---- > [!definition] Definition. ([[categorical colimit]]) > Let $\mathscr{F}: \mathsf{I} \to \mathsf{C}$ be a [[covariant functor]], where one thinks of $\mathsf{I}$ as a [[category]] 'of indices'. The **colimit** of $\mathscr{F}$ is (if it exists) an object $C$ of $\mathsf{C}$, endowed with morphisms $\lambda_{I}:\mathscr{F}(I) \to C$ for all objects $I$ of $\mathsf{I}$, such that $\lambda_{I}= \mathscr{F}(\alpha) \circ \lambda_{J}$ for all $\alpha:I \to J$: > > > ```tikz > \usepackage{tikz-cd} > \usepackage{amsmath} > \usepackage{amsfonts} > \usepackage[mathscr]{euscript} > \begin{document} > % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZARgBpiBdUkANwEMAbAVxiRABkQBfU9TXfIRQAGUsKq1GLNgB0ZAWzo4AFnADGAJ2AAxLgAoAkgEpuvEBmx4CRAExiJ9Zq0Qg5ileq269AKRNcJGCgAc3giUAAzDQh5JDIQHAgkO0knWRlGNGU6U0jo2MR4xKRRVOkXOQY6eQAjKDoAfQMQaiqamAYABX4rIRANLGDlHFyQKJjk6mLEUsdy1xkq2vqGn24KLiA > \begin{tikzcd} > \mathscr{F}(I) \arrow[rr, "\mathscr{F}(\alpha)"] \arrow[rd, "\lambda_I"'] & & \mathscr{F}(J) \arrow[ld, "\lambda_J"] \\ > & C & > \end{tikzcd} > \end{document} > ``` > > and that $C$ is [[terminal object|initial]] with respect to this requirement. > > The notion is dual to that of a [[categorical limit]]. The colimit of $\mathscr{F}$ is denoted $\lim\limits_{{\longrightarrow}} \mathscr{F}$ and is also called the **direct limit** or **injective limit** of $\mathscr{F}$. Suppose $\mathsf{I}$ is a [[filtered category]]. Any [[diagram]] in $\mathsf{Set}$ indexed by $\mathsf{I}$ (a **direct system**) has the following, with the obvious maps $A_{i} \hookrightarrow \coprod_{i \in \mathsf{I}} A_{i} \twoheadrightarrow L$ to it, as a colimit: $\left\{ (a_{i}, i) \in \coprod_{i \in \mathsf{I}} A_{i} \right\} / \left(\begin{align}& (a_{i}, i) \sim (a_{j}, j) \text{ if and only if there are } f:A_{i} \to A_{k} \\ & g:A_{j} \to A_{k} \text{ in a diagram for which } f(a_{i})=g(a_{j}) \text{ in } A_{k} \end{align} \right)$ (The "filtered" hypothesis is there to ensure $\sim$ is an [[equivalence relation]])[^1] Call the prospective colimit $L$. **First** we show the factorization condition. Consider mappings $A_{i} \to L$ and $A_{j} \to L$. Let $f_{i \to j}:A_{i} \to A_{j}$ be a morphism indexed by some arrow $i \to j$ in $\mathsf{I}$. We want to show that ```tikz \usepackage{tikz-cd} \usepackage{amsmath} \begin{document} % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAEEB9LEAX1PSZc+QigBM5KrUYs2XAFZ8BIDNjwEiARlKap9Zq0QgAMnykwoAc3hFQAMwBOEALZIyIHBCRj+9p68RtDy9EH2VHFzdqTyQg-VkjO05gLAACAB10z1T5XjNeIA \begin{tikzcd} A_i \arrow[rd] \arrow[rr, "f_{i \to j}"] & & A_j \arrow[ld] \\ & L & \end{tikzcd} \end{document} ``` commutes. This amounts to evaluating whether $(a_{i}, i) \sim^{?