nLab abrupt category

Each category with star-morphisms 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 MM 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 aritym\operatorname{arity}m for some mMm \in M) families of objects of the category CC and an (arbitrarily choosen) object None\operatorname{None} not in this set

% There are the following disjoint sets of morphisms:

  • indexed (by aritym\operatorname{arity} m for some mMm \in M) families of morphisms of CC

  • elements of MM

  • the identity morphism id None\operatorname{id}_{\operatorname{None}} on None\operatorname{None}

% Source and destination of morphims are defined by the formulas:

  • Srcf=λidomf:Srcf i\operatorname{Src}f = \lambda i \in \operatorname{dom}f : \operatorname{Src}f_i;

  • Dstf=λidomf:Dstf i\operatorname{Dst}f = \lambda i \in \operatorname{dom}f : \operatorname{Dst}f_i

  • Srcm=None\operatorname{Src}m =\operatorname{None}

  • Dstm=Obj m\operatorname{Dst}m =\operatorname{Obj}_m.

% Compositions of morphisms are defined by the formulas:

  • gf=λidomf:g if ig \circ f = \lambda i \in \operatorname{dom}f : g_i \circ f_i

  • fm=StarProd(m;f)f \circ m =\operatorname{StarProd} \left( m ; f \right)

  • mid None=mm \circ \operatorname{id}_{\operatorname{None}} = m

  • id Noneid None=id None\operatorname{id}_{\operatorname{None}} \circ \operatorname{id}_{\operatorname{None}} = \operatorname{id}_{\operatorname{None}}

% Identity morphisms for an object XX are:

  • λiX:id X i\lambda i \in X : \operatorname{id}_{X_i} if XNoneX \neq \operatorname{None}

  • id None\operatorname{id}_{\operatorname{None}} if X=NoneX =\operatorname{None}

We need to prove it is really a category.

Proof We need to prove:

  • Composition is associative

  • Composition with identities complies with the identity law.

Really:

  • (hg)f=λidomf:(h ig i)f i=λidomf:h i(g if i)=h(gf)\left( h \circ g \right) \circ f = \lambda i \in \operatorname{dom} f : \left( h_i \circ g_i \right) \circ f_i = \lambda i \in \operatorname{dom} f : h_i \circ \left( g_i \circ f_i \right) = h \circ \left( g \circ f \right);

g(fm)=StarComp(StarComp(m;f);g)=StarComp(m;λiaritym:g if i)=StarComp(m;gf)=(gf)mg \circ \left( f \circ m \right) = \operatorname{StarComp} \left( \operatorname{StarComp} \left( m ; f \right) ; g \right) = \operatorname{StarComp} \left( m ; \lambda i \in \operatorname{arity} m : g_i \circ f_i \right) = \operatorname{StarComp} \left( m ; g \circ f \right) = \left( g \circ f \right) \circ m;

f(mid None)=fm=(fm)id Nonef \circ \left( m \circ \operatorname{id}_{\operatorname{None}} \right) = f \circ m = \left( f \circ m \right) \circ \operatorname{id}_{\operatorname{None}}.

  • mid None=mm \circ \operatorname{id}_{\operatorname{None}} = m; id Dstmm=StarComp(m;λiaritym:id Obj mi)=m\operatorname{id}_{\operatorname{Dst} m} \circ m = \operatorname{StarComp} \left( m ; \lambda i \in \operatorname{arity} m : \operatorname{id}_{\operatorname{Obj}_m i} \right) = m.

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

Categories with star-morphisms and abrupt categories arise in research of cross-composition product.

Created on June 14, 2012 at 22:58:53. See the history of this page for a list of all contributions to it.