# Abrupt categories induced by categories with star-morphisms

In this blog post I introduced the notion of category with star-morphisms, a generalization of categories which have aroused in my research.

Each star category gives rise to a category (abrupt category, see a remark below why I call it “abrupt”), as described below. Below for simplicity I assume that the set ${M}$ and the set of our indexed families of functions are disjoint. The general case (when they are not necessarily disjoint) may be easily elaborated by the reader.

• Objects are indexed (by ${\mathrm{arity}m}$ for some ${m \in M}$) families of objects of the category ${C}$ and an (arbitrarily choosen) object ${\mathrm{None}}$ not in this set
• There are the following disjoint sets of morphims:
1. indexed (by ${\mathrm{arity} m}$ for some ${m \in M}$) families of morphisms of ${C}$
2. elements of ${M}$
3. the identity morphism ${\mathrm{id}_{\mathrm{None}}}$ on ${\mathrm{None}}$
• Source and destination of morphims are defined by the formulas:
• ${\mathrm{Src}f = \lambda i \in \mathrm{dom}f : \mathrm{Src}f_i}$
• ${\mathrm{Dst}f = \lambda i \in \mathrm{dom}f : \mathrm{Dst}f_i}$
• ${\mathrm{Src}m =\mathrm{None}}$
• ${\mathrm{Dst}m =\mathrm{Obj}_m}$.
• Compositions of morphisms are defined by the formulas:
• ${g \circ f = \lambda i \in \mathrm{dom}f : g_i \circ f_i}$
• ${f \circ m =\mathrm{StarProd} \left( m ; f \right)}$
• ${m \circ \mathrm{id}_{\mathrm{None}} = m}$
• Identity morphisms for an object ${X}$ are:
• ${\lambda i \in X : \mathrm{id}_{X_i}}$ if ${X \neq \mathrm{None}}$
• ${\mathrm{id}_{\mathrm{None}}}$ if ${X =\mathrm{None}}$

We need to prove it is really a category.

Proof We need to prove:

1. Composition is associative
2. Composition with identities complies with the identity law.

Really:

1. ${\left( h \circ g \right) \circ f = \lambda i \in \mathrm{dom} f : \left( h_i \circ g_i \right) \circ f_i = \lambda i \in \mathrm{dom} f : h_i \circ \left( g_i \circ f_i \right) = h \circ \left( g \circ f \right)}$; $g \circ \left( f \circ m \right) = \mathrm{StarComp} \left( \mathrm{StarComp} \left( m ; f \right) ; g \right) = \\ \mathrm{StarComp} \left( m ; \lambda i \in \mathrm{arity} m : g_i \circ f_i \right) = \mathrm{StarComp} \left( m ; g \circ f \right) = \left( g \circ f \right) \circ m$; ${f \circ \left( m \circ \mathrm{id}_{\mathrm{None}} \right) = f \circ m = \left( f \circ m \right) \circ \mathrm{id}_{\mathrm{None}}}$.
2. ${m \circ \mathrm{id}_{\mathrm{None}} = m}$; ${\mathrm{id}_{\mathrm{Dst} m} \circ m = \mathrm{StarComp} \left( m ; \lambda i \in \mathrm{arity} m : \mathrm{id}_{\mathrm{Obj}_m i} \right) = m}$.

Remark I call the above defined category abrupt category because (excluding identity morphisms) it allows composition with an $m\in M$ only on the left (not on the right) so that the morphism $m$ is “abrupt” on the right.