- Tags
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Refs:
- ncatlab intro: https://golem.ph.utexas.edu/category/2014/01/ends.html
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Links:
- Kan extension
- tensored
coend
- Analogy: limits are right adjoints to diagonals, and ends are right adjoints to homs.
Ends and Coends
Definitions:
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End of a functor \(F: \mathsf{C}^{\operatorname{op}}\times \mathsf{C} \to X\): an equalizer \begin{align*} \int_{\mathsf{C}} F \coloneqq\int_{x} F(x, x) \rightarrow \prod_{x \in \mathsf{C}} F(x, x) \rightrightarrows \prod_{\mathsf{C}(x, y) } F\left(x, y\right) .\end{align*}
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Coend: a coequalizer \begin{align*} \int^{\mathsf{C}} F \coloneqq\int^{y} F(y, y) \leftarrow \coprod_{y \in \mathsf{C}} F(y, y) \leftleftarrows \coprod_{\mathsf{C}(x, y) } F\left(y, x \right) .\end{align*}
Examples of (co)ends
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Can realize global sections: \begin{align*} {{\Gamma}\qty{X; {\mathcal{F}}} } = \int_{U \in {\mathsf{Open}}(X)^{\operatorname{op}}} {\mathcal{F}}(U) .\end{align*}
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Can realize natural transformations as ends: \begin{align*} \mathop{\mathrm{Mor}}_{{\mathsf{Fun}}}(F, G) = \int_c \mathsf{C}(F(c), G(c)) ,\end{align*} realizing them as a coherent family of morphisms.
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Idea: given the singular set functor \(S({-}): {\mathsf{Top}}\to{\mathsf{sSet}}\) where \(S({-})([n]) = {\mathsf{Top}}(\Delta^n, {-})\), construct a left adjoint \(L\) (geometric realization). This should give a bijection \begin{align*} {\mathsf{Top}}(LX, Y) { \, \xrightarrow{\sim}\, }{\mathsf{sSet}}(X, S(Y)) ,\end{align*} where homs on the right-hand side are natural transformations.
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Do this by bending natural transformations: \begin{align*} X([n]) \to {\mathsf{Top}}(\Delta^n, Y) \leadsto {\mathsf{Top}}(X[n] \times \Delta^n, R(X)) ,\end{align*} where for every map on the right-hand side there is a map \(R(X)\to Y\) making a diagram commute. The solution: a coend \begin{align*} R(X) \coloneqq\int^n X([n]) \times \Delta^n .\end{align*}
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Think of functors like modules and coends like tensor products.
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Think of ends as generalizations of limits to profunctors.
- Need to replace cones of functors with wedges of profunctors.
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Alternative (co)end characterization:
Grothendieck construction
Misc
