202505111205

Tags : Category Theory

Element Category

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Consider the setup of Universal Property (Riehl), where we have defined a universal property.

Gautham

For me what helped is noting that universal properties are (almost always) just stating that some object is terminal/initial in some category. almost always I think actually.

The definition of Universal property gives us $(c,e)$ to be the object, where $c$ is the representation of some functor $F$ and $e\in Fc$ be the element that represents the isomorphism given by the representation.

This leads us to the following construction

Definition

The Category of Elements $\int F$ of a covariant functor $F:C\to \text{Set}$ has:

• An objects $(c,e)$ where $e\in Fc$
• a morphism $(c,e)\to (c^{\prime },e^{\prime })$ if $f:c\to c^{\prime }$ is a morphism in $C$ and $Ff(e)=e^{\prime }$.

For a contravariant functor $G$ we have a morphism $(c,e)\to (c^{\prime },e^{\prime })$ if $f:c\to c^{\prime }$ in $C$ and $Ff(e^{\prime })=e$.

There is an evident forgetful functor $\prod$ for the element category.

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References

• Examples of Element Categories
• Element Category is Isomorphic to Comma Category
• Universal Elements are Universal Elements
• Universal Property (Riehl)