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I was rereading the paper Directoids: algebraic models of up-directed sets by Ježek and Quackenbush, this time with category theory in mind. When I tried to describe what the results in that paper mean from a categorial perspective, I arrived at the following simple construction.

Let $P$ be a poset with $0$, not necessarily directed. Write $\nabla$ for the relation $P\times P\to P$, where $(a_1,a_2)\nabla b$ if and only if $a_1,a_2\leq b$. Then $(P,\nabla)$ is a semigroup in the monoidal category $(\mathbf{Rel},\times,1)$. However, it is not a monoid: the arrow $0:1\to P$ does not satisfy the triangle axiom strictly, for $a\in P$ we only have $a\in\nabla(0,a)$.

Question: What is the approriate categorial setting to describe such "monoids with lax unit"?

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  • $\begingroup$ Are you looking for more than just "a lax monoid object in a 2-category whose associativity is strict"? $\endgroup$ Feb 24, 2015 at 5:44
  • $\begingroup$ @MikeSchulman I do not know what I am looking for. That is the problem. $\endgroup$ Feb 24, 2015 at 8:35
  • $\begingroup$ @MikeSchulman That is the setting I am working in now. I am even assuming that the Hom-sets are posets; there are other examples besides the one in question (try $(\mathbf{Rel},\sqcup,\emptyset)$, it is fun). The problem is that everything works too nice, so it seems to me that I am working on a particular case of some theory that is known for decades. $\endgroup$ Feb 24, 2015 at 8:47
  • $\begingroup$ How much of your theory depends on the associativity being strict? The theory of ordinary lax monoids is of course well-known. $\endgroup$ Feb 25, 2015 at 17:40
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    $\begingroup$ One possible reference is maths.mq.edu.au/~street/Multicats.pdf. (Also, note that there is no "c" in my name.) $\endgroup$ Feb 26, 2015 at 5:17

1 Answer 1

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Saavedra's (beautiful!) theory of units might help: instead of the triangle for the unit object $0$, you only demand that maps $1 \times P \stackrel{0 \times 1}{\to} P \times P \stackrel{\nabla}{\to} P$ and $P \times 1 \stackrel{1 \times 0}{\to} P \times P \stackrel{\nabla}{\to} P$ are monomorphisms. A semimonoid with this property is called a Saavedra monoid. See e.g. section 4 of this nice paper by Kock.

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  • $\begingroup$ Thanks, the paper is really interesting. Alas, the maps are not monic in this case :-(. $\endgroup$ Feb 23, 2015 at 8:30

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