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In reading about model theory (David Marker's Book), a witness has a specific meaning, i.e. an element $t$ making $\exists x\ \phi (x)$ true. Some theories have the the witness property. In reading Category Theory In Context, Riehl writes (p 50)

The universal element witnessing the universal property of the complete graph is an $n$-coloring $K_n$, an element of the set $n\text{-}\mathrm{Color}(K_n)$.

which has a clearly related meaning, although it doesn't seem obvious that there is some formula being witnessed. My question is that I am wondering if there is a formal idea here, or if witness just means in effect "makes true". I don't mean to be pedantic, the meaning is understood, and sorry if this is too vague.

Edit: Martin Brandenburg pointed out in the comments that it is clearly a formula, be it in a language I am not familiar with. I think my question more accurately stated is whether the informal usage of "witness" always coincides with its formal usage?

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    $\begingroup$ IMO it is not "formal": it only means "an onject instantiating a general property". $\endgroup$
    – Mauro ALLEGRANZA
    Commented Jan 24 at 7:17
  • $\begingroup$ @MauroALLEGRANZA ah that makes sense. I was secretly hoping there'd be some broad formal theory of witnesses permeating my intro to category theory :) $\endgroup$
    – 4u9ust
    Commented Jan 24 at 7:32
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    $\begingroup$ "it doesn't seem obvious that there is some formula being witnessed" - why do you think so? Clearly what comes after ... is a formula, in the language of category theory or an extension of it. Just phrased in an informal way. Please clarify. Also, give a specific example where you think that it's not a formula. This will help to formulate better answers. $\endgroup$
    – Martin Brandenburg
    Commented Jan 24 at 9:38
  • $\begingroup$ @MartinBrandenburg you are completely. I edited the question to complete the quote, but it is definitely a formula in some sufficiently sorted language, although I wouldn't be exactly sure what language that would be, but that speaks more to being new to category theory. I suppose this changes my question to: does the informal usage of witness always coincide with its formal usage? $\endgroup$
    – 4u9ust
    Commented Jan 26 at 0:04
  • $\begingroup$ completely right* $\endgroup$
    – 4u9ust
    Commented Jan 26 at 1:26

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