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title: Cocommas in quantale enriched categories

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# Cocommas in quantale enriched categories

Let $\Omega$ be a quantale.

**Cocommas in Posets**

If $\Omega=2$, then a cocomma of a span 
$$A \stackrel{p}{\longleftarrow} B \stackrel{q}{\longrightarrow} C$$

is the colimit weighted by 
- $\phi(A)=\phi(C)=1=\{0\}$ and 
- $\phi(B)=\{0<1\}$ and 
- $\phi(p)(0)=0$ and 
- $\phi(q)(0)= 1$. 

Explicitely, the cocomma is the disjoint union $D=A+C$ with 
- $D(a,c) = \textsf{ if } \exists b(a=pb\wedge c=pb) \textsf{ then } 1 \textsf{ else } 0$
- $D(c,a)=0$
- $D(a,a') = A(a,a')$
- $D(c,c') = C(c,c')$

**Cocommas in Quantale enriched categories**

In principle, there could be more than one way to define the weight of $B$. But the following seems the obvious generalisation of the poset case.

Modify the definition of the poset case so that $\phi(B)= \{0,1\}$ with 

|$x$ | $y$ |$\phi(B)(x,y)$|
|:---:|:---:|:---:|
|0|0| $e$|
|0|1| $e$|
|1|0| $\bot$ |
|1|1| $e$|

Then the cocomma of $A \stackrel{p}{\longleftarrow} B \stackrel{q}{\longrightarrow} C$ is the disjoint union $D=A+C$ with 
- $D(a,c) = \textsf{ if } \exists b(a=pb\wedge c=pb) \textsf{ then } e \textsf{ else } \bot$
- $D(c,a)= \bot$
- $D(a,a') = A(a,a')$
- $D(c,c') = C(c,c')$

**Composing cospans of automata**

If $\Omega$ is the powerset of the free monoid over an alphabet $\Sigma$, then a cospan of $\Omega$-categories is a non-determinstic automaton in which transitions are labelled not with symbols from $\Sigma$ but with languages over $\Sigma$.

While composing these cospans via pushouts corresponds to glueing final and initial states, composing via cocommas corresponds to inserting $\epsilon$-transitions between final states and initial states (labelled with the same symbol from $B$).