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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.

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  • $\begingroup$ 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
    Commented Dec 8, 2016 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
    Commented Dec 8, 2016 at 13:19
  • $\begingroup$ Is a good theory. $\endgroup$
    – Benjamin Steinberg
    Commented Dec 8, 2016 at 13:19
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    $\begingroup$ A relevant amusing quotation from Lawvere (Some thoughts on the future of category theory): "Quite non-trivial in fact is also the idea that there must be definite domains and codomains and that there must be identity maps; even today there are many who think one could usefully "generalize" by omitting those requirements, sometimes on grounds of dislike for the "stasis" they think they imply. However, in modern Greek "stasis" means "bus-stop"; how useless an intricate network of speeding buses would be without them, and how disembodied would be processes without states." $\endgroup$
    – Alexander Campbell
    Commented Dec 8, 2016 at 21:08
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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
    Commented Dec 9, 2016 at 1:16

1 Answer 1

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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.

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  • $\begingroup$ 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
    Commented Dec 14, 2016 at 1:11
  • $\begingroup$ I'll try to contact Salvatore. For the moment, refer to v1 $\endgroup$
    – fosco
    Commented Dec 14, 2016 at 8:52
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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
    Commented Dec 29, 2018 at 7:21
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    $\begingroup$ Is any result contained in v1 wrong? It looks pretty interesting to me. $\endgroup$
    – user57432
    Commented Jun 25, 2019 at 6:45
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    $\begingroup$ I just re-opened this post after an upvote; sorry for not having answered before :) To my knowledge, nothing in Salvo's paper was wrong. The entire framework he was building clarifies, at least to me, many phenomena in universal algebra and CT, and yet, at the times, his work was considered "too abstract" also for category theory: he struggled for a while to have it published, but to no avail. I suspect this has had a role in the withdrawal. imsc.uni-graz.at/tringali as you can see, Salvo has, and has had, a pretty wide variety of mathematical interests. $\endgroup$
    – fosco
    Commented Jan 31, 2020 at 8:30

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