Since I'm behind in my series of posts on fields, quantum or otherwise, I will instead talk today about some linear algebra, and not define most of my terms.
The category Vect of vector spaces (over generic field \R = "real numbers") nicely generalizes the category Set of sets. Indeed, there is a "forgetful" functor in which each set forgets that it has a basis. Yes, that's the direction I mean. A vector space generalizes ("quantizes") in a natural way the notion of "set": rather than having definite discrete elements — two elements in a set either are or are not the same — a vector space allows super-positions of elements. A set is essentially "a vector space with a basis": morphisms of sets are morphisms of vector spaces that send basis elements to basis elements. So our forgetful functor takes each set X to the vector space Hom(X,\R) (Hom taken in the category of sets). But, I hear you complain, Hom(-,\R) is contravariant! Yes, but in this case, where I forgot to tell you that all sets are finite and all vector spaces finite-dimensional, we can make F = Hom(-,\R) covariant by F(\phi): f \mapsto g(y) = \sum_{x\in X s.t. \phi(x)=y} f(x). Actually, of course, if I'm allowing infinite sets, then I should specify that I don't quite want X \to Hom(X,\R), but the subspace of functions that send cofinitely many points in X to zero.
Anyhoo, so Set has an initial object 1 = {one element} and a terminal object 0 = {empty set}, and well-defined (up to canonical isomorphism) addition and multiplication (respectively disjoint union and cartesian product). These generalize in Vect to 1 = \R and 0 = {0}, and to direct sum and tensor product; if we identify n = "\R^n" (bad notation, because it's really n\R; I want n-dimensional space with a standard basis, so the space of column vectors), then it's especially clear that sums and products are as they should be. So Vect is, well, not quite a rig (ring without negation), because nothing is defined uniquely, but some categorified version, where all I care is that everything be defined up to canonical isomorphism (so, generically, given by a universal property).
But I can do even better. To each vector space V is associated a dual space V^* = Hom_{Vect}(V,\R), and most of the time V^{**} = V. (I need to learn more linear algebra: I think that there are various kinds of vector spaces, e.g. finite-dim ones, for which this is true, and I think that there's something like V^* = V^{***}. If so, then I should certainly always pass immediately from V to V^{**}, or some such; I really want dualing to be involutive.) By equals, of course, I always mean "up to a canonical isomorphism". Now, V\times V^* = Hom(V,V) is rather large, but there is a natural map Trace:Hom(V,V)\to\R, and this allows us to define a particular product "." which multiplies an element v\in V with w\in V^* by v.w = Tr(v\tensor w). Then . is multi-linear, as a product ought to be, and we can thus consider V.V^* = \R. Indeed, we can imagine some object 1/V that looks like V^* — a physicists wouldn't be able to tell the difference, because their elements are the same — so that V \tensor 1/V = \R. (Up to canonical isomorphism. It's not, of course, clear which copy of V we should contract 1/V with in V\tensor V. But either choice is the same up to canonical isomorphism.) There is even a natural trace from, say, \Hom(2,4) \to 2 — take the trace of the two 2x2 squares that make up the 4x2 matrices — "proving" that 4/2 = 2.
So it seems that, well, Vect is not a division rig, but it naturally extends to one. But what about that n in "ring"? What about negative dimensions? This I don't know.
See, it's an important question. Because, consider the tensor algebra T^{.}(V) = \R + V + V\tensor V + ... — this is an \N-graded algebra of multilinear functions on V^*. This looks an awful lot like the uncategorified 1+x+x^2+..., which we know is equal to 1/(1-x). (Why? Because (1-x)(1+x+...) = 1-x+x-x^2+x^2-... = 1, since every term cancels except for the -x^\infty, which is way off the page.) Anyhoo, so we ought to write the tensor algebra as 1/(1-V).
Which doesn't make any sense at all. 1-V? Well, we might as well define 1-V as dual to the tensor algebra: there should be a natural way to contract any element of 1-V with any multilinear function on V^*. But this has a much shorter algebraic expression, which ought to have Platonic meaning. So, what's a natural object that we can construct out of V that contracts (linearly) with all multilinear functions to give real-valued traces?
If we could answer this, then perhaps we could find out what -V is. How? Not, certainly, by subtracting 1=\R from 1-V. No, I suggest that whatever our proposal be, we then try it on 1-2V = (T^.(V+V))^* = 1/(\R + V+V + (V+V)\tensor(V+V) + ...), and compare. What out to happen is that there should be some natural object W such that 1-2V = W + 1-V, and it should turn out that 1-V = 1 + W. Whatever the case is, there should be a natural operation that "behaves like +" such that 1-V + V = 1. It's certainly not standard direct sum, just like how V \times 1/V is not the standard tensor product. But it should be like it in some appropriate sense. Most necessarily, it should satisfy linearity: if v_1,v_2\in V and w_1,w_2\in W, then v_1+w_1 and v_2+w_2 \in V+W should sum to (v_1+v_2)+(w_1+w_2). And, of course, if you have