I prefer a somewhat different view of ends and coends, with the intuition stemming more from classical linear algebra and functional analysis. So for me an end is really an integral in a categorical sense.

Let me explain it in more detail. The first fact about ends that show me their power and is basic in enriched category theory is the natural transformation lemma. Basically, $$Nat(F\cdot; G\cdot) = \int_c Hom_C(Fc; Gc)$$ This equations shows that a global natural transformation is as if "summed up" from the local, "differential" transformations on each object. That's exactly what a natural transformation is: a coherent family of morphisms. In this way an end allows us to pass from the local to the global picture, just like a real integral. Let's consider a global section functor for sheaves: $$ \Gamma(X; \mathcal{F}) = Nat(\Bbb{Z}, \mathcal{F}) = \int_U Hom_{Ab}(\Bbb{Z}; \mathcal{F}(U)) = \int_{U\in Ouv^{op}} \mathcal{F}(U)$$ Compare it with measure integration, where you have a (non-negative) measure defined for all measurable subsets of $X$ and you can, in principle, define the measure of $X$ analysing its subsets. At least for not-too-bad measure spaces you can find the measure of $X$ as the supremum of the subsets' measures. This can be also viewed as an end, if you consider a functor $M: Ouv \to \Bbb{R}_{+}$, where $\Bbb{R}_{+}$ is a poset category with objects $[0;\infty]$, $f:a\to b \iff a \leqslant b$. However, I don't understand at the moment how can nontrivial general integrals be treated in this conext.

Even more enlightening is the composition of distributors. A good account of distributors is in J. Benabou's article "Distributors at work". Informally, it is like a "generalized functor", the most important property being the existence of right adjoint for any functor considered as a distributor. Kan extensions also emerge miraculously. The name itself hints of this connection. A distributor is to a functor what a distribution is to a function. Formally, a (Set-valued) distibutor $F:A \nrightarrow B$ from category $A$ to category $B$ is a functor $$\hat F: A\times B^{op} \to Set$$ A composition of distributors can be defined via Kan extensions along the Yoneda embedding, or much more neatly as a coend $$G\circ F (a;c) = \int^b \hat G(b; c) \times \hat F(a;b)$$ This clearly reminds of matrix composition law. A simple example of (identity) distributor is the hom-functor in a V-category: $$[a;c] = \int^b [b; c] \otimes [a;b]$$

Clearly we just integrate out the dummy variable and the inner hom is just a change-of-coordinates Jacobian! A special case of these identities is the Yoneda lemma, which I will write as a left Kan extension: $$F(a) = \int_c [c;a] \otimes F(c) $$ and the Kan extension itself: $$\mathrm{Lan}_K(F)(a) = \int_c [K(c);a] \otimes F(c) $$

Clearly it's just a change of integration variables!

Another enlightening example comes from the theory of metric spaces, considered by F.W.Lawvere in "Metric spaces, generalized logic, and closed categories". A metric space is considered as a category enriched over $\Bbb{R}_+$, defined above. The objects are points, hom from a to b is the distance from a to b (the metric need not be symmetric). In this case for $\Bbb{R}_+$-valued functor $F$ it's limit is clearly it's supremum. A limit is an end $$\mathrm{Lim} F = \int_d F(d)$$ where $F$ is considered as a bifunctor constant in its first variable. So $$\sup_{x\in X} F(x) = \int_{x\in X} F(x)$$

A person familiar with tropical geometry and idempotent analysis will instantly recognise this formula as a tropical integral! Simple as it is, it is another exact shot for categorical integration. A Kan extension of $\phi: X \to \Bbb{R}$ along $f:X\to Y$ is $$\mathrm{Ran}_{f} \phi (y) = \sup_x [ \phi(x) - \lambda Y(y, f(x)) ] $$

If not for the nonlinear hom nature, it would be immediately recognisable as a tropical Fourier transform, aka Legendre transform.

Even the common integral itself can be considered as a kind of transfinite tensor product, but the construction is somewhat clumsy and eventually reduces to common measure, so it's not of much use practically but fits nicely in the categorical integration picture. As of yet I do not know any neat way to incorporate common integrals into the categorical framework, like tropical integrals do.

Sorry for a big post, but I just couldn't resist sharing these examples.

