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Background: If $\mathcal{C}$ is a cocomplete category and $f : I \to J$ is a functor between small categories, then $f^* : \mathrm{Hom}(J,\mathcal{C}) \to \mathrm{Hom}(I,\mathcal{C})$ has a left adjoint $\mathrm{Lan}(f)$, the left Kan extension along $f$. One may express this as the following coend: $$\mathrm{Lan}(f)(F) = \int^i \hom(f(i),-) \otimes F(i).$$

What are some toy examples for left Kan extensions? I know that left Kan extensions are generalizations of colimits, that they are useful for constructing pullback functors of presheaves and the definition of left derived functors in the context of model categories as well as homological algebra. But I would like to see some specific easy examples which are perhaps not really important, but show what's going on.

Here is an example: Consider the inclusion $f : \{0,1\} \hookrightarrow \{0<1\}$. The left Kan extension corresponds to the functor $\mathcal{C} \times \mathcal{C} \to \mathrm{Mor}(\mathcal{C})$ which maps a pair of objects $(A,B)$ to the morphism $(A \to A \oplus B)$.

I am not looking for well-known general classes of examples (geometric realization, tensor products, etc.).

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    $\begingroup$ Hi Martin :) In the unlikely case you don't already know it, you might be interested in the notion of derivators as an abstraction of the formal properties of left and right Kan extensions. It can be used to develop axiomatic homotopy theory midway between the sometimes too weak triangulated categories and the sometimes too technical $\infty$-categories. See arxiv.org/abs/1112.3840 By its very nature, it involves many examples of left and right Kan extensions. $\endgroup$ – Hanno Jan 17 '15 at 21:14
  • $\begingroup$ Thank you. Can you extract some specific examples for left Kan extensions from this abstract theory? $\endgroup$ – Martin Brandenburg Jan 17 '15 at 21:33
  • $\begingroup$ Check out the paper on the "Additivity of Derivator K-theory": math.univ-toulouse.fr/~dcisinsk/addkth.pdf . They give a nice description of kernels and cokernels using Kan extensions in a derivator-theoretic context. $\endgroup$ – 54321user May 27 '17 at 18:32
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(I hope I didn't mess any of these up.)

  • For $f : B\mathbb{N} \to B\mathbb{Z}$ (where $BM$ is the one-object category corresponding to the monoid $M$) given by the inclusion, $\mathrm{Lan}(f)$ sends an endomorphism $g:X \to X$ to the endomorphism of $Y := \mathrm{colim}(X \xrightarrow{g} X \xrightarrow{g} \cdots)$ that is given by $g$ on every copy of $X$. This is the usual construction for "inverting $g$".

  • Let $I = \{0 \to 1\}$. For $f : I \to B \mathbb{N}$, the morphism that picks out of the generator of $\mathbb{N}$, $\mathrm{Lan}(f)$ sends a morphism $g : X \to Y$ to the shift endomorphism on $\coprod_\mathbb{N} Y$ (completely ignoring $X$ and $g$).

  • Let $I$ be as above and $J$ be the category with two objects and a unique isomorphism between them. For the inclusion $f : I \to J$, again $\mathrm{Lan}(f)$ sends a morphism $g: X \to Y$ to the shift on $\coprod_\mathbb{Z} Y$ (which is now an isomorphism).

  • Let $P = I \coprod_{\{0,1\}} I$ be the pair of parallel arrows. For the fold map $f : P \to I$, $\mathrm{Lan}(f)$ sends a pair $g, h : X \to Y$ to the canonical morphism $X \to \mathrm{coeq}(g,h)$ (note the domain is $X$, not $Y$).

  • (Before you scold me for this one, consider that one person's "proposition" is another's "family of examples".) If $f : C \to D$ is an opfibration, then $\mathrm{Lan}(f)$ is computed by taking colimits over the fibres: it sends a functor $g : C \to E$ to $d \mapsto \mathrm{colim}(g|_{f^{-1}(d)})$, where $f^{-1}(d)$ is category consisting of the objects of $C$ mapping to $d \in D$ and those morphisms between them that map to the identity on $d$.

