### Algebraic Geometry 3: Some more definitions

Index categories: These are categories in which the objects are essentially elements of a partially ordered set, and there exists at most one morphism between two objects. One example would be $\Bbb{N}$, where $\exists f\in$ Hom$(x,y)$ iff $x\leq y$.

Let $\mathcal{I}$ be an index category. A functor $F:\mathcal{I}\to\mathcal{K}$ is said to be indexed by $\mathcal{I}$. What does such a functor look like? If this functor is fully faithful, then $\mathcal{K}$ will also be an index category. However, even if it is neither faithful nor full, there is some structure that is imposed. For example, if $\mathcal{I}$ is $\Bbb{Z}$, then we can index a subset of objects in $\mathcal{K}$, and ensure that morphisms exist between all such indexed objects (mapping objects of lower indices to those with higher indices).

Limit: Let $\mathcal{K}$ be indexed by $\Bbb{Z}$. Then the limit is an object $\varprojlim\limits_{\mathcal{\Bbb{Z}}} A_i$ such that there exists a morphism between $\varprojlim\limits_{\mathcal{\Bbb{Z}}} A_i$ and every object (including itself), and if $A_m$ and $A_n$ are two such objects with $F(f)\in$ Hom$(A_m,A_n)$ (remember that $F$ is the functor between $\Bbb{Z}$ and $\mathcal{K}$), then the three morphisms under consideration commute.

So what exactly is happening here? It is easy to see that $\varprojlim\limits_{\Bbb{Z}} A$ is the initial object in $F(\Bbb{Z})\subset \mathcal{K}$. For example, let $F:\Bbb{N}\to$Set be a functor which maps $n\to \{1,2,3,\dots,n\}$, and the morhism $m\to n$ is mapped to the morphism $x\to x, \forall x\in\{1,2,3,\dots,m\}$. Then $\emptyset$ or $F(0)$ is the unique limit.

Similarly, the colimit is the final object of $F(P)\subset \mathcal{K}$, where $P$ is any partially ordered set.

Filtered set: A nonempty partially ordered set $S$ is said to be filtered is for any $x,y\in S$, there exists a $z\in S$ such that $x\geq z$ and $y\geq z$. For example, $\Bbb{Z}$ is fitered, and so is any subset of it. However, if $S=\{apple,orange,banana\}$, and $apple is the only relation we know, then this is not a filtered poset (partially ordered set).

Adjoints: Two covariant functors $F:A\to B$ and $G:B\to A$, where $A,B$ are categories, are considered to be adjoint if there is a natural bijection $\tau_{AB}:$ Mor$_A(F(A),B)\to$ Mor$(A,G(B))$, and for all $f:A\to A'$ in $A$ we require

Philosophy behind adjoint functors: These ensure that at least the set of morhisms between objects (both in the domain and range) are “isomorphic” to each other, whilst not being concerned about the objects themselves.