# Are there such things as cyclically presented (arrows-only) categories?

## Motivation:

Let $$w$$ be an element of the free group $$F_n$$ on the generators $$\{x_i\}_{i=0}^{n-1}$$. Define a function $$\theta: F_n\to F_n$$ by $$x_i\mapsto x_{i+1}$$ (and extend this to all elements of $$F_n$$), where the subscripts are taken modulo $$n$$. Consider the cyclic group presentation

$$\mathscr{P}_n(w):=\langle x_0, \dots , x_{n-1}\mid w, \theta(w), \dots , \theta^{n-1}(w)\rangle.$$

Then the group $$\mathscr{G}_n(w)$$ defined by $$\mathscr{P}_n(w)$$ is called a cyclically presented group.

Similarly, one can construct a cyclically presented semigroup.

## The Question:

Are there such things as "cyclically presented (arrows-only) categories"?

## Thoughts:

It might be necessary to restrict ourselves to arrows-only categories to make sense of the analogue of the "shift" $$\theta$$ in the case of group presentations.

I think I understand the definition of a presentation of a category:

Let $$G$$ be a directed graph, and let $$R$$ be a function that assigns to each pair $$a,b$$ of objects of the free category $$F(G)$$ a binary relation $$R_{a,b}$$ on the hom-set $$F(G)(a,b)$$. The category with generators $$G$$ and relations $$R$$ is the quotient category. For a category $$C$$, an isomorphism $$C\to F(G)/R$$ is called a presentation of $$C$$.

This definition seems to require that the binary relations, which in groups define the relators of the presentation, need to be "in two generators only", for lack of a better way of putting it. That might prevent us from making sense of a "cyclically presented (arrows-only) category".

Based on what it says about (the definition of) quotient categories being "evil" (which I don't fully understand), perhaps a cyclically presented category would, too, be "evil".

It's not necessarily the case that $$x_n\mapsto x_{n+1}$$ gives a well defined function on the relations. You need a condition that if $$x_mx_n$$ is defined then so is $$x_{n+1}x_{m+1}$$. Given that, there's no issue.