Diagonal functor

In <a href="/facts/Category_theory/6zS3DXPa">category theory</a>, a branch of <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, the diagonal functor 
 
 
 
 
 
 C
 
 
 →
 
 
 C
 
 
 ×
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}
 
 is given by 
 
 
 
 Δ
 (
 a
 )
 =
 ⟨
 a
 ,
 a
 ⟩
 
 
 {\displaystyle \Delta (a)=\langle a,a\rangle }
 
, which maps <a href="/facts/Object_(category_theory)/7WzxUThf">objects</a> as well as <a href="/facts/Morphism/Kdpuyz3S">morphisms</a>. This <a href="/facts/Functor/l2u3rIBE">functor</a> can be employed to give a succinct alternate description of the product of objects within the <a href="/facts/Category_(mathematics)/7WzxUThf">category</a> 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
: a product 
 
 
 
 a
 ×
 b
 
 
 {\displaystyle a\times b}
 
 is a universal arrow from 
 
 
 
 Δ
 
 
 {\displaystyle \Delta }
 
 to 
 
 
 
 ⟨
 a
 ,
 b
 ⟩
 
 
 {\displaystyle \langle a,b\rangle }
 
. The arrow comprises the projection maps.
More generally, given a <a href="/facts/Small_category/7WzxUThf">small</a> <a href="/facts/Index_category/kGTlM945">index category</a> 
 
 
 
 
 
 J
 
 
 
 
 {\displaystyle {\mathcal {J}}}
 
, one may construct the <a href="/facts/Functor_category/r7sWdb2h">functor category</a> 
 
 
 
 
 
 
 C
 
 
 
 
 J
 
 
 
 
 
 {\displaystyle {\mathcal {C}}^{\mathcal {J}}}
 
, the objects of which are called <a href="/facts/Diagram_(category_theory)/kGTlM945">diagrams</a>. For each object 
 
 
 
 a
 
 
 {\displaystyle a}
 
 in 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
, there is a <a href="/facts/Constant_functor/l2u3rIBE">constant diagram</a> 
 
 
 
 
 Δ
 
 a
 
 
 :
 
 
 J
 
 
 →
 
 
 C
 
 
 
 
 {\displaystyle \Delta _{a}:{\mathcal {J}}\to {\mathcal {C}}}
 
 that maps every object in 
 
 
 
 
 
 J
 
 
 
 
 {\displaystyle {\mathcal {J}}}
 
 to 
 
 
 
 a
 
 
 {\displaystyle a}
 
 and every morphism in 
 
 
 
 
 
 J
 
 
 
 
 {\displaystyle {\mathcal {J}}}
 
 to 
 
 
 
 
 1
 
 a
 
 
 
 
 {\displaystyle 1_{a}}
 
. The diagonal functor 
 
 
 
 Δ
 :
 
 
 C
 
 
 →
 
 
 
 C
 
 
 
 
 J
 
 
 
 
 
 {\displaystyle \Delta :{\mathcal {C}}\rightarrow {\mathcal {C}}^{\mathcal {J}}}
 
 assigns to each object 
 
 
 
 a
 
 
 {\displaystyle a}
 
 of 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
 the diagram 
 
 
 
 
 Δ
 
 a
 
 
 
 
 {\displaystyle \Delta _{a}}
 
, and to each morphism 
 
 
 
 f
 :
 a
 →
 b
 
 
 {\displaystyle f:a\rightarrow b}
 
 in 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
 the <a href="/facts/Natural_transformation/xwPumTHt">natural transformation</a> 
 
 
 
 η
 
 
 {\displaystyle \eta }
 
 in 
 
 
 
 
 
 
 C
 
 
 
 
 J
 
 
 
 
 
 {\displaystyle {\mathcal {C}}^{\mathcal {J}}}
 
 (given for every object 
 
 
 
 j
 
 
 {\displaystyle j}
 
 of 
 
 
 
 
 
 J
 
 
 
 
 {\displaystyle {\mathcal {J}}}
 
 by 
 
 
 
