We continue describing some examples of categories from the last post. This will be a very small list of examples and many books on the subject list long tables of categories.
I am attaching two slides to quote examples of some common categories in mathematics and physics. Unless you already have a mathematics background it will be OK to not understand what these categories are.
But if you have at least some mathematics background then you might understand that the first slide describes what structure we are considering on objects. All the objects in these categories are constructed from plain sets along with additional well-defined axioms on their elements.
Using above objects we can form corresponding categories as shown below.
Don't worry if you are not able to understand these examples since they need some mathematical background. However I will try to give some visual intuition regarding what these structures are and how an arrow preserves these structures thereby setting up a correspondence between objects and leading to a category.
Our first example are the categories FiniteSet and Set. We already saw in the last post general arrows of FiniteSet. I am attaching again a sample arrow in this category for refreshing your memory.
The category Set is a further enlargement of FiniteSet which will include any general set as an object (not necessarily finite) and all the functions between such sets as arrows. It will satisfy the axioms of a category similar to the example of four sets we saw in the last post.
Next example is a category called Top. It is a category with topological spaces as its objects and continuous functions as its arrows. I will illustrate the structure of a general object and an arrow preserving this structure.
The structure of a topological space is precisely a topology on a set. Intuitively it is how the elements of a set are arranged spatially and which elements are near each other forming neighbourhoods (or open sets). The arrow in Top must preserve this structure which roughly means it is a special function between elements of X and Y which preserves the neighbourhoods of the elements. Refer the book An Introduction to Topology and Homotopy by Alan Sieradski for an intuitive yet rigorous approach to understand neighbourhoods and topology structure.
Next example is a category called Meas. It is a category with measurable spaces as its objects and measurable functions as its arrows. I will illustrate the structure of a general object and an arrow preserving this structure.
The structure of a measurable space is precisely a sigma algebra of subsets of a set X which includes all subsets of X that can be measured. Intuitively it is how the elements of a set are measured and which subsets can be measured. The arrow in Meas must preserve this structure which roughly means it is a special function between elements of X and Y which preserves the measurable subsets of the X. Refer the book Real mathematical analysis by Pugh, Charles Chapman for an intuitive approach to measure theory and analysis. The measure theory volumes by D. H. Fremlin are rigorous and exhaustive and were my favourite for research work but they are not visual.
Next example is a category called Vect. It is a category with vector (or linear) spaces as its objects and linear transformations as its arrows. I will illustrate the structure of a general object and an arrow preserving this structure.
The structure of a vector space is precisely the linearity of elements of a set V which roughly looks like a straight space of multiple dimensions. Intuitively it is how the elements of a set are linearly arranged forming many parallel lines to their basis (linearly independent x and y axes). The arrow in Vect must preserve this structure which roughly means it is a special function between elements of V and W which preserves the straight lines or in other words transforms parallel lines into parallel lines. Refer the famous video lectures by Gilbert Strang for an intuitive approach to Linear Algebra.
Next example is a category called Grp. It is a category with groups as its objects and group homomorphisms as its arrows. I will illustrate the structure of a general object and an arrow preserving this structure.
The structure of a group (abstractly capturing the symmetry) is represented by a cayley digram. The arrow in Grp must preserve this structure which roughly means it is a special function between elements of groups C3 and S3 which preserves the structure of cayley digram. Refer the book Visual group theory by Nathan Carter for an intuitive approach to Group Theory.
Our final example is a category called Cat. It is a category with categories themselves as its objects and functors as its arrows ! I will illustrate the structure of a general object and an arrow preserving this structure.
The structure of a category is characterized by the axioms of its definition as we saw in earlier posts. Intuitively it is depicted by commutative diagrams as shown in the figure. The arrow in Cat must preserve this structure of category which means objects are mapped to objects while arrows are mapped to arrows, the identity arrows, compositions, unit laws and associativity are all preserved by F which I have shown in the figure.
I illustrated some examples of categories showing how in general structured objects and structure preserving arrows form various examples of categories.
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