Just as a monoid is a category with a single object, a semigroup may be seen as a non-unital category, still with associative composition. Then an $S$-set for $S$ a semi-group can be seen as a functor from the category corresponding to $S$ into the category of sets.


One of the nice things about the functor category of $M$-sets for $M$ a fixed monoid is that it's a topos, so in particular extensive i.e admitting a nice notion of connectedness. In this context an $M$-set $X$ is connected iff $\forall x,y\in X \exists s,t\in M$ such that $sx=y$ or $ty=x$.
I think this topos is also locally connected (there's a functorial assignment of connected components, left adjoint to discrete stuff) since we can just "take" these connected components for any monoid.


I was wondering whether not having units gets in the way of repeating the same terms with meaning for semigroups?


More generally, which portions of category theory still hold without identities? The most crucial thing that fails, I think, is Yoneda.



Category without identities?

> 
> which portions of category theory still hold without identities?
> 
> 
> 


Almost everything. <https://arxiv.org/abs/1311.3524v1>. (The link is to v1 of the paper, because the author has removed v2 from the arXiv.)


I'm honestly astonished by the fact that this paper hasn't the impact it deserves.


