P²-irreducible

In category theory, if C is a category and $F:C\to \mathbf{S}\mathbf{e}\mathbf{t}$ is a set-valued functor, the category of elements of F $\mathrm{e}\mathrm{l}(F)$ (also denoted by ∫^CF) is the category defined as follows:

• Objects are pairs $(A,a)$ where $A\in \mathrm{O}\mathrm{b}(C)$ and $a\in FA$.
• An arrow $(A,a)\to (B,b)$ is an arrow $f:A\to B$ in C such that $(Ff)a=b$.

A more concise way to state this is that the category of elements of F is the comma category $\ast \downarrow F$, where $\ast$ is a one-point set. The category of elements of F comes with a natural projection $\mathrm{e}\mathrm{l}(F)\to C$ that sends an object (A,a) to A, and an arrow $(A,a)\to (B,b)$ to its underlying arrow in C.

The category of elements of a presheaf

Somewhat confusingly in some texts (e.g. Mac Lane, Moerdijk), the category of elements for a presheaf is defined differently. If $P\in \hat{C}:={\mathbf{S}\mathbf{e}\mathbf{t}}^{C^{op}}$ is a presheaf, the category of elements of P (again denoted by $\mathrm{e}\mathrm{l}(P)$, or, to make the distinction to the above definition clear, ∫_C P) is the category defined as follows:

• Objects are pairs $(A,a)$ where $A\in \mathrm{O}\mathrm{b}(C)$ and $a\in P(A)$.
• An arrow $(A,a)\to (B,b)$ is an arrow $f:A\to B$ in C such that $(Pf)b=a$.

As one sees, the direction of the arrows is reversed. One can, once again, state this definition in a more concise manner: the category just defined is nothing but $(\ast \downarrow P{)}^{\mathrm{o}\mathrm{p}}$. Consequentially, in the spirit of adding a "co" in front of the name for a construction to denote its opposite, one should rather call this category the category of coelements of P.

For C small, this construction can be extended into a functor ∫_C from $\hat{C}$ to $\mathbf{C}\mathbf{a}\mathbf{t}$, the category of small categories. In fact, using the Yoneda lemma one can show that ∫_CP $\cong \mathbf{\text{y}}\downarrow P$, where $\mathbf{\text{y}}:C\to \hat{C}$ is the Yoneda embedding. This isomorphism is natural in P and thus the functor ∫_C is naturally isomorphic to $\mathbf{\text{y}}\downarrow -:\hat{C}\to \mathbf{\text{<mrow data-mjx-texclass="ORD">Cat</mrow>}}$.

References

• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
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• 20 year-old Real Estate Agent Rusty from Saint-Paul, has hobbies and interests which includes monopoly, property developers in singapore and poker. Will soon undertake a contiki trip that may include going to the Lower Valley of the Omo.
  
  My blog: http://www.primaboinca.com/view_profile.php?userid=5889534

External links

• Template:Nlab