Category theory MOC

Diagonal functor

Let 𝖩 and 𝖢 be categories. The corresponding diagonal functor cat

Δ𝖩:𝖢→𝖢𝖩

is the functor into the functor category 𝖢𝖩 sending each object 𝑋 ∈𝖢 to the constant functor 𝑋 :𝖩 →𝖢 and each morphism 𝑓 ∈𝖢(𝑋,𝑌) to the natural transformation whose components are all 𝑓.

In the case 𝖩 =𝟤 =1―― ⊕1――, we have 𝖢𝟤 ≅𝖢 ×𝖢, giving the typical binary diagonal functor.

Properties

• The adjoints, if they exist, are limits and colimits of shape 𝖩 in 𝖢 — See Limits and colimits as adjoints to the diagonal

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