I prefer a somewhat different view of ends and coends, with the intuition stemming more from classical linear algebra and functional analysis. So for me an end is really an integral in a categorical sense.

Let me explain it in more detail. The first fact about ends that show me their power and is basic in enriched category theory is the natural transformation lemma. Basically, $$Nat(F\cdot; G\cdot) = \int_c Hom_C(Fc; Gc)$$ This equations shows that a global natural transformation is as if "summed up" from the local, "differential" transformations on each object. That's exactly what a natural transformation is: a coherent family of morphisms. In this way an end allows us to pass from the local to the global picture, just like a real integral. Let's consider a global section functor for sheaves: $$ \Gamma(X; \mathcal{F}) = Nat(\Bbb{Z}, \mathcal{F}) = \int_U Hom_{Ab}(\Bbb{Z}; \mathcal{F}(U)) = \int_{U\in Ouv^{op}} \mathcal{F}(U)$$ Compare it with measure integration, where you have a (non-negative) measure defined for all measurable subsets of $X$ and you can, in principle, define the measure of $X$ analysing its subsets. At least for not-too-bad measure spaces you can find the measure of $X$ as the supremum of the subsets' measures. This can be also viewed as an end, if you consider a functor $M: Ouv \to \Bbb{R}_{+}$, where $\Bbb{R}_{+}$ is a poset category with objects $[0;\infty]$, $f:a\to b \iff a \leqslant b$. However, I don't understand at the moment how can nontrivial general integrals be treated in this conext.

Even more enlightening is the composition of distributors. A good account of distributors is in J. Benabou's article "Distributors at work". Informally, it is like a "generalized functor", the most important property being the existence of right adjoint for any functor considered as a distributor. Kan extensions also emerge miraculously. The name itself hints of this connection. A distributor is to a functor what a distribution is to a function. Formally, a (Set-valued) distibutor $F:A \nrightarrow B$ from category $A$ to category $B$ is a functor $$\hat F: A\times B^{op} \to Set$$ A composition of distributors can be defined via Kan extensions along the Yoneda embedding, or much more neatly as a coend $$G\circ F (a;c) = \int^b \hat G(b; c) \times \hat F(a;b)$$ This clearly reminds of matrix composition law. A simple example of (identity) distributor is the hom-functor in a V-category: $$[a;c] = \int^b [b; c] \otimes [a;b]$$

Clearly we just integrate out the dummy variable and the inner hom is just a change-of-coordinates Jacobian! A special case of these identities is the Yoneda lemma, which I will write as a left Kan extension: $$F(a) = \int^c [c;a] \otimes F(c) $$ and the Kan extension itself: $$\mathrm{Lan}_K(F)(a) = \int^c [K(c);a] \otimes F(c) $$

Clearly it's just a change of integration variables!

Another enlightening example comes from the theory of metric spaces, considered by F.W.Lawvere in "Metric spaces, generalized logic, and closed categories". A metric space is considered as a category enriched over $\Bbb{R}_+$, defined above. The objects are points, hom from a to b is the distance from a to b (the metric need not be symmetric). In this case for $\Bbb{R}_+$-valued functor $F$ it's limit is clearly it's supremum. A limit is an end $$\mathrm{Lim} F = \int_d F(d)$$ where $F$ is considered as a bifunctor constant in its first variable. So $$\sup_{x\in X} F(x) = \int_{x\in X} F(x)$$

A person familiar with tropical geometry and idempotent analysis will instantly recognise this formula as a tropical integral! Simple as it is, it is another exact shot for categorical integration. A Kan extension of $\phi: X \to \Bbb{R}$ along $f:X\to Y$ is $$\mathrm{Ran}_{f} \phi (y) = \sup_x [ \phi(x) - \lambda Y(y, f(x)) ] $$

If not for the nonlinear hom nature, it would be immediately recognisable as a tropical Fourier transform, aka Legendre transform.

Even the common integral itself can be considered as a kind of transfinite tensor product, but the construction is somewhat clumsy and eventually reduces to common measure, so it's not of much use practically but fits nicely in the categorical integration picture. As of yet I do not know any neat way to incorporate common integrals into the categorical framework, like tropical integrals do.

Sorry for a big post, but I just couldn't resist sharing these examples.