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  • $\begingroup$ Thank you. I really like your compilation. Further toy examples appear in my book "Einführung in die Kategorientheorie", Example 9.5.11, Exercises 9.21 -- 9.24. (This book was also the motivation for asking here.) $\endgroup$ – Martin Brandenburg Jun 4 '17 at 10:32
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    $\begingroup$ Meanwhile I really love the idea of getting used with a mathematical concept by looking at lots of concete examples. $\endgroup$ – Martin Brandenburg Jun 4 '17 at 10:43
  • $\begingroup$ @MartinBrandenburg: Getting intuition for a concept by looking at concrete examples has the disadvantage that it can sometimes be a lot of work, but for me it also has the advantage that it seems to be the only method that reliably works. :) $\endgroup$ – Omar Antolín-Camarena Jun 4 '17 at 16:36
  • $\begingroup$ @MartinBrandenburg: I managed to get a hold of your book (and could figure out what your examples where from looking at the picture and identifying the one or two keywords in German I could guess at :)). Nice examples! I particulary liked the inclusion of the span into the commuting triangle and the "fold" from the non-commuting triangle to the commuting one. $\endgroup$ – Omar Antolín-Camarena Jun 6 '17 at 17:14
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If $J$ is only locally small, the definition of $\mathrm{Lan}(f)(F)$ still makes sense even if $f^\ast$ is not definable without jumping to a bigger universe. It satisfies the 'local' adjointness property : there is a natural transformation $\eta \colon F \to \mathrm{Lan}(f)(F) \circ f$ universal in the sense that any $F \to G \circ f$ factors through $\eta$.

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In this framework, it is very useful to consider the left Kan extension of a functor $F$ along the Yoneda embedding. For example, if you consider the standard cosimplicial space $\Delta^\bullet \colon \boldsymbol\Delta \to \mathsf{Top}$, then its left Kan extension along the Yoneda embedding is precisely the geometric realization functor $|\!-\!| \colon \hat{\boldsymbol\Delta} \to \mathsf{Top}$.

Still with simplicial sets, the left Kan extension of the full inclusion $i\colon \boldsymbol\Delta \to \mathsf{Cat}$ is the fundamental category functor $\tau_1 \colon \hat{\boldsymbol\Delta} \to \mathsf{Cat}$ (that is the functor mapping $X$ to the category whose objects are the elements of $X_0$ and whose morphims are freely generated by $X_1$ under the relation of composition given by the three face maps $X_2 \to X_1$).

In the two previous examples, $\eta$ is actually $\mathrm{id}_F$ (I mean the lax commutative triangle is actually commutative) and the functor $\mathrm{Lan}(f)(F)$ admits a right adjoint (the singular functor in the first case and the nerve functor in the second case). More generally, any functor $F \colon \mathcal I \to \mathcal C$ from a small category $\mathcal I$ to a cocomplete category $\mathcal C$ admits a left Kan extension $\mathrm{Lan}(\mathfrak h^\mathcal I)(F)$ along the Yoneda embedding $\mathfrak h^\mathcal I \colon \mathcal I \to \hat{\mathcal I}$ with structural map $\eta = \mathrm{id}_F$. Moreover this left Kan extension admits $c \mapsto \hom_{\mathcal C}(F-,c)$ as a right adjoint.

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    $\begingroup$ The downvoter is more than welcome to help with constructed criticism. $\endgroup$ – Pece Jan 18 '15 at 17:00
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    $\begingroup$ You have answered the question "What is Kan extension and what are canonical applications", but my question is "What are toy examples for Kan extensions". $\endgroup$ – Martin Brandenburg Jan 19 '15 at 10:18
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    $\begingroup$ I really wonder why it happens so often that I ask questions on math.SE or MO and they are misunderstood. Answers are given, but they don't address the question. Maybe your answer is useful for others, but not for me, sorry. $\endgroup$ – Martin Brandenburg Jan 19 '15 at 10:25
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    $\begingroup$ (It is my impression that pure mathematicians tend to forget the meaning of the word "example" and confuse it with "proposition".) $\endgroup$ – Martin Brandenburg Jan 19 '15 at 10:30
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    $\begingroup$ @MartinBrandenburg Sorry my answer don't meet your expectations. I leave it nevertheless for others (the big-list tag often means the answers can be useful to others). $\endgroup$ – Pece Jan 19 '15 at 10:37
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Here's one you might know. If $f: H \to G$ is a group homomorphism, then $f^*: [G,\mathsf{Vect}] \to [H,\mathsf{Vect}]$ is restriction of group representations, denoted $\operatorname{Res}_f$. The left adjoint $\operatorname{Ind}_f: [H,\mathsf{Vect}] \to [G,\mathsf{Vect}]$ is the induced representation functor. If $G$ and $H$ are finite, then $\operatorname{Ind}_f$ is also right adjoint to $\operatorname{Res}_f$. Induced representations can be written down in an explicit formula which really comes from the general formula you give, but in some ways feels more concrete.

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