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This question was inspired by the Homotopy Type Theory Book.

Might we define a weak $\omega$-category as described below?

Is any similar approach already considered in the literature?

Let $\def\Ob{{\rm Ob}\,} \Ob\mathcal C$ be a set [class] with partial functions $\def\dom{\rm dom} \def\cod{\rm cod} \dom,\cod$ defined on a subset $\mathcal C'\subseteq\Ob\mathcal C$ (i.e. $\dom,\cod:\mathcal C'\to\Ob\mathcal C$). We write of course $f:A\to B$ for $\dom(f)=A,\ \cod(f)=B$ and elements of $\mathcal C'$ are also called arrows.
We require a composition of consecutive arrows and identity arrows, satisfying associativity and unit constraintments with coherence isomorphisms such that the coherence diagrams commute up to equivalence.
(Objects $A$ and $B$ are equivalent if there is a binary tree of objects $X_{\bf t}$ and arrows $f_{\bf t}$, ${\bf t}\in\{0,1\}^*$ such that $X_0=A,\ \ X_1=B,\ \ \ f_{{\bf t}0}:X_{{\bf t}0}\to X_{{\bf t}1},\ \ f_{{\bf t}1}:X_{{\bf t}1}\to X_{{\bf t}0}$ where objects $X_{{\bf t}00},\ X_{{\bf t}01},\ X_{{\bf t}10},\ X_{{\bf t}11}$ are defined to be $f_{{\bf t}1}\circ f_{{\bf t}0} $, $\ 1_{X_{{\bf t}0}}$, $f_{{\bf t}0}\circ f_{{\bf t}1} $, $\ 1_{X_{{\bf t}1}}$; so that $A\simeq B \iff \exists f:A\to B,\ g:B\to A\,$ s.t. $\,g\circ f\simeq 1_A,\ f\circ g\simeq 1_B$)
To formulate these, we also need a horizontal composition functor, from the full substructure of $\mathcal C'\times\mathcal C'$ generated by consecutive pairs of arrows as objects, to $\mathcal C'$.
(For first,) globularity is assumed (i.e., $\alpha:f\to g$ with $f,g\in\mathcal C'$ implies $\cod(f)=\cod(g)$ and $\dom(f)=\dom(g)$).

Objects are $0$-cells, and an arrow $f:A\to B$ is regarded an $n+1$-cell whenever $A$ and $B$ are $n$-cells. [Note that an $n$-cell is automatically also $k$-cell for any $k<n$.]

More details can be found in my note.

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    $\begingroup$ The central unsolved problem of all these "algebraic" definitions of weak omega-categories is to give the right space of morphisms between these objects. The obvious morphisms that preserve all the algebraic structure are typically too strict. If one wants to use them to model weak omega-functors, one needs to have a method to resolve the domain, hence one needs a minimum of homotopy theory on the given naive category of weak omega-categories. $\endgroup$ – Urs Schreiber Dec 31 '13 at 13:44
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    $\begingroup$ Once one has a candidate definition of this, one needs to check that the homotopy type of maps between any two weak n-categories for finite n is equivalent to what it is supposed to be, as known from the non-algebraic definitions that work. This has not been done for any of the many proposals for algebraically defined weak omega-categories, as far as I am aware. $\endgroup$ – Urs Schreiber Dec 31 '13 at 13:45
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    $\begingroup$ To add to what Urs wrote, I think a main proposed technique is a method of cofibrant replacement for the domain so that weak morphisms are identified with strict morphisms out of the cofibrant replacement. It is quite true that this area is sadly underdeveloped (and it's something I mean to return to, someday). $\endgroup$ – Todd Trimble Dec 31 '13 at 15:54
  • $\begingroup$ A major problem with this method is that compositions in the 3rd, 4th... directions don't automatically arise, which would be expected in higher categories. $\endgroup$ – Berci Feb 17 '14 at 0:07
  • $\begingroup$ Isn't this almost exactly the same as strict $\omega$ categories (defined by R. Street)? $\endgroup$ – Adam Gal Mar 3 '14 at 7:15

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