I prefer a somewhat different view of ends and coends, with the intuition stemming more from classical linear algebra and functional analysis. So for me an end is really an integral in a categorical sense.

Let me explain it in more detail. The first fact about ends that show me their power and is basic in enriched category theory is the natural transformation lemma. Basically, $$Nat(F\cdot; G\cdot) = \int_c Hom_C(Fc; Gc)$$ This equations shows that a global natural transformation is as if "summed up" from the local, "differential" transformations on each object. That's exactly what a natural transformation is: a coherent family of morphisms. In this way an end allows us to pass from the local to the global picture, just like a real integral. Let's consider a global section functor for sheaves: $$ \Gamma(X; \mathcal{F}) = Nat(\Bbb{Z}, \mathcal{F}) = \int_U Hom_{Ab}(\Bbb{Z}; \mathcal{F}(U)) = \int_{U\in Ouv^{op}} \mathcal{F}(U)$$ Compare it with measure integration, where you have a (non-negative) measure defined for all measurable subsets of $X$ and you can, in principle, define the measure of $X$ analysing its subsets. At least for not-too-bad measure spaces you can find the measure of $X$ as the supremum of the subsets' measures. This can be also viewed as an end, if you consider a functor $M: Ouv \to \Bbb{R}_{+}$, where $\Bbb{R}_{+}$ is a poset category with objects $[0;\infty]$, $f:a\to b \iff a \leqslant b$. However, I don't understand at the moment how can nontrivial general integrals be treated in this conext.

Even more enlightening is the composition of distributors. A good account of distributors is in J. Benabou's article "Distributors at work". Informally, it is like a "generalized functor", the most important property being the existence of right adjoint for any functor considered as a distributor. Kan extensions also emerge miraculously. The name itself hints of this connection. A distributor is to a functor what a distribution is to a function. Formally, a (Set-valued) distibutor $F:A \nrightarrow B$ from category $A$ to category $B$ is a functor $$\hat F: A\times B^{op} \to Set$$ A composition of distributors can be defined via Kan extensions along the Yoneda embedding, or much more neatly as a coend $$G\circ F (a;c) = \int^b \hat G(b; c) \times \hat F(a;b)$$ This clearly reminds of matrix composition law. A simple example of (identity) distributor is the hom-functor in a V-category: $$[a;c] = \int^b [b; c] \otimes [a;b]$$

Clearly we just integrate out the dummy variable and the inner hom is just a change-of-coordinates Jacobian! A special case of these identities is the Yoneda lemma, which I will write as a left Kan extension: $$F(a) = \int_c [c;a] \otimes F(c) $$ and the Kan extension itself: $$\mathrm{Lan}_K(F)(a) = \int_c [K(c);a] \otimes F(c) $$

Clearly it's just a change of integration variables!

Another enlightening example comes from the theory of metric spaces, considered by F.W.Lawvere in "Metric spaces, generalized logic, and closed categories". A metric space is considered as a category enriched over $ \Bbb{R}_{+} $, defined above. The objects are points, hom from a to b is the distance from a to b (the metric need not be symmetric). In this case for $\mathbb{R}_+$-valued functor $F$ it's limit is clearly it's supremum. A limit is an end $$\mathrm{Lim} F = \int_d F(d)$$ where $F$ is considered as a bifunctor constant in its first variable. So $$\sup_{x\in X} F(x) = \int_{x\in X} F(x)$$

A person familiar with tropical geometry and idempotent analysis will instantly recognise this formula as a tropical integral! Simple as it is, it is another exact shot for categorical integration. A Kan extension of $\phi: X \to \Bbb{R}$ along $f:X\to Y$ is $$\mathrm{Ran}_{f} \phi (y) = \sup_x \left( \phi(x) - \lambda Y(y, f(x)) \right) $$

If not for the nonlinear hom nature, it would be immediately recognisable as a tropical Fourier transform, aka Legendre transform.

Even the common integral itself can be considered as a kind of transfinite tensor product, but the construction is somewhat clumsy and eventually reduces to common measure, so it's not of much use practically but fits nicely in the categorical integration picture. As of yet I do not know any neat way to incorporate common integrals into the categorical framework, like tropical integrals do.

Sorry for a big post, but I just couldn't resist sharing these examples.

