# General colimits and filtered colimits in the category of sets

A category $$\mathsf{I}$$ is filtered if

• $$\mathsf{Ob(I)} \neq \varnothing$$,

• for any $$i,j \in \mathsf{Ob(I)}$$ there is $$k \in \mathsf{Ob(I)}$$ and morphisms $$f\colon i\to k$$ and $$g\colon j\to k$$,

• for any pair $$f,g\colon i\to j$$ of parallel morphisms in $$\mathsf{I}$$ there is $$k \in \mathsf{Ob(I)}$$ together with a morphism $$h\colon j\to k$$ so that $$h\circ f = h\circ g$$.

Let $$F\colon\mathsf{I}\to\mathsf{Set}$$ be a functor with $$\mathsf{I}$$ being small. It is known that a colimit of such a functor is the quotient $$\bigsqcup_{i \in \mathsf{Ob(I)}} F(i)/\sim$$ together with a colimit cocone $$\lambda\colon F\Rightarrow\bigsqcup_{i \in \mathsf{Ob(I)}} F(i)/\sim$$ such that for any $$i \in \mathsf{Ob(I)}$$ and for any $$x \in F(i)$$ we have $$\lambda_i(x) = [(i,x)]_{\sim}$$ (the equivalence class of $$(i,x)$$ with respect to $$\sim$$) where $$\sim$$ is the equivalence relation generated by the relation $$\{ ((i,x),(j,y)) \in (\bigsqcup_{i \in \mathsf{Ob(I)}} F(i))\times(\bigsqcup_{i \in \mathsf{Ob(I)}} F(i)) \mid$$ there is $$f\colon i\to j$$ so that $$y = F(f)(x) \}$$, that is, such an equivalence relation on $$\bigsqcup_{i \in \mathsf{Ob(I)}}$$ so that for any $$(i,x), (j,y) \in \bigsqcup_{i \in \mathsf{Ob(I)}} F(i)$$ there is $$n \in \mathbb{N}_{>0}$$ and there are $$(i_1,x_1), ..., (i_n,x_n) \in \bigsqcup_{i \in \mathsf{Ob(I)}} F(i)$$ so that $$(i_1,x_1) = (i,x), (i_n,x_n) = (j,y)$$ and for any $$1 \leq k < n$$ there is a there is either a morphism $$f\colon i_k \to i_{k+1}$$ for which we have $$x_{k+1} = F(f)(x_k)$$ and there is a morphism $$f\colon i_{k+1} \to i_k$$ for which we have $$x_k = F(f)(x_{k+1})$$.

It is also known that if $$\mathsf{I}$$ is a filtered small category, then a colimit of a funtor $$F\colon\mathsf{I}\to\mathsf{Set}$$ is the quotient $$\bigsqcup_{i \in \mathsf{Ob(I)}}F(i)/\sim$$ (together with a colimit cocone $$\lambda\colon F\Rightarrow\bigsqcup_{i \in \mathsf{Ob(I)}}F(i)$$ such that for any $$i \in \mathsf{Ob(I)}$$ and for any $$x \in F(i)$$ we have $$\lambda_i(x) = [(i,x)]_{\sim}$$) so that we have $$(i,x) \sim (j,y)$$ precisely when there is $$k \in \mathsf{Ob(I)}$$ and there are morphisms $$f\colon i\to k, g\colon j\to k$$ so that $$F(f)(x) = F(g)(y)$$.

My question is whether we can deduce the second result (about filtered colimits in $$\mathsf{Set}$$) from the first result (about general colimits in $$\mathsf{Set}$$) by reducing the first equivalence relation to the second in the case of $$\mathsf{I}$$ being filtered or do we have to start from scratch.

• If you have a general construction for colimits, it will also work for filtered colimits. – leibnewtz Jan 20 at 22:36
• A different but equivalent way of defining a filtered category that I think is more helpful is a filtered category is one where there is a cocone for every finite diagram. – Derek Elkins Jan 21 at 0:54
• This is the exact definition of the relation defining a filtered colimit, which just happens to be an equivalence relation in that case. – Kevin Carlson Jan 21 at 15:42