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Mirrors > Home > MPE Home > Th. List > df-diag | Structured version Visualization version GIF version |
Description: Define the diagonal functor, which is the functor 𝐶⟶(𝐷 Func 𝐶) whose object part is 𝑥 ∈ 𝐶 ↦ (𝑦 ∈ 𝐷 ↦ 𝑥). The value of the functor at an object 𝑥 is the constant functor which maps all objects in 𝐷 to 𝑥 and all morphisms to 1(𝑥). The morphism part is a natural transformation between these functors, which takes 𝑓:𝑥⟶𝑦 to the natural transformation with every component equal to 𝑓. (Contributed by Mario Carneiro, 6-Jan-2017.) |
Ref | Expression |
---|---|
df-diag | ⊢ Δfunc = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ (〈𝑐, 𝑑〉 curryF (𝑐 1stF 𝑑))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cdiag 17720 | . 2 class Δfunc | |
2 | vc | . . 3 setvar 𝑐 | |
3 | vd | . . 3 setvar 𝑑 | |
4 | ccat 17167 | . . 3 class Cat | |
5 | 2 | cv 1542 | . . . . 5 class 𝑐 |
6 | 3 | cv 1542 | . . . . 5 class 𝑑 |
7 | 5, 6 | cop 4547 | . . . 4 class 〈𝑐, 𝑑〉 |
8 | c1stf 17676 | . . . . 5 class 1stF | |
9 | 5, 6, 8 | co 7213 | . . . 4 class (𝑐 1stF 𝑑) |
10 | ccurf 17718 | . . . 4 class curryF | |
11 | 7, 9, 10 | co 7213 | . . 3 class (〈𝑐, 𝑑〉 curryF (𝑐 1stF 𝑑)) |
12 | 2, 3, 4, 4, 11 | cmpo 7215 | . 2 class (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ (〈𝑐, 𝑑〉 curryF (𝑐 1stF 𝑑))) |
13 | 1, 12 | wceq 1543 | 1 wff Δfunc = (𝑐 ∈ Cat, 𝑑 ∈ Cat ↦ (〈𝑐, 𝑑〉 curryF (𝑐 1stF 𝑑))) |
Colors of variables: wff setvar class |
This definition is referenced by: diagval 17748 |
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