Definition of Initial&Terminal Objects in an Object-Free'' Category - MathOverflow most recent 30 from http://mathoverflow.net 2013-06-18T21:46:23Z http://mathoverflow.net/feeds/question/60833 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/60833/definition-of-initialterminal-objects-in-an-object-free-category Definition of Initial&Terminal Objects in an Object-Free'' Category SHK 2011-04-06T16:09:42Z 2011-04-07T01:56:41Z <p>I'm reading "Abstract and Concrete Categories" and, in the Chap. I (Definition 3.52 on Page 41), there's an Object-Free'' definition of a Category which, through the isomorphism $A \rightarrow \textit{id}_A$ turns out to be equivalent to the usual one. But all the other definitions in the book are only given in the usual "Objects+Morphisms" fashion. What is the definition of Initial&amp;Terminal Objects in the "Object-Free" version of a Category?</p> http://mathoverflow.net/questions/60833/definition-of-initialterminal-objects-in-an-object-free-category/60838#60838 Answer by Chris Heunen for Definition of Initial&Terminal Objects in an Object-Free'' Category Chris Heunen 2011-04-06T17:25:27Z 2011-04-06T17:25:27Z <p>In this view, objects are equated with morphisms that are identities, or "units" in their terminology. So a morphism $x$ is initial when it is a unit and for every unit $y$ there is a unique morphism $f$ for which $f \circ x$ and $y \circ f$ are defined. Similarly, a morphism $y$ is terminal when it is a unit and for every unit $x$ there is a unique morphism $f$ for which $f \circ x$ and $y \circ f$ are defined.</p> http://mathoverflow.net/questions/60833/definition-of-initialterminal-objects-in-an-object-free-category/60870#60870 Answer by Hurkyl for Definition of Initial&Terminal Objects in an Object-Free'' Category Hurkyl 2011-04-07T01:56:41Z 2011-04-07T01:56:41Z <p>As the others have said, the object-free definition can always define objects later and do everything normally. One might ask if there was a different-than-normal definition that was more arrow-like. For example:</p> <p>A morphism $f$ is "terminating" if, for every morphism $g$, there exists a unique morphism $h$ such that $h \circ g$ and $f$ have the same target.</p> <p>So, the target of $f$ is a terminal object, and $f$ itself is the unique projection from its source. But this seems like a superficial change to me. Maybe someone else knows something better?</p>