 
 η
 
 j
 
 
 =
 f
 
 
 {\displaystyle \eta _{j}=f}
 
). Thus, for example, in the case that 
 
 
 
 
 
 J
 
 
 
 
 {\displaystyle {\mathcal {J}}}
 
 is a <a href="/facts/Discrete_category/Tk31I89n">discrete category</a> with two objects, the diagonal functor 
 
 
 
 
 
 C
 
 
 →
 
 
 C
 
 
 ×
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}
 
 is recovered.
Diagonal functors provide a way to define <a href="/facts/Limit_(category_theory)/JoLugLs3">limits</a> and <a href="/facts/Colimit/JoLugLs3">colimits</a> of diagrams. Given a <a href="/facts/Diagram_(category_theory)/kGTlM945">diagram</a> 
 
 
 
 
 
 F
 
 
 :
 
 
 J
 
 
 →
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {F}}:{\mathcal {J}}\rightarrow {\mathcal {C}}}
 
, a natural transformation 
 
 
 
 
 Δ
 
 a
 
 
 →
 
 
 F
 
 
 
 
 {\displaystyle \Delta _{a}\to {\mathcal {F}}}
 
 (for some object 
 
 
 
 a
 
 
 {\displaystyle a}
 
 of 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
) is called a <a href="/facts/Cone_(category_theory)/ST4dKlGE">cone</a> for 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
. These cones and their factorizations correspond precisely to the objects and morphisms of the <a href="/facts/Comma_category/wTa3tURp">comma category</a> 
 
 
 
 (
 Δ
 ↓
 
 
 F
 
 
 )
 
 
 {\displaystyle (\Delta \downarrow {\mathcal {F}})}
 
, and a limit of 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 is a terminal object in 
 
 
 
 (
 Δ
 ↓
 
 
 F
 
 
 )
 
 
 {\displaystyle (\Delta \downarrow {\mathcal {F}})}
 
, i.e., a <a href="/facts/Universal_arrow/5XJBYMNz">universal arrow</a> 
 
 
 
 Δ
 →
 
 
 F
 
 
 
 
 {\displaystyle \Delta \rightarrow {\mathcal {F}}}
 
. Dually, a <a href="/facts/Colimit/JoLugLs3">colimit</a> of 
 
 
 
 
 
 F
 
 
 
 
 {\displaystyle {\mathcal {F}}}
 
 is an initial object in the comma category 
 
 
 
 (
 
 
 F
 
 
 ↓
 Δ
 )
 
 
 {\displaystyle ({\mathcal {F}}\downarrow \Delta )}
 
, i.e., a universal arrow 
 
 
 
 
 
 F
 
 
 →
 Δ
 
 
 {\displaystyle {\mathcal {F}}\rightarrow \Delta }
 
.
If every functor from 
 
 
 
 
 
 J
 
 
 
 
 {\displaystyle {\mathcal {J}}}
 
 to 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
 has a limit (which will be the case if 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
 is <a href="/facts/Complete_category/wpQzAbpB">complete</a>), then the operation of taking limits is itself a functor from 
 
 
 
 
 
 
 C
 
 
 
 
 J
 
 
 
 
 
 {\displaystyle {\mathcal {C}}^{\mathcal {J}}}
 
 to 
 
 
 
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}}
 
. The limit functor is the <a href="/facts/Adjoint_functors/bJ1P7AN2">right-adjoint</a> of the diagonal functor. Similarly, the colimit functor (which exists if the category is cocomplete) is the left-adjoint of the diagonal functor. For example, the diagonal functor 
 
 
 
 
 
 C
 
 
 →
 
 
 C
 
 
 ×
 
 
 C
 
 
 
 
 {\displaystyle {\mathcal {C}}\rightarrow {\mathcal {C}}\times {\mathcal {C}}}
 
 described above is the left-adjoint of the binary <a href="/facts/Product_(category_theory)/evFecz0t">product functor</a> and the right-adjoint of the binary <a href="/facts/Coproduct/0bJ8SCdu">coproduct functor</a>.

Diagonal functor open-in-new

Diagonal functor