# Category Theory/(Co-)cones and (co-)limits

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Definition (cone):

Let ${\displaystyle {\mathcal {C}}}$ be a category and let ${\displaystyle G=(V,A)}$ be a diagram in ${\displaystyle {\mathcal {C}}}$. A cone over ${\displaystyle G}$ is an object ${\displaystyle a}$ of ${\displaystyle {\mathcal {C}}}$, together with morphisms ${\displaystyle \phi _{a,b}:a\to b}$ for each ${\displaystyle b\in V}$, so that for each ${\displaystyle \psi :b\to b'\in A}$ (so that ${\displaystyle b,b'\in V}$) we have

${\displaystyle \phi _{a,b'}=\psi \circ \phi _{a,b}}$.

Definition (limit):

Let ${\displaystyle {\mathcal {C}}}$ be a category, let ${\displaystyle G=(V,A)}$ be a diagram in ${\displaystyle {\mathcal {C}}}$ and let

Definition (diagonal):

Let ${\displaystyle {\mathcal {C}}}$ be a category, and let ${\displaystyle X}$ be an object of ${\displaystyle {\mathcal {C}}}$. Further, suppose that the object ${\displaystyle X\times X}$ and the morphisms ${\displaystyle \pi _{1}:X\times X\to X}$ and ${\displaystyle \pi _{2}:X\times X\to X}$ constitute a product of ${\displaystyle X}$ with itself in ${\displaystyle {\mathcal {C}}}$. Then the diagonal (also called "diagonal morphism") is the unique morphism

${\displaystyle \Delta :X\to X\times X}$

such that ${\displaystyle \pi _{1}\circ \Delta =\pi _{2}\circ \Delta =\operatorname {Id} _{X}}$.

Definition (anti-diagonal):

Let ${\displaystyle {\mathcal {C}}}$ be a category, and let ${\displaystyle X}$ be an object of ${\displaystyle {\mathcal {C}}}$. Further, suppose that the object ${\displaystyle X\sqcup X}$ and the morphisms ${\displaystyle \iota _{1}:X\to X\sqcup X}$ and ${\displaystyle \iota _{2}:X\to X\sqcup X}$ constitute a coproduct of ${\displaystyle X}$ with itself in ${\displaystyle {\mathcal {C}}}$. Then the anti-diagonal (also called "anti-diagonal morphism") is the unique morphism

${\displaystyle \nabla :X\sqcup X\to X}$

such that ${\displaystyle \nabla \circ \iota _{1}=\nabla \circ \iota _{2}=\operatorname {Id} _{X}}$.

## Exercises

1. Let ${\displaystyle {\mathcal {D}}}$ be a category such that every two objects of ${\displaystyle {\mathcal {D}}}$ have a product. Suppose further that ${\displaystyle {\mathcal {C}}}$ is another category, and that ${\displaystyle T:{\mathcal {C}}\to {\mathcal {D}}}$ is a functor. Let ${\displaystyle Y}$ be an object of ${\displaystyle {\mathcal {D}}}$. Use the universal property of the product in order to show that there exists a functor ${\displaystyle S_{Y}:{\mathcal {C}}\to {\mathcal {D}}}$ that sends an object ${\displaystyle X}$ of ${\displaystyle {\mathcal {C}}}$ to the object ${\displaystyle T(X)\times Y}$ of ${\displaystyle {\mathcal {D}}}$.
2. Prove that any morphism ${\displaystyle g:Y\to Z}$ in ${\displaystyle {\mathcal {D}}}$ gives rise to a natural transformation ${\displaystyle S_{Y}\to S_{Z}}$.
3. Can we weaken the assumption that every two objects of ${\displaystyle {\mathcal {D}}}$ have a product? (Hint: Consider the image of the class function on objects associated to the functor ${\displaystyle T}$.)