# categorías: en Herstein y otros encontramos…

 CATEGORÍA OBJETOS Y MORFISMOS $\mathfrak{C}$ ${\rm {obj}}\mathfrak{C}$, $\hom{\mathfrak{C}}$ SET Conjuntos y mapeos GP Grupos y homomorfismos RING Anillos y morfismos de anillos FIELD Campos y … MOD Módulos y … EV espacios vectoriales y transformaciones lineales ALG Álgebras … sGP Semigrupos … MON Monoides … GPDE Grupoides … TOPO Espacios topológicos y funciones continuas BANACH Espacios de Banach … HILBERT Espacios de Hilbert … CHAIN COMPLEXES Cadenas complejas y cadeno morfismos

Little more on categories at PlanetMath.org

Universal algebra links in John Baez’s page

It is possible to relate two categories: A functor is a transformation

$F:{\rm{CAT}}_1\to {\rm{CAT}}_2$

which send objects in ${\rm{CAT}}_1$ to objects in ${\rm{CAT}}_2$ and if $\xi :X_1\to X_2$ is a morphism inter two objects $X1,X_2\in{\rm{CAT}}_1$ then

$F(\xi):F(X1)\to F(X_2)$

is a morphism $\in{\rm{CAT}}_2$. It is a common practice to mane ${\rm{Hom}}(X,Y)$ to the set of morphisms among the objects $X,Y$ in a category. Then a functor maps ${\rm{Hom}}(X,Y)\to {\rm{Hom}}(FX,FY)$

• The simplest example is the forgetful one $GP\to SET$ which gives to each group the underliying set, i.e. forgets the binary operation in the group
• Other example of a Functor is: ${\rm{Ab}}: GP\to GPA$ given by $G\to G/[G,G]$, which is the “abelianization of the group”
• Another: ${\rm{TOPOAC}}\to {\rm{GP}}$ given by $X\to \pi_1(X)$ the fundamental group functor, where TOPOAC is the category of arc-connected topo-spaces
• Cohomology functor. ${\rm{TOPOPAIRS}}\stackrel{H^*}\to {\rm{CHAINCOMPLEXES}}$ given by $(X,A)\mapsto \{H^0(A)\to H^0(X)\to H^0(X,A)\to H^1(A)\to ...\}$ which assigns to each topo-pair its cohomology-long-exact-sequence

Category in EOM More Pro reference

more on cat here in WP