Examples of (co)ends

Question

I am reading texts about (co)ends, and everywhere I see a lack of examples. I am not an expert in this area, and without examples it is difficult for me to use my intuition to grasp the idea. MacLane and Loregian mention some examples in passing, in particular, the so-called "geometric realization" $$ \int^n (Sn)\cdot\Delta n, $$ and the "integral against measure" $$ \int_X f(x) d\mu, $$ but they don't give details, and I don't understand what is meant here.

Can anybody enlighten me,

What are typical examples of (co)ends?

Not necessarily the examples that MacLane and Loregian mention, but just examples with accurate formulations: "if we take this bifunctor, then this construction will be the (co)end"... Of course, the more examples, the better. Thank you.

Edit. From people's comments here I see that I have to clarify that by examples I mean examples for non-specialists, namely, the constructions from other fields of mathematics that could be interpreted as (co)ends. The examples that I see up to now in the texts are methodical, they explain to specialists the details of the definition, and do not rouse interest of non-specialists.

At the same time, what could rouse this interest, are not examples of (co)ends: as an illustration, in Example 1.4.5 Fosco Loregian mentions the Stokes theorem, but, as far as I understand the construction he considers there is a cowedge, not a coend. So it remains unclear why (co)ends are important.

Gregory Arone gave a link to a thread where some informative examples are discussed. It will take me some time to analyze this, but I hope, people will add some more examples here if there indeed are some.

Answer 1

From what I understand, the OP wants a list of examples of (co)ends, where the notion is something familiar that doesn't require a background in category theory to understand. Something like this list of examples of adjoint functors, including free functors, functors of the form $-\otimes X$, induction $Ind_H^G$ from a subgroup $H<G$, and the discrete topology (left adjoint to the forgetful functor from $Top$ to $Set$). Since this question has no answer, I'll pull together a list, most of which have already appeared in the links in the comments to the OP.

1. Let $R$ be a ring, and $A$ (resp. $B$) be a right (resp. left) $R$-module. Then $A\otimes_R B = \int^R A\otimes B$ is a coend, fundamental to algebra. See Example 3.2.12 of Loregian's book.

2. The geometric realization functor from topological spaces to simplicial sets, left adjoint to the singularization functor, is a coend $\int^{n\in \Delta} \Delta^n \times X_n$. Similarly, the categorical realization of a simplicial set, arising from $\tau_1: sSet\leftrightarrows Cat: N_i$, is a co-end, as is the coherent nerve. See Sec 3.1.2 and Examples 3.2.5, 3.2.6, 3.2.7 of Loregian's book. This is also a left Kan extension.

3. Given two functors $F,G:C\to D$, the collection of natural transformations $Nat(F,G)$ is an end, built from pieces of the form $Hom_C(Fc,Gc)$ over all $c\in C$. As a special case, the global section functor of a sheaf $\mathcal{F}$ is $\Gamma(X,\mathcal{F}) = \int_U \mathcal{F}(U)$ over a particular collection of $U$. See also Theorem 1.4.1 in Loregian's book.

4. Viewing a metric space as a category $X$ enriched over $\mathbb{R}_+$, for any $F:X\to \mathbb{R}_+$, the supremum of $F$ is the end $\sup F = \int_{x\in X} F(x)$.

5. Stokes' Theorem has already been mentioned in the comments, and appears as Example 1.4.5 of Loregian's book.

6. Reconstructing a $G$-space $X$ from its fixed points is a co-end $\int^{H\in Orb(G)} X^H \times G/H \cong X$. Similarly, one can apply co-ends to extensions in Galois theory; see Exercise 1.8 of Loregian's book.

7. Let $V$ be a finite dimensional $k$-vector space and let $V^\vee$ be the dual vector space of linear transformations $V\to k$. Then the co-end reconstructs $k$ from the vector spaces, as $\int^V V^{\vee} \otimes_k V \cong k$. This is Example 2.3.12 of Loregian's book.

8. The Fourier transform is a co-end. See Exercise 5.19 of Loregian's book.

9. As Loregian points out in a comment, "(Pointwise) Kan extensions are ends (on the right) or coends (on the left)" - see Chapter 2 of Loregian's book. So, the Dold-Kan correspondence from simplicial abelian groups to non-negatively graded chain complexes, is a left Kan extension, hence a co-end. See Example 3.2.10 of Loregian's book. Similarly, subdivision functors in simplicial contexts.

10. Building on the above, there are many examples of Kan extensions, including limits of functors (as right Kan extensions, hence ends), colimits of functors, direct sum, kernels and cokernels, extending presheaves (Example 2.7 in the link), etc. As the link says "any list is necessarily wildly incomplete."

11. The convolution product is a co-end; see Loregian Prop 6.2.1. Also, it's a left Kan extension. The composition product for symmetric sequences (used to define operads) is a co-end; see Chapter 6 of Loregian, and exercise 6.8.

12. Homotopy Kan extensions are examples of homotopy (co)ends. These see application in physics, including extended quantum field theories and path integral quantization.

13. The induced representation functor, coming from a group homomorphism $f: H\to G$ (or, an inclusion $H\leq G$), is a left Kan extension. Also, taking the connected components of a simplicial object, or its Cech nerve.

14. Weighted (co)limits are examples of (co)ends; see Loregian chapter 4 and Sec 2.4.2. Here's a concrete example. If $R$ is a ring and $f$ is a map of chain complexes of $R$-modules, then the mapping cone of $f$ is the co-end $C(f) \cong \int^i W(i)\otimes f(i)$ where $W$ is the collection of weights in this weighted colimit. The mapping cone is fundamental to homological algebra. See Example 4.2.1 of Loregian's book.

Obviously, there are many, many more examples. But, I've run out of steam for now. I may add more later. Ends and co-ends are also useful for what they can prove, e.g., the Bousfield-Kan formulas for homotopy (co)limits.