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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.

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  • $\begingroup$ If you have a general construction for colimits, it will also work for filtered colimits. $\endgroup$ – leibnewtz Jan 20 at 22:36
  • $\begingroup$ 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. $\endgroup$ – Derek Elkins Jan 21 at 0:54
  • $\begingroup$ This is the exact definition of the relation defining a filtered colimit, which just happens to be an equivalence relation in that case. $\endgroup$ – Kevin Carlson Jan 21 at 15:42

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