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Does the category that consists of two objects and exactly one non-identity morphism that connects both objects have a specific name?

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  • $\begingroup$ Welcome to math.SE! I think that we've already met on the MP. ;-) $\endgroup$ Apr 9, 2015 at 16:02
  • $\begingroup$ Correct ;) Seems like here are more people answering on category theory questions (and my miserable language skills will benefit as well I hope ;)) $\endgroup$ Apr 9, 2015 at 16:49
  • $\begingroup$ @MartinBrandenburg what is MP? $\endgroup$
    – magma
    Apr 10, 2015 at 0:49

3 Answers 3

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The interval category. It is often denoted $\mathbf{I}$. Note that if $\mathbf{C}$ is category, then $\mathbf{C}^\mathbf{I}$ is the arrow category of $\mathbf{C}$.

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  • $\begingroup$ Thanks! I have used it to show that the functor from Cat to Set that sends a small category to its arrow set is representable so your remark was helpful as well (in the sense that it confirmed my result). $\endgroup$ Apr 9, 2015 at 14:59
  • $\begingroup$ The geometric realization of the interval category is the interval $[0,1]$. Therefore the name. $\endgroup$ Apr 9, 2015 at 17:18
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As goblin suggests, "the interval category" is good, and it has the imprimatur of the nlab. I might also call it "the arrow category". If I say "the interval category", I somehow feel the need to clarify that I'm talking about the category with one nontrivial arrow, rather than one nontrivial isomorphism. I might also use the notation $\uparrow$, although this is hard to pronounce.

Another possibility is "the walking arrow" (as opposed to "the walking isomorphism"). In general, if you have a functor $F$ which is isomorphic to $\mathrm{Hom}(X,-)$, then you might call $X$ "the walking $F$" (in this case we have an isomorphism of 2-functors $\mathrm{Arr} \cong \mathrm{Hom}(\uparrow, - ): \mathsf{Cat} \to \mathsf{Cat}$ where $\mathrm{Arr}$ sends a category $C$ to its arrow category $\mathrm{Arr} C$). This terminology probably feels a little too informal in some contexts, although it's fairly precise.

I suppose you could also say "the universal arrow" (as opposed to "the universal isomorphism").

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    $\begingroup$ Another good option would be: "let $\mathbf{I}$ denote the generic arrow." I agree that "interval category" can give the wrong impression. $\endgroup$ Apr 9, 2015 at 15:02
  • $\begingroup$ I like this option with "the walking arrow" as it encodes further information on the usage of this category. Thanks for this. $\endgroup$ Apr 9, 2015 at 15:06
  • $\begingroup$ I've been using $\textbf{I}_+$ for the directed interval category and $\textbf{I}$ for the interval groupoid. Kind of conflicts with notation for monos though. But it is a bit more extendable for stuff like delooping of the integers/naturals under addition $\textbf{S}$, $\textbf{S}_+$. "Circle groupoid" and "directed circle category" are kind of abusive terms anyway though. $\endgroup$ Jun 9, 2022 at 1:28
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It might also be denoted $\mathbf 2$ as it is the category associated to the cardinal poset $\mathbf 2 = \{0<1\}$.

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    $\begingroup$ This notation is also used by Mac Lane and therefore by many other authors. It is easy to confuse with $2=\{0,1\}$, the discrete category on two objects, which is also the reason why I don't like this notation. $\endgroup$ Apr 9, 2015 at 16:01
  • $\begingroup$ In this vein, I might also use $[1]$, viewing it as the 1-simplex. $\endgroup$
    – tcamps
    Apr 9, 2015 at 16:48
  • $\begingroup$ My first idea was also to use 2 and my second thought was what Martin said. It's some kind of funny, I came with to less options and now I have to many. $\endgroup$ Apr 9, 2015 at 18:46

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