# A new category $C^*$ from a given category $C$

For a given category $$C$$ define the category $$C^*$$ as follows: the objects of $$C^*$$ are those of $$C$$; for given objects $$u,v$$, the $$C^*$$-morphisms $$u\to v$$ are all finite sequences $$(a_1,\dots,a_n)$$ of morphisms $$a_i\in Mor(C)$$ such that $$a_1:u\to x$$ and $$a_n:y\to v$$ where $$x$$ and $$y$$ are arbitrary objects (no other restrictions). The composition of two composable morphisms is $$(a_1,\dots,a_m)(b_1,\dots,b_n)=(a_1,\dots,a_mb_1,\dots,b_n)$$ (where $$a_mb_1$$ are composed in $$C$$). My question: is there an established name for the category $$C^*$$?

Update:

1) Motivation: the construction $$C\mapsto C^*$$ appears as a tool of proof in my research and I wanted to know if, where and for which purpose this construction appears in literature, in order to insert some references. Intuitively I would say this is somehow the free category generated by $$C \cup (C\times C)$$ modulo the relations which hold in $$C$$.

2) Intuitive background: I have algebraic terms of a certain type which can be interpreted as instructions what to do in a certain category $$C$$. There are two sorts of instructions:

a) type $$w$$: they tell me I should compose certain morphisms of $$C$$ the result of which is the morphism $$a_1$$, say, and thereby I run through the underlying graph $$\Gamma$$ of $$C$$

b) type $$w^{\mathfrak m}$$: they say I should jump elsewhere in the graph $$\Gamma$$

The application of such instructions alternatingly ends up with a tuple $$(a_1,a_2,\dots,a_n)$$ as in the question. The category $$C^*$$ seems to model exactly this behaviour. For technical reasons, I allow in type b) "empty jumps", that is, even if two consecutive morphisms $$a_i$$ and $$a_{i+1}$$ are composable in $$C$$ I would like to distinguish between $$\dots a_i,a_{i+1}\dots$$ and $$\dots a_ia_{i+1}\dots$$.

• Does the composition have $n+m-1$ terms? Jan 15, 2019 at 19:16
• @Dog_69 Yes, the adjacent morphisms $a_m$ and $b_1$ are composed in $C$, so $a_mb_1$ is one entry of the sequence. Jan 15, 2019 at 19:23
• And I suppose that two morphisms if $m=n$ and they agree in each component, isn't it? Jan 15, 2019 at 20:17
• Do you have a particular reason to think that this category may have an established name? The definition looks a bit arbitrary to me... Jan 15, 2019 at 21:45
• @Dog_69 No, for every $n$ there can be many tuples of length $n$ (I'm not sure if I understood the question correctly). Jan 16, 2019 at 10:42

• This is not the case. For example, if $C$ is finite, $C^*$ will be infinite (if $C$ has at least two different morphisms). Jan 15, 2019 at 20:03
• I don't see this. Among the morphisms in $C^*$ from $A$ to $B$, you have all the morphisms in $C$ from $A$ to $B$ (regarded as sequences of length 1), and you also have sequence of greater length. Note that a directed path n $C$ from $A$ to $B$ is a morphism in $C^*$ and has not been identified with the composite of its members. Furthermore, the terms $a_i$ in a $C^*$-morphism needn't even line up to form a path in $C$. Jan 15, 2019 at 20:05