Very general object-free categories? Is there a name for categories $\mathcal C$ such that $\text{Obj}(\mathcal C)$ coinside with $\text{Mor}(\mathcal C)$?
In the diagram below the morphisms are 
$a\overset{a}{\to}a$, $a\overset{b}{\to}c$, $d\overset{c}{\to}c$, 
$a\overset{d}{\to}e$, $e\overset{e}{\to}f$, $c\overset{f}{\to}e$.

I don't know if this is interesting mathematically. My idea is modelling. In the models all objects should be of the arrow-type in the same category. Nothing in the definition of categories exclude this. Domains and codomains can be defined as usual.
 A: Interpreted in a certain way what you are asking is standard. A category  $C$ is a collection $C_1$ together with two maps $s,t:C_1\to C_1$ and a map $m$ from $C_1\times_{\langle s,t\rangle} C_1 =\{(f,g)\,|\,s(f)=t(g)\}$ to $C_1$ satisfying:
$st=t$, $ts=s$, $m(m(f,g),h))=m(f,m(g,h))$ and $m(f,s(f))=f=m(t(f),f)$.
Note that normally we write $m(f,g)=f\circ g$.
To obtain the usual description of category set $C_0=\{f\in C_1\,|\,s(f)=f\}$, keep the same composition, and let the domain and codomain maps $c,d :C_1\to C_0$ be defined by $c(f)=t(f)$ and $d(f)=s(f)$ respectively.
Conversely given a category $C$ forget the set $C_0$, keep the same composition, and let $s$ and $t$ be the maps defined by $t(f) = 1_{c(f)}$ and $s(f)=1_{d(f)}$.
A: Depending on what precisely you are trying to model, you will need to specify some more axioms. You may then find out you are running into trouble. In any case, you may be interested in monoidal categories, or more generally in 2-categories. There is plenty online to read.
