Algebraic Geometry 3: Some more definitions

by ayushkhaitan3437

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}\toSet 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

Natural transformation-bmp

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.