Category


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"Category Theory allows you to work on structures without the need first to pulverize them into set theoretic dust" (Corfiel). The above quote explains, in a rather pictorial way, what Category Theory and in particular Topos Theory are really about. In fact, Category Theory and in particular Topos Theory allow to abstract from the specification of points (elements of a set) and functions between these points to a universe of discourse in which the basic elements are arrows, and any property is given in terms of compositions of arrows.
Let us analyze in a more rigorous way what a Category is.

Definition 1.1   [5] [6] [4] A category consists of two things:
  1. a collection of objects
  2. a collection of morphisms between these objects such that the following conditions hold:
  • composition condition: given two morphisms $ f:a\rightarrow b$ and $ g:b\rightarrow c$ with dom g=cod f then there exists the composite map $ gof:a\rightarrow c$
  • associative law: given $ a\xrightarrow{f}b\xrightarrow{g}c$ then (h o(g o f))=((h o g)o f) i.e. the following diagram commutes

    $\displaystyle \xymatrix{ \ar[dd]_{(h o g)o f}&a\ar[rr]^f\ar@{-->}[rrdd]^{g o f}\ar[dd]_{h o (g o f)}&&b\ar[dd]^g\ar[lldd]^{h o g}\\ &&&\\ &d\ar[rr]^h&&c\\ }$

  • identity law: for any object b in the category there exists a morphism $ 1_b:b\rightarrow b$ called identity arrow such that, given any other two morphisms $ f:a\rightarrow b$ and $ g:b\rightarrow c$ we then have $ 1_b o f=f$ and $ g o 1_b=g$, i.e. the following diagram commutes

$\displaystyle \xymatrix{ a\ar[rr]^f\ar[rrdd]^f&&b\ar[rrdd]^g\ar[dd]^{1_b}&&\\ &&&&\\ &&b\ar[rr]^g&&c\\ }$


Subsections
Cecilia Flori 2006-11-20