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.

  • $\begingroup$ For an M-set being connected is much weaker. It means the equivalence relation generated by x relates to mx has a single class. $\endgroup$ – Benjamin Steinberg Dec 8 '16 at 13:15
  • $\begingroup$ S-sets for a semigroup S is the same as M-sets for the monoid obtained by adjoining an identity but notice this changes your topos if S is already a monoid. If S has enough idempotents like having local units there $\endgroup$ – Benjamin Steinberg Dec 8 '16 at 13:19
  • $\begingroup$ Is a good theory. $\endgroup$ – Benjamin Steinberg Dec 8 '16 at 13:19
  • $\begingroup$ Ignoring units yields the theory of Semigroupoids and a lot of work has been done on them, for example by Wolfram Kahl. $\endgroup$ – Musa Al-hassy Dec 8 '16 at 19:10
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    $\begingroup$ Arrow, one of the nLab regulars named Thomas Holder asked me to comment that the following reference may be quite useful for you, on the question of categories without units: Moens, Berni-Canani, Borceux: On regular presheaves and regular semi-categories , Cah.Top.Diff.Géom.Diff.Cat. XLIII-3 (2002) pp.163-190. ( numdam.org/item?id=CTGDC_2002__43_3_163_0 ) $\endgroup$ – Todd Trimble Dec 9 '16 at 1:16

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.

  • $\begingroup$ The linked page mentions an April 2016 revision, but I see no pdf available. Could you post a link to the revised version? $\endgroup$ – Arrow Dec 14 '16 at 1:11
  • $\begingroup$ I'll try to contact Salvatore. For the moment, refer to v1 $\endgroup$ – Fosco Dec 14 '16 at 8:52
  • $\begingroup$ Any news on the theory of plots? $\endgroup$ – eternaleye Nov 23 '18 at 22:52
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    $\begingroup$ Unfortunately Salvo removed the paper from arxiv. You can reach him by email anyway, asking for more info! $\endgroup$ – Fosco Dec 29 '18 at 7:21
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    $\begingroup$ Is any result contained in v1 wrong? It looks pretty interesting to me. $\endgroup$ – user 170039 Jun 25 '19 at 6:45

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