By Rafael Villarroel Flores

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**Extra resources for Notes on Categories [Lecture notes]**

**Example text**

Remember that a sink from F to X can be identified with a natural transformation from F to ∆(X). 13) Hence, if C is cocomplete and I is small, we obtain a functor colim : CI → C, defined on the object F to be colim F, and if η : F → F is a natural transformation, we get a map colim η : colim F → colim F induced by the natural transformation F → F → ∆(colim F ). 14) 40 Chapter 4. Limits and Colimits for all i ∈ obj I. Similarly if C is complete and I is small, we can define a functor lim : CI → C, sending F to lim F, and if F → F is a natural transformation, we get a map lim F → lim F induced by the natural transformation ∆(lim F) → F → F .

We say that a functor F : C → Set is representable if there is ∼ homC (A, −). 11 Theorem. (Yoneda Lemma) Let C be any category, F : C → Set a functor, and A ∈ obj C. Then there is a bijection FA ↔ homSetC homC (A, −), F . Proof. Given a ∈ A, we define a natural transformation η(a) : homC (A, −) → F. Its component at B ∈ obj C is η(a)B : homC (A, B) → FB sending f → F(f)(a). 14) η(a)B G FB homC (A, B ) And that follows, since evaluating at f ∈ homC (A, B) it comes down to Fφ F(f)(a) = F(φ ◦ f)(a).

3). So {εi } is a sink. Step 4. 1. Limits in a Functor Category for each C ∈ obj C, since this is precisely the condition that defines the maps εC i . Step 5. We show that the maps εi : T (i) → coliml T (l)(−) form a limiting sink, that is, the functor coliml T (l)(−) : C → D is the colimit of T : I → DC . Let εi : T (i) → Z be a natural sink. 7) We want to show that the MC are the components of a natural transformation coliml T (l)(−) → Z, that is, that the following diagram is commutative for all C-maps f : C → C .

### Notes on Categories [Lecture notes] by Rafael Villarroel Flores

by Thomas

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