(Left and Right) Kan Extensions

To introduce Kan extensions, let's define functors $u$ and $F$ as so: $A \xrightarrow{\,u\,} B \quad\text{and}\quad A \xrightarrow{\,F\,} C$, where $A,B,C$ are (locally small) categories. The definitions and properties we'll define below capture the essential notion of how to **extend** a functor $F\colon A\to C$ to a larger domain $B$ **along** (or **via**) $u\colon A\to B$. The left Kan extension can be thought of as a “best approximation from above” (and is colimit-based). The right Kan extension can be conceived as a “best approximation from below” (and is limit-based). To be clear: left and right Kan extensions are both functors from $B$ to $C$, i.e. objects in the functor category $[B,C]$. They represent two ways to approximate or extend a functor along another functor. --- A left Kan extension of $F$ along $u$ is, by definition, a functor $\mathrm{Lan}_u(F)\colon B \longrightarrow C$ together with a natural transformation $\eta \colon F \;\Longrightarrow\; \mathrm{Lan}_u(F)\circ u$ satisfying a universal property. ![[lan.png]]  --- A right Kan extension of $F$ along $u$ is a functor $\mathrm{Ran}_u(F)\colon B \longrightarrow C$ together with a natural transformation $\varepsilon \colon \mathrm{Ran}_u(F)\circ u \;\Longrightarrow\; F$ satisfying a corresponding universal property. ![[ran.png]]  --- ## 1. Universal Property of the Left Kan Extension Given functors $u\colon A\to B$ and $F\colon A\to C$, a **left Kan extension** of $F$ along $u$ consists of: 1. A functor $\mathrm{Lan}_u(F)\colon B \to C$. 2. A natural transformation $\eta \colon F \Rightarrow \mathrm{Lan}_u(F)\circ u$. These data are required to make the following universal property hold: > For every functor $G\colon B\to C$ and every natural transformation $\alpha\colon F \Rightarrow G\circ u$, there exists a **unique** natural transformation $\widetilde{\alpha}\colon \mathrm{Lan}_u(F)\Rightarrow G$ such that > > $\begin{array}{ccc} F & \xRightarrow{\eta} & \mathrm{Lan}_u(F)\circ u \\ & \searrow\alpha & \Big\Downarrow{\widetilde{\alpha}\,\circ\, u} \\ & & G\circ u \end{array}$ > > commutes (as a diagram of functors and natural transformations). Equivalently, there is a **natural isomorphism**: $\operatorname{Nat}\bigl(\mathrm{Lan}_u(F), G\bigr) \;\;\cong\;\; \operatorname{Nat}\bigl(F,\,G\circ u\bigr)$, natural in $G \in [B,C]$. In 2-categorical language, $\mathrm{Lan}_u$ is a left bi-adjoint (in the 2-category of functors, natural transformations, and modifications) to the pullback functor $(-)\circ u$. When the relevant categories are locally small and $C$ is cocomplete enough, the pointwise (or conical) formula for left Kan extensions says: $(\mathrm{Lan}_u(F))(b) \;\;\cong\;\; \mathrm{colim}\Bigl( (u \downarrow b) \;\xrightarrow{\;\pi\;} A \;\xrightarrow{F}\; C \Bigr)$ where $(u \downarrow b)$ is the comma category whose objects are morphisms $u(a)\to b$ in $B$, and $\pi$ is the obvious forgetful functor $(u(a)\to b)\mapsto a$. Concretely, one takes the colimit in $C$ of the diagram given by $F\circ\pi$. --- ## 2. Universal Property of the Right Kan Extension Given functors $u\colon A\to B$ and $F\colon A\to C$, a **right Kan extension** of $F$ along $u$ consists of: 1. A functor $\mathrm{Ran}_u(F)\colon B \to C$. 2. A natural transformation $\varepsilon \colon \mathrm{Ran}_u(F)\circ u \Rightarrow F$. These data must make the following universal property hold: > For every functor $G\colon B\to C$ and every natural transformation $\beta\colon G\circ u \Rightarrow F$, there exists a **unique** natural transformation $\widetilde{\beta}\colon G\Rightarrow \mathrm{Ran}_u(F)$ such that > > $\begin{array}{ccc} G\circ u & \xRightarrow{\beta} & F \\ \Big\Downarrow{\widetilde{\beta}\,\circ\,u} & \swarrow\epsilon & \\ \mathrm{Ran}_u(F)\circ u \end{array}$ > > commutes (again as a diagram of functors and natural