@MISC{CAT_6.MIZ,
  AUTHOR = {Riccardi, Marco},
  TITLE = {Object Free Category},
  SECTION1 = {Yet Another Definition of Category},
  SECTION2 = {Transform a Category in the Other},
  DAY = {7},
  MONTH = {October},
  YEAR = {2013},
  ADDRESS1 = {Via del Pero 102\\54038 Montignoso\\Italy},
  SUMMARY = {Category theory was formalized in Mizar with two different approaches
  \cite{CAT_1.ABS}, \cite{ALTCAT_1.abs} that correspond to those most commonly used
  \cite{MacLane:1}, \cite{Borceaux}.
  Since there is a one-to-one correspondence between objects and identity morphisms, some authors have
  used an approach that does not refer to objects as elements of the theory, and are usually indicated
  as object-free category \cite{Adame:2009} or as arrows-only category \cite{MacLane:1}.
  In this article is proposed a new definition of an object-free category, introducing the two properties
  left-composable and right-composable, and a simplification of the notation through a symbol,
  a binary relation between morphisms, that indicates whether the composition is defined.
  In the final part we define two functions that allow to switch from the two definitions, with and
  without objects, and it is shown that their composition produces isomorphic categories.}}

@BOOK{Borceaux,
       AUTHOR = {Borceaux, Francis},
        TITLE = {Handbook of Categorical Algebra I. {B}asic Category Theory},
    PUBLISHER = {Cambridge University Press},
         YEAR = {1994},
       VOLUME = {50},
       SERIES = {Encyclopedia of Mathematics and its Applications},
      ADDRESS = {Cambridge}}

@BOOK{Adamek:2009,
       AUTHOR = {Adamek, Jiri and Herrlich, Horst and Strecker, George E.},
        TITLE = {Abstract and Concrete Categories: The Joy of Cats},
    PUBLISHER = {Dover Publication},
         YEAR = {2009},
      ADDRESS = {New York}}
