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In category theory, I have seen "weakly initial object" used as follows: $X$ is weakly initial iff for all objects $Y,$ there is at least one arrow $X \rightarrow Y$. Of course, another way of weakening the definition of "initial object" would be to require that for all objects $Y$, there is at most one arrow $X \rightarrow Y$. Let us call such objects "coweakly initial."

Is there an accepted term for "coweakly initial" objects? (This particular phrase returns no hits on Google.)

Examples and motivation.

  1. Every object of a poset (viewed as a category) is coweakly initial. Indeed, this characterizes posets; the posetal categories are precisely those categories whose every object is coweakly initial.

  2. In the category of fields, following objects are coweakly initial: $$\{\mathbb{Q}\} \cup \{\mathbb{F}_p \mid p \mbox{ is prime}\}.$$

    (I'm not 100% sure whether or not there are other coweakly initial objects in $\mathbf{Field}$ besides these.)

    More generally, if $\mathbf{C}$ is an $I$-indexed family of categories, then for all $i \in I,$ the inclusion $\eta_i : \mathbf{C}_i \rightarrow \coprod_{i \in I} \mathbf{C}_i$ both preserves and reflects coweak initiality.

  3. Take all semirings to be unital, and all semiring homomorphisms to preserve $1$. Then $\mathbb{Z}$ is coweakly initial in the category of semirings. (There is an arrow $\mathbb{Z} \rightarrow S$ iff $S$ is in fact a ring.) Similarly, let $\mathbb{B}$ denote the semiring whose underlying set is $\{0,1\}$ that satisfies $1+1=1$. Then $\mathbb{B}$ is coweakly initial. (There is an arrow $\mathbb{B} \rightarrow S$ iff $S$ has idempotent addition.)

    More generally, if $N$ is coweakly initial, and if there is an epimorphism $N \rightarrow M,$ then $M$ is also coweakly initial. The above two examples are obtained by taking $N$ equal to $\mathbb{N}$ and $M$ equal to $\mathbb{Z}$ and $\mathbb{B}$ respectively.

    I think that, in general, if $\mathbf{C}$ is a category, it is sometimes interesting to study the subcategory $\mathbf{C}^\diamond$ of coweakly initial objects, since this is always posetal. Furthermore, given any element $N$ of $\mathbf{C}^\diamond$, we may consider all the objects of $\mathbf{C}$ that $N$ has an arrow to, thereby getting a full subcategory of $\mathbf{C}$ in which $N$ is initial. This establishes an antitone mapping from $\mathbf{C}^\diamond$ into the poset of full subcategories of $\mathbf{C}$. I think the image of this mapping consists of all full subcategories of $\mathbf{C}$ that happen to have initial objects.

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    $\begingroup$ According to the nlab, the dual notion is a subterminal object (the idea being that if there is a terminal object, then the property is equivalent to being a subobject of the terminal object). How to dualize the terminology is not clear to me. "Subinitial object" has the unfortunate connotation that the object should be a subobject of the initial object. You could say "co-subterminal", I suppose. What you really need is a prefix that means "a quotient object of", and put it before "initial object". $\endgroup$
    – tcamps
    Jan 5, 2015 at 21:42
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    $\begingroup$ Also related: if a category $\mathcal{C}$ has a set of objects such $S$ that for every $X \in \mathcal{C}$ there is a unique object $I \in S$ admitting an arrow $I \to X$, and moreover that arrow $I \to X$ is unique, then the set $S$ is called a multi-initial object (in particular, the objects in $S$ are "co-weakly intial" / co-subterminal). For example, you exhibit a multi-initial object in the category of fields. $\endgroup$
    – tcamps
    Jan 5, 2015 at 21:52
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    $\begingroup$ The closest thing I can think of to a prefix meaning "a quotient object of" is the suffix "-ly indexed", in analogy with "finitely-indexed" objects. So you could try "initially-indexed object". A variant would be "initially-generated" object, in analogy with e.g. finitely generated groups. But maybe these possibilities are taking the notion of "that's what it would be if there were an initial object" too far. Maybe something simple like "pre-initial object" would do... $\endgroup$
    – tcamps
    Jan 5, 2015 at 22:11
  • $\begingroup$ @tcamps, thanks for all the info man. I guess "epi-initial object" or some variant thereof would be workable. $\endgroup$ Jan 5, 2015 at 23:51
  • $\begingroup$ Glad I could help. But how do you know I am not a woman?:) $\endgroup$
    – tcamps
    Jan 6, 2015 at 3:57

1 Answer 1

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I was hoping to find something more definitive, but it doesn't look like it's going to happen. And I suppose it's good hygiene to have questions that have been more-or-less answered more visibly "marked" as answered. So, prodded by Martin's comment, here are my comments in answer form:

According to the nlab, the dual notion is a subterminal object (the idea being that if there is a terminal object, then the property is equivalent to being a subobject of the terminal object). How to dualize the terminology is not clear to me. "Subinitial object" has the unfortunate connotation that the object should be a subobject of the initial object. You could say "co-subterminal", I suppose. What you really need is a prefix that means "a quotient object of", and put it before "initial object".

The closest thing I can think of to a prefix meaning "a quotient object of" is the suffix "-ly indexed", in analogy with "finitely-indexed" objects. So you could try "initially-indexed object". A variant would be "initially-generated" object, in analogy with e.g. finitely generated groups. But maybe these possibilities are taking the notion of "that's what it would be if there were an initial object" too far. Maybe something simple like "pre-initial object" would do...

Also related: if a category $\mathcal{C}$ has a set of objects $S$ such that for every $X \in \mathcal{C}$ there is a unique object $I \in S$ admitting an arrow $I\to X$, and moreover that arrow $I \to X$ is unique, then the set S is called a multi-initial object (in particular, the objects in S are "co-weakly initial" / co-subterminal/ initially indexed). For example, you exhibit a multi-initial object in the category of fields.

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