I prefer a somewhat different view of ends and coends, with the intuition stemming more from classical linear algebra and functional analysis. So for me an end is really an integral in a categorical sense.

Let me explain it in more detail. The first fact about ends that show me their power and is basic in enriched category theory is the natural transformation lemma. Basically, $$Nat(F\cdot; G\cdot) = \int_c Hom_C(Fc; Gc)$$ This equations shows that a global natural transformation is as if "summed up" from the local, "differential" transformations on each object. That's exactly what a natural transformation is: a coherent family of morphisms. In this way an end allows us to pass from the local to the global picture, just like a real integral. Let's consider a global section functor for sheaves: $$ \Gamma(X; \mathcal{F}) = Nat(\Bbb{Z}, \mathcal{F}) = \int_U Hom_{Ab}(\Bbb{Z}; \mathcal{F}(U)) = \int_{U\in Ouv^{op}} \mathcal{F}(U)$$ Compare it with measure integration, where you have a (non-negative) measure defined for all measurable subsets of $X$ and you can, in principle, define the measure of $X$ analysing its subsets. At least for not-too-bad measure spaces you can find the measure of $X$ as the supremum of the subsets' measures. This can be also viewed as an end, if you consider a functor $M: Ouv \to \Bbb{R}_{+}$, where $\Bbb{R}_{+}$ is a poset category with objects $[0;\infty]$, $f:a\to b \iff a \leqslant b$. However, I don't understand at the moment how can nontrivial general integrals be treated in this conext.

Even more enlightening is the composition of distributors. A good account of distributors is in J. Benabou's article "Distributors at work". Informally, it is like a "generalized functor", the most important property being the existence of right adjoint for any functor considered as a distributor. Kan extensions also emerge miraculously. The name itself hints of this connection. A distributor is to a functor what a distribution is to a function. Formally, a (Set-valued) distibutor $F:A \nrightarrow B$ from category $A$ to category $B$ is a functor $$\hat F: A\times B^{op} \to Set$$ A composition of distributors can be defined via Kan extensions along the Yoneda embedding, or much more neatly as a coend $$G\circ F (a;c) = \int^b \hat G(b; c) \times \hat F(a;b)$$ This clearly reminds of matrix composition law. A simple example of (identity) distributor is the hom-functor in a V-category: $$[a;c] = \int^b [b; c] \otimes [a;b]$$

Clearly we just integrate out the dummy variable and the inner hom is just a change-of-coordinates Jacobian! A special case of these identities is the Yoneda lemma, which I will write as a left Kan extension: $$F(a) = \int_c [c;a] \otimes F(c) $$ and the Kan extension itself: $$\mathrm{Lan}_K(F)(a) = \int_c [K(c);a] \otimes F(c) $$

Clearly it's just a change of integration variables!

Another enlightening example comes from the theory of metric spaces, considered by F.W.Lawvere in "Metric spaces, generalized logic, and closed categories". A metric space is considered as a category enriched over $\Bbb{R}_+$, defined above. The objects are points, hom from a to b is the distance from a to b (the metric need not be symmetric). In this case for $\Bbb{R}_+$-valued functor $F$ it's limit is clearly it's supremum. A limit is an end $$\mathrm{Lim} F = \int_d F(d)$$ where $F$ is considered as a bifunctor constant in its first variable. So $$\sup_{x\in X} F(x) = \int_{x\in X} F(x)$$

A person familiar with tropical geometry and idempotent analysis will instantly recognise this formula as a tropical integral! Simple as it is, it is another exact shot for categorical integration. A Kan extension of $\phi: X \to \Bbb{R}$ along $f:X\to Y$ is $$\mathrm{Ran}_{f} \phi (y) = \sup_x [ \phi(x) - \lambda Y(y, f(x)) ] $$

If not for the nonlinear hom nature, it would be immediately recognisable as a tropical Fourier transform, aka Legendre transform.

Even the common integral itself can be considered as a kind of transfinite tensor product, but the construction is somewhat clumsy and eventually reduces to common measure, so it's not of much use practically but fits nicely in the categorical integration picture. As of yet I do not know any neat way to incorporate common integrals into the categorical framework, like tropical integrals do.

Sorry for a big post, but I just couldn't resist sharing these examples.