Any number of constructions guarantee the existence of maps $f$ without guaranteeing their uniqueness. Some time ago, I was introduced to the terminology "versal" for such a construction.

I wonder: is this widespread? Is the origin of the terminology known?

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    $\begingroup$ It's a pun on "universal". $\endgroup$ – Dan Petersen Nov 24 '10 at 16:03
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    $\begingroup$ Right. Universal without uni(queness) $\endgroup$ – Francesco Polizzi Nov 24 '10 at 16:11
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    $\begingroup$ Arnold uses "versal" in his book on classical mechanics, but also "miniversal", to speak of models of minimal dimension. $\endgroup$ – Denis Serre Nov 24 '10 at 16:14
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    $\begingroup$ And yes it is widespread. (I guess if both Arnold and hard core stack people use it it must be...) $\endgroup$ – Torsten Ekedahl Nov 24 '10 at 16:57
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    $\begingroup$ And just as the noun "denial" comes from the verb "deny", and "reversal" from "reverse", and "acquittal" from "acquit", and similary for about three dozen verbs in the hodge-podge of a language that is English, we have here an opportunity to create a new back-formed verb. $\endgroup$ – Michael Hardy Nov 24 '10 at 18:44

coined in the 60's from "universal" by René Thom to describe "unfoldings" of singularities:

An $r$-unfolding of a function $f: \mathbb{R}^n \to \mathbb{R}$ is a function $F: \mathbb{R}^{n+r} \to \mathbb{R}$ such that $F(x_1,..., x_n, 0,..., 0) = f(x_1,..., x_n)$. An $r$-unfolding of $f$ is versal if all other unfoldings of $f$ can be induced from it. It is universal if $r$ is the smallest dimension for which a versal $r$-unfolding of $f$ exists.

Further comment (due credit to John Mather):

Going back to Thom's complete works, I came upon the following unpublished comment:

« En mathématique pure, mes propres résultats n'allèrent guère au-delà de développements limités de certaines singularités de potentiel. Il fallut la pertinence de mathématiciens américains (Milnor) ou européens (théorie du déploiement universel, Grauert, J. Martinet) pour sortir la théorie de son marasme initial. Mon seul apport à la théorie mathématique fut d'introduire la notion de « déploiement universel » - corrigé peu après en versel par les collègues algébristes (Mather). Il n'y a pas de doute que des mathématiciens américains (Mather,Milnor), puis soviétiques (Arnold) ont apporté à la théorie des singularités des progrès décisifs. La vision de ces mathématiciens m'a fait comprendre combien la théorie des singularités a des origines profondes en mathématiques. C'est la rencontre de mathématiciens soviétiques comme Arnold (souvent férocement critique de mes procédés rustres) qui m'a fait comprendre à quel point la théorie des singularités tire son origine de structures profondes (Polynômes de Dynkin, carquois de Gabriel, théorie des tresses, immeubles de Tits). L'intérêt de la T.C. est bien d'avoir attiré l'attention sur ces théories « profondes » dont la source reste (pour moi) bien mystérieuse.»

So, when Thom did introduce the terminology "d\'eploiement universel" (universal unfolding), John Mather is, according to Thom himself, responsible for the (relevant) alteration to "versal".

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    $\begingroup$ I took the liberty to TeXify the answer for readability. $\endgroup$ – José Figueroa-O'Farrill Nov 24 '10 at 18:10

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