Mathematics Prelims

February 22, 2009

Category Theory – Basic Definitions

Filed under: Category Theory — cjohnson @ 10:41 pm

There are certain ideas that are common across many areas of mathematics.  Regardless of whether we’re talking about vector spaces, topological spaces, groups, or simply sets, we have some notion of a mapping between entities in that space.  In the case of sets we may require nothing more than that the maps be well-defined, whereas with groups or vector spaces we may also require that the map preserves some structure.  For example, when discussing maps between vector spaces we may require linearity (e.g., a linear transformation); for groups we may require that the map respects the group operation (homomorphisms).  In either case, we may place some other requirements on the maps that do not require the structure of the space.  In set-theoretic terms, we may concern ourselves with maps that are injective (one-to-one; for each output, there is only one input); that are surjective (there is an input that will give each possible output), or both (bijective).  The language and ideas of category theory allow us to generalize this common notion and also to explore consequences of the requirements we place on these maps.

A category \mathcal{C} is a class of objects and a class of morphisms between those objects that satisfy certain properties.  Before discussing those properties it should be noted that though in many instances morphisms will be actual maps or functions, they don’t necessarily have to be.  We may in fact think of a category simply as a directed graph, where the vertices are objects and the edges are morphisms.  Here there is no structure or mapping quality of the morphisms, they are just “arrows” that start at one object and end at another.

Collectively, the class of objects of the category \mathcal{C} is referred to as \text{Ob}(\mathcal{C}), and the class of morphisms as \text{Hom}(\mathcal{C}).  To refer to the collection of morphisms whose source is the object A and whose target is the object B, we write \text{Hom}_{\mathcal{C}}(A, B) (or just \text{Hom}(A, B) if the category \mathcal{C} is clear from the context).  The morphisms of \mathcal{C} must satisfy the following.

  1. For each A \in \text{Ob}(\mathcal{C}) there is an identity morphism from that object to itself.  That is, there is a 1_A \in \text{Hom}(A, A) with the property that for any g \in \text{Hom}(B, A) and h \in \text{Hom}(A, C) such that 1_A \circ g = g and h \circ 1_A = h.
  2. Composition of morphisms yields a morphism:  if f \in \text{Hom}(A, B) and g \in \text{Hom}(B, C), then there is a morphism, g \circ f \in \text{Hom}(A, C).
  3. Composition of morphisms is associative:  if f, g, and h are morphisms, then (f \circ g) \circ h = f \circ (g \circ h).

For example, the following diagram could be taken as the definition of a category \mathcal{C}.

The category $latex \mathcal{C}$

In this category we have three objects, A, B, and C.   In addition to the identity morphisms, we have an f \in \text{Hom}(A, B), a g \in \text{Hom}(B, C), and their composition.

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4 Comments »

  1. Would you mind telling me what is the LaTeX code, if there is any, that generates the diagram above.

    (I know that my question has nothing to do with the content of your post).

    Best regards,

    Comment by Américo Tavares — February 23, 2009 @ 9:32 am | Reply

  2. Sure. Though here I just generated a PDF and took a screenshot of that to post here. But anyway the code is

    \documentclass[12pt]{article}
    \usepackage{xypic}
    \xyoption{curve}
    \begin{document}
    $$
    \xymatrix{
    A \ar@(l,u)[]^{1_A}
    \ar@{->}[r]^f
    \ar@{->}[dr]_{g \circ f} &
    B \ar@(u,r)[]^{1_B}
    \ar@{->}[d]^g \\
    & C \ar@(d,r)[]_{1_C}
    }
    $$
    \end{document}

    I pretty much just copied, pasted and played around with examples in the xypic user guide: http://www.ctan.org/tex-archive/macros/generic/diagrams/xypic/xy/doc/xyguide.pdf

    Comment by cjohnson — February 23, 2009 @ 10:38 am | Reply

  3. Many thanks.

    Comment by Américo Tavares — February 23, 2009 @ 11:18 am | Reply

  4. […] are, of course, categories which are not concrete.  In the last post we saw a very simple category with three objects.  These objects aren’t sets, they’re […]

    Pingback by Concrete and Non-Concrete Categories (Informally) « Mathematics Prelims — February 24, 2009 @ 4:02 pm | Reply


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