theory.category J] (j : category_theory.connected_components J) (self : {X // (λ (k : J), quotient.mk' k = j) X}) :
@[instance]

Each connected component of the category is nonempty.

@[instance]

Each connected component of the category is connected.

def category_theory.decomposed (J : Type u₁) [category_theory.category J] :
Type u₁

The disjoint union of Js connected components, written explicitly as a sigma-type with the category structure. This category is equivalent to J.

The inclusion of each component into the decomposed category. This is just sigma.incl but having this abbreviation helps guide typeclass search to get the right category instance on decomposed J.

The forward direction of the equivalence between the decomposed category and the original.

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@[instance]
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This gives that any category is equivalent to a disjoint union of connected categories.

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