} \big( f_{i \to j}(a_{i}), j \big)$ holds. Clearly, it does hold, by considering $f=f_{i \to j}$ and $g=\id_{A_{j}}$ in the definition of $\sim$. **Now** we show [[universal property|universality]]. Suppose $M$ is a set which, accompanied by morphisms $\varphi_{*}:A_{*} \to M$, satisfies the factorization condition from the colimit definition. Contemplating the desired commutative diagram, a portion of which looks like: ```tikz \usepackage{tikz-cd} \usepackage{amsmath} \begin{document} % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAEEB9LEAX1PSZc+QigBM5KrUYs2XAFZ8BIDNjwEiARlKap9Zq0QgAMksFqRW0mL0zDIALJ8pMKAHN4RUADMAThABbJDIQHAgkCWkDNgAdGLQsAAI4-Bw6bhBqBjoAIxgGAAUhdVEQXyw3AAscMxA-QKRtUPDEMX4ffyDEELCkAGZqPLAoJABaPpD9WSNvDKzc-KKLDSNyqpr2us7G6l7W6in7OISUiDTORU36rqa9gZAhkcQJg7s2Wcvla4jdlvuGLBgexQOhwSquTJRaYgOIwAAeWDgODgiQAhMl4that9ur8dlD7LNgEk4mFEvJeM5eEA \begin{tikzcd} A_i \arrow[rd, "\pi \iota_i"'] \arrow[rdd, "\varphi_i"', bend right] \arrow[rr, "f_{i \to j}"] & & A_j \arrow[ld] \arrow[ld, "\pi\iota_j"] \arrow[ldd, "\varphi_j", bend left] \\ & L \arrow[d, "\exists ! \psi", dashed] & \\ & M & \end{tikzcd} \end{document} ``` we see that *if* $\psi:L \to M$ exists, commutativity forces that it must be given by $\psi([a_{\ell}, \ell])= \varphi_{\ell}(a_{\ell}).$ So one just has to check well-definition, and this is where 'all of' $\sim$ has to be used. Suppose $(a_{p}, p) \sim (a_{\ell}, \ell)$; we have to show $\varphi_{p}(a_{p})=\varphi_{\ell}(a_{\ell})$. The equivalence provides morphisms $A_{p} \xrightarrow{g}A_{k}$ and $A_{\ell} \xrightarrow{h}A_{k}$ satisfying $g(a_{p})=h(a_{\ell})$. Because $M$ satisfies that factorization condition, the following diagram commutes: ```tikz \usepackage{tikz-cd} \usepackage{amsmath} \begin{document} % https://tikzcd.yichuanshen.de/#N4Igdg9gJgpgziAXAbVABwnAlgFyxMJZABgBpiBdUkANwEMAbAVxiRAEEB9NEAX1PSZc+QigCM5KrUYs2XANZ8BIDNjwEiAJknV6zVog6cAOsZgMGSwWpFEJYqXtmGAsnykwoAc3hFQAMwAnCABbJAkQHAgkAGZdGQMQU3pAtAALLE5FfgDgsMQySOjEOOl9NmS6VIzuEGoGOgAjcwAFIXVREECsLzScKxAg0KRtItj48sNK6szTc0scwbykQqjwiecQLwGh-NG1xAinRLS6kAbmhjabDUNu3v7eCl4gA \begin{tikzcd} A_p \arrow[rd, "\varphi_p"'] \arrow[r, "g"] & A_k \arrow[d, "\varphi_k"] & A_\ell \arrow[ld, "\varphi_\ell"] \arrow[l, "h"'] \\ & M & \end{tikzcd} \end{document} ``` which tells us that $\varphi_{p}=\varphi_{k} \circ g$ and $\varphi_{\ell}=\varphi_{k} \circ h$, and since we know that $g(a_{p})=h(a_{\ell})$, it follows that $\varphi_{p}(a_{p})=\varphi_{k} \circ g(a_{p})=\varphi_{k} \circ h(a_{\ell})=\varphi_{\ell}(a_{\ell})$ as desired. > [!specialization] > If there are no $f_{i \to j}$s at all ($\mathsf{I}$ is a [[category|discrete category]]), then there is no need to [[quotient set|quotient]] for the colimit conditions to hold, and indeed $\sim$ evaporates in this case. We just get the usual [[categorical coproduct|coproduct]] $\coprod$ of sets. > > [!basicnonexample] > > But if even one $f_{i \to j}$ is present, it is clear that at least one of the requisite diagrams will not commute unless we make *some* identification. ^specialization $\{ (x_{i}, i) \in \coprod_{i \in I} \}/(x_{i} \sim f_{ij}(x_{i}) \text{ for all }i \leq j)$ $\{ (x_{i}, i) \in \coprod_{i \in I} \}/ \big( (x_{i}, i) \sim (x_{j}, j) \iff \exists k^{\geq i}_{\geq j} \text{ s.t. } f_{ik}(x_{i})=f(jk)(x_{j}) \big)$. ---- #### [^1]: ![[CleanShot 2025-01-30 at [email protected]]] ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```