Elements and their relations in a Category


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Subobjects

Definition 1.6   A subobject of a C-object d is an equivalence class of C-arrow which are monics with codomain, d i.e. of the form

$\displaystyle \xymatrix{ *++{a}\ar@{>->}[rr]&&d\\ }$

This definition implies that the inclusion relation between subobjects of d is defined as follows: given

$\displaystyle \xymatrix{ *++{f:a}\ar@{>->}[rr]d&&*++{g:b}\ar@{>->}[rr]&&d\\ }$

$ f\subseteq g$ iff $ \exists$ a C-arrow

$\displaystyle \xymatrix{ *++{h:a}\ar@{>->}[rr]&&b\\ }$

such that the following diagram commutes

$\displaystyle \xymatrix{ *++{b}\ar@{>->}[rrd]^g&&\\ &&*++{d}\\ *++{a}\ar@{>->}[uu]^h\ar@{>->}[rru]_f&&\\ }$

i.e f= g o h. Since f and g are monic, it follows that h is monic, therefore h is a subobject of d. We have then showed that $ f\subseteq g$ iff f factors through g. It follows that the collection Sub(d) forms a partial ordered set where $ [f]\leq[g]$ iff f=gh.

Elements

Definition 1.7   Given a category $ \mathscr{C}$, with terminal object 1, then an element of a $ \mathscr{C}$-object b is a C-arrow $ x:1\rightarrow b$

Example 1.1   In set, an element $ x\in A$, can be identified with the singleton subset $ \{*\}$ therefore, with an arrow $ \{*\}\rightarrow A$ from the terminal object to A (see definition of terminal object)

Products

Definition 1.8   A product of two objects A and B in a category $ \mathscr{C}$ is a third $ \mathscr{C}$-object $ A\times B$ together with a pair of $ \mathscr{C}$-projection arrows:
$ pr_A:A\times B\rightarrow A$ and $ pr_B:A\times B\rightarrow B$
such that, given any other pair of $ \mathscr{C}$-arrows $ f:C\rightarrow A$ and $ g:C\rightarrow B$, there exists a unique arrow $ \langle f,g\rangle:C\rightarrow A\times B$ such that the following diagram commute

$\displaystyle \xymatrix{ &&C\ar[rrdd]^g\ar@{-->}[dd]^{\langle f,g\rangle}\ar[lldd]_f&&\\ &&&&\\ A&&A\times B\ar[rr]_{pr_B}\ar[ll]^{pr_A}&&B\\ }$

i.e.

$\displaystyle pr_A o\langle f,g\rangle=f\hspace{.2in}and\hspace{.2in}pr_b o\langle f,g\rangle=g$    

Co-products

Definition 1.9   A co-product of two objects A and B in a category $ \mathscr{C}$ is a third $ \mathscr{C}$-object $ A + B$ together with a pair of $ \mathscr{C}$-arrows:
$ i_A:A\rightarrow A + B$ and $ i_B:B\rightarrow A + B$
such that, given any other pair of $ \mathscr{C}$-arrows $ f:A\rightarrow C$ and $ g:B\rightarrow C$, there exists a unique arrow $ [f,g]:A + B\rightarrow C$ which makes the following diagram commutes

$\displaystyle \xymatrix{ A\ar[rr]^{i_A}\ar[rrdd]_f&&A+B\ar[dd]^{[f,g]}&&B\ar[ll]_{i_B}\ar[ddll]^g\\ &&&&\\ &&C&&\\ }$

i.e. the co-product is the dual of the product


Next: Example of Categories in Up: Category Previous: Arrows in a category
Cecilia Flori 2006-11-20