1. Feb 20, 2013

### Jim Kata

I have been trying to teach myself some category theory, and have been working through some proofs. I didn't understand the proofs I read about proving colimits (in the case of a small categories) can be given in terms of coproducts and coequalizers. Here is my attempt at a proof. I would appreciate someone to correct my mistakes and explain the aspects I don't understand. I'm sorry if it is a bit disconbobulating since I'm not sure how to draw diagrams in latex. Let $$\mathcal{F} : \mathcal{B} \rightarrow \mathcal{A}$$ be a diagram where $$\mathcal{B}$$ is a small category. Let $$X_j$$ be an object in $$\mathcal{B}$$ (we can index it i guess because $$\mathcal{B}$$ is a small category?)

let $$\varphi : X_{j} \rightarrow X_{l}$$

so $$\mathcal{F}(\varphi) : \mathcal{F}(X_{j}) \rightarrow \mathcal{F}(X_{l})$$

Since the coproduct exists for every $$X_j$$

there exists $$i_j:X_j \rightarrow \coprod_{Obj \mathcal{B}}B$$

so there are two morphisms $$\mathcal{F}(i_j):\mathcal{F}(X_j) \rightarrow \mathcal{F}(\coprod_{Obj \mathcal{B}}B)$$
and $$\mathcal{F}(i_l\varphi):\mathcal{F}(X_j) \rightarrow \mathcal{F}(X_l)\rightarrow \mathcal{F}(\coprod_{Obj \mathcal{B}}B)$$

Using the existence of the coequalizer we have the cocone $$(\phi,Q)$$

where $$\phi(X_j)= q\circ\mathcal{F}(i_j): \mathcal{F}(X_j) \rightarrow Q$$ and by the universal property of the coequalizer we get the universal property of the cocones. I guess my problem is I don't see how I ever used the universal property of the coproduct and I'm not sure I used the small category part right?