# mathlibdocumentation

category_theory.limits.shapes.finite_limits

# Categories with finite limits. #

A typeclass for categories with all finite (co)limits.

@[class]
structure category_theory.limits.has_finite_limits (C : Type u)  :
Prop
• out : ∀ (J : Type ?) [𝒥 : [_inst_2 : ,

A category has all finite limits if every functor J ⥤ C with a fin_category J instance has a limit.

This is often called 'finitely complete'.

Instances
@[instance]

If C has all limits, it has finite limits.

@[class]
structure category_theory.limits.has_finite_colimits (C : Type u)  :
Prop
• out : ∀ (J : Type ?) [𝒥 : [_inst_2 : ,

A category has all finite colimits if every functor J ⥤ C with a fin_category J instance has a colimit.

This is often called 'finitely cocomplete'.

Instances
@[instance]

If C has all colimits, it has finite colimits.

@[instance]
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@[class]
structure category_theory.limits.has_finite_wide_pullbacks (C : Type u)  :
Prop
• out : ∀ (J : Type ?) [_inst_2 : [_inst_3 : fintype J],

has_finite_wide_pullbacks represents a choice of wide pullback for every finite collection of morphisms

@[instance]
@[class]
structure category_theory.limits.has_finite_wide_pushouts (C : Type u)  :
Prop
• out : ∀ (J : Type ?) [_inst_2 : [_inst_3 : fintype J],

has_finite_wide_pushouts represents a choice of wide pushout for every finite collection of morphisms

@[instance]

Finite wide pullbacks are finite limits, so if C has all finite limits, it also has finite wide pullbacks

Finite wide pushouts are finite colimits, so if C has all finite colimits, it also has finite wide pushouts

@[instance]
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