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 $apple<orange$ 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.

One thought on “Algebraic Geometry 3: Some more definitions”

Nice post, Ayush. Just two minor points :
1.) Any functor from an index category has to be faithful. So you may like to replace ‘even if it is neither faithful nor full’ by ‘even if it is not full’ in line 3 para 2.
2.) (para 2 under heading ‘limit’) I think it is better to say ‘an initial object’ rather than ‘the initial object’ to emphasize that a category may have more than one initial object. Of course, any two initial objects have to be isomorphic of course.