transformations). Equivalently, there is a **natural isomorphism**: $\operatorname{Nat}\bigl(G,\mathrm{Ran}_u(F)\bigr) \;\;\cong\;\; \operatorname{Nat}\bigl(G\circ u,\,F\bigr)$, natural in $G \in [B,C]$. In 2-categorical language, $\mathrm{Ran}_u$ is a right bi-adjoint to the pullback functor $(-)\circ u$. When $C$ is sufficiently complete, the pointwise (or conical) formula for right Kan extensions is: $(\mathrm{Ran}_u(F))(b) \;\;\cong\;\; \mathrm{lim}\Bigl( (b \downarrow u) \;\xrightarrow{\;\pi\;} A \;\xrightarrow{F}\; C \Bigr),$ where $(b \downarrow u)$ is the comma category whose objects are morphisms $b \to u(a)$ in $B$, and $\pi$ is again the obvious forgetful functor. --- ## 1. The Basic “Extension” Diagram A high-level view of extending $F\colon A\to C$ along $u\colon A\to B$:  <iframe class="quiver-embed" src="https://q.uiver.app/#q=WzAsMyxbMCwwLCJBIl0sWzEsMCwiQyJdLFswLDEsIkIiXSxbMCwyLCJ1Il0sWzAsMSwiRiJdLFsyLDEsIlxcbWF0aHJte0xhbn1fdShGKSIsMix7InN0eWxlIjp7ImJvZHkiOnsibmFtZSI6ImRhc2hlZCJ9fX1dXQ==&embed" width="304" height="304" style="border-radius: 8px; border: none;"></iframe> Here, $\mathrm{Lan}_u(F)$ is the new functor $B\to C$ making the triangle commute _up to_ the natural transformation $\eta\colon F \Rightarrow (\mathrm{Lan}_u(F))\circ u$. --- ## 2. The Universal Property Triangle For the universal property, we show how $\eta$ and any given $\alpha$ factor uniquely through $\widetilde{\alpha}$. This is often drawn as a commutative triangle of **natural transformations**:  <iframe class="quiver-embed" src="https://q.uiver.app/#q=WzAsMyxbMCwwLCJGIl0sWzEsMCwiXFxtYXRocm17TGFufV91KEYpXFxjaXJjIHUiXSxbMSwxLCJHXFxjaXJjIHUiXSxbMCwxLCJcXGV0YSIsMCx7ImxldmVsIjoyfV0sWzAsMiwiXFxhbHBoYSIsMCx7ImxldmVsIjoyfV0sWzEsMiwie1xcd2lkZXRpbGRle1xcYWxwaGF9XFxjaXJjIHV9IiwwLHsibGV2ZWwiOjJ9XV0=&embed" width="404" height="304" style="border-radius: 8px; border: none;"></iframe> The data: 1. $\eta\colon F \Rightarrow \mathrm{Lan}_u(F)\circ u$ (given by the Kan extension), 2. $\alpha\colon F \Rightarrow G\circ u$ (any other candidate extension), 3. $\widetilde{\alpha}\colon \mathrm{Lan}_u(F)\Rightarrow G$ (the unique natural transformation making the diagram commute). This says precisely that for **any** $G\colon B\to C$ and $\alpha\colon F\Rightarrow G\circ u$, there is a **unique** $\widetilde{\alpha}\colon \mathrm{Lan}_u(F)\Rightarrow G$ such that the above triangle of 2-cells commutes. Equivalently, in formula form: $\operatorname{Nat}\bigl(\mathrm{Lan}_u(F),\,G\bigr) \;\;\cong\;\; \operatorname{Nat}\bigl(F,\,G\circ u\bigr)$. --- ## 3. The Pointwise (Colimit) Description Diagram When C is cocomplete enough, $\mathrm{Lan}_u(F)$ can be computed **pointwise** as a colimit. Concretely, for each $b\in B$, $(\mathrm{Lan}_u(F))(b) \;\;\cong\;\; \mathrm{colim}\Bigl( (u \downarrow b) \;\xrightarrow{\;\pi\;} A \;\xrightarrow{F}\; C \Bigr).$  <iframe class="quiver-embed" src="https://q.uiver.app/#q=WzAsMyxbMCwwLCIodSBcXGRvd25hcnJvdyBiKSJdLFsxLDAsIkEiXSxbMSwxLCJDIl0sWzAsMSwiXFxwaSJdLFswLDIsIkYgXFxjaXJjIFxccGkiXSxbMSwyLCJGIl1d&embed" width="329" height="304" style="border-radius: 8px; border: none;"></iframe> - The **comma category** $(u \downarrow b)$ has objects which are morphisms $u(a)\to b$ in $B$. - $\pi\colon (u(a)\to b)\mapsto a$ is the obvious projection back to $A$. - Composing with $F$ yields the functor $F\circ \pi\colon (u \downarrow b)\to C$. - The colimit of $F\circ \pi$ in $C$ is exactly $(\mathrm{Lan}_u(F))(b)$. --- ### Putting It All Together A typical presentation of the left Kan extension includes: 1. **The extension diagram** showing $A\to B$ and $A\to C$ along with the dashed arrow $B\to C$. 2. **The universal property** triangle of 2-cells ($\eta, \alpha, \widetilde{\alpha}$). 3. **The pointwise colimit** diagram for when $C$ is cocomplete. These three diagrams combine to give the full picture of $\mathrm{Lan}_u(F)$.