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I have been encountering the word "formal" in several math books, and I don't understand what it means in context. For example, Hatcher describes n-chains as "finite formal sums" on p. 105 and p. 108. What is a "formal sum?" Are there "informal sums?"

Similarly, Steve Awodey on p.15 of "Category Theory" says opposite (or dual) categories are derived from other categories "with all the arrows formally turned around." What does "formally" mean here?

This may be more of a language question than a math question, but I'd appreciate any clarification.

PossumP
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  • From the duplicate answer: ""Formal manipulation," that is, manipulating expressions according to certain rules without caring about convergence, etc."

    In your example of the opposite category, the arrows are "formally" reversed: This means if you're starting with a concrete category, the morphisms have a clear direction, namely domain to codomain. Reversing the morphisms doesn't suddenly mean that the functions have domain and codomain interchanged; it just means that the category is changed.

     – Dustan Levenstein Jun 03 '17 at 23:04
  • You misquoted me - I said the category is changed. The functions aren't changed - there is no meaningful sense in which a function can be manipulated in order to interchange the domain and codomain. So it's only the categorical data that's changed. – Dustan Levenstein Jun 04 '17 at 13:38

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"Formal" means, roughly, "without semantic content". For example, in category theory an arrow is usually a function; if we just say "reverse the arrows", there arises a natural question of "wait, what's the reversal of a function?" Saying "formally reverse the arrows" means that we don't need to answer that question - a formally reversed arrow is just an arrow going backwards, nothing else. Likewise, a "formal sum" of two objects is just the two of them written with a $+$ between them - the formal sum of $a$ and $b$ is "$a+b$", the formal sum of "apple" and "orange" is "apple $+$ orange", and the formal sum of $1$ and $1$ is "$1+1$" - not $2$, just the string "$1 + 1$".

Basically, we use "formal" when we don't want to do anything other than just let an operation make sense - when we want to be able to add elements of a set, for example, without wanting to introduce any new relationships between them. We don't impose any semantics, any "meaning" to "sums" or "reversals" or whatever we're talking about; we just do the operation we want to do, and leave it there.

One characteristic of formal operations is that there's no "other way" to get the same result. For example, the only way to use a formal sum to get "$1 + 1$" is to take the formal sum of $1$ and $1$; you can't sum $2$ and $0$, or $-1$ and $3$, to get the same answer. A formal sum carries absolutely no information beyond what was necessary to build it.

Reese Johnston
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  • Yes and no... I think that "formal sum" is most often meant in a way that things like $P + Q$ and $Q + P$ mean the same thing. Or that $P + P = 2P$. Or that $(P+Q)+R = P+(Q+R)$. –  Jun 04 '17 at 00:18
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"Formal" here is meant in the sense of "relating to form".

In regards to the first example, given a set $X$, a "Formal sum of things in $X$" means an element of the free abelian group generated by $X$.

There's a general construction in universal algebra of free algebraic structures where the element of a free algebra on $n$ elements are precisely the algebraic expressions one can write down in $n$ variables. (modulo the relations such structures satisfy)

For example, elements of the free commutative ring generated by one element are polynomials — the kinds of expressions you can construct from a single variable using just addition and multiplication (and $0$, $1$, and negation), simplified by imposing the algebraic identities defining rings.

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Sometimes (especially in older texts) this (the Hatcher sums) are called "abstract sums." It often happens in algebraic geometry (especially when talking about the Riemann-Roch theorem). Let's think about formal sums of points on a curve

If $C$ is a curve, a formal sum of points on a curve is a sum of the form $$ \sum_{i=1}^n a_ip_i. $$ Where $a_i\in\mathbb{Z}$ and $p_i\in C$. There isn't a physical sense to adding points together, but you work as if you could add points. Although surprising, this turns out to be quite useful in many situations.

Michael Burr
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