Deformation theory tag info https://mathoverflow.net/tags/deformation-theory/info is very minimal and says almost nothing:

deformation theory is the study of infinitesimal conditions associated with varying a solution P of a problem to slightly different solutions $P_\epsilon$, where $\epsilon$ is a small number, or vector of small quantities.

Can some experienced user consider editing the tag info to make it better?

What I understood (from nlab) is the following.

A typical problem in deformation theory is the following :

Given a map $f:X\rightarrow Y$ (in a category), injective maps (monomorphisms) of special kind $i_X:X\rightarrow \tilde{X}$ and $i_Y:Y\rightarrow \tilde{Y}$ can we find a map $g:\tilde{X}\rightarrow \tilde{Y}$ such that $g\circ i_X=i_Y\circ f$. This map $g$ is called an infinitesimal deformation of $f$.

Does adding this content for tag info make it any better?

  • $\begingroup$ I will mention that this is one of the problematic tag-infos mentioned quite a long time ago in another meta posts (created by copying from another source without any attribution): Recent suggested tag wiki edits. (Also related to this problem: Should tag-wiki include a source where it is taken from?) It has been improved since then a bit - at least to include some links to the sources. $\endgroup$ Dec 27, 2018 at 10:46
  • $\begingroup$ @MartinSleziak I do not know what to reply for you comment. I just wanted to say I read your comment :) Thanks. $\endgroup$ Dec 27, 2018 at 12:53
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    $\begingroup$ The nlab's definition is weird and idiosyncratic (like their definitions often are). Adding it to this tag would just make things confusing. $\endgroup$ Dec 27, 2018 at 19:20
  • $\begingroup$ Also, the sentence in the tag info at present (I will edit it into the question above) is just bad writing. $\endgroup$ Dec 27, 2018 at 20:04
  • $\begingroup$ @DavidRoberts it is taken from Wikipedia article en.m.wikipedia.org/wiki/Deformation_theory $\endgroup$ Dec 28, 2018 at 2:27
  • $\begingroup$ @AndyPutman I am not experienced enough to comment about definitions given in nlab.. Can you take some time to write down tag wiki for this.. $\endgroup$ Dec 28, 2018 at 6:00
  • $\begingroup$ No, I don't have the time to edit tag info's -- what little time I have to spend here on MO I prefer to spend engaging in mathematics rather than moderation. $\endgroup$ Dec 28, 2018 at 17:39
  • $\begingroup$ (I realize that participating in meta violates that policy somewhat; however, spending lots of time on minutia like tag info's would be far more of a time sink) $\endgroup$ Dec 28, 2018 at 17:41
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    $\begingroup$ @AndyPutman Thanks for those wise words.. I have wasted so much time on this... I did not had any motivation/necessity to learn deformation theory nor any of the statements in wikipedia/n lab I understand.. I think it better to spend time on things that I find interesting (knowing what it is).. Thanks again.. :) $\endgroup$ Dec 29, 2018 at 3:53

1 Answer 1


I changed it to:

Deformation theory is the study of how the properties of an object change as the parameters defining it are changed by a small, possibly infinitesimal amount.

Is that reasonable?

  • $\begingroup$ So, what you are saying is.. we have a collection of objects $\{X(t)\}$ over some index set $\Lambda$ with $t\in \Lambda$.... There is also some compatibility depending on what $\Lambda$ is... if $\Lambda$ is a topological space then we want $X(t)$ to vary continuously and if $\Lambda$ is smooth we want $X(t)$ to vary smoothly and similarly in other cases.., now suppose $X(0)$ is a connected manifold we see if $X(\epsilon)$ is also connected manifold.. Is it something like that? If I don’t understand it is just that I did not understand, not that what you said is not reasonable... :) :) $\endgroup$ Dec 28, 2018 at 11:22
  • $\begingroup$ @PraphullaKoushik That is one reasonable way to interpret it. $\endgroup$
    – S. Carnahan Mod
    Dec 28, 2018 at 14:23
  • $\begingroup$ Sir, I do not want to keep on asking questions.. last one :) can you think of some paper/book that says something about deformation theory, gives a rough idea of what it is and something that is not in first two pages of google search for “deformation theory”... I have checked almost all references (Hartshorne’s Deformation theory book, Nitin Nitsure’s notes) but did not understand anything :) :) :) You can atleast say what background do I need to understand deformation theory,, I am familiar with schemsand some cohomology as in Hartshorne Algebriac geometry,, $\endgroup$ Dec 28, 2018 at 14:52
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    $\begingroup$ @PraphullaKoushik I do not know of any general references, and I doubt any exist. I chose this definition based on how I see mathematicians using the term. However, I don't think there is a unifying theory that encompasses all examples from all fields. I note that neither the Wikipedia article nor nLab say anything about, e.g., Popa's work on deformations of von Neumann algebras. $\endgroup$
    – S. Carnahan Mod
    Dec 28, 2018 at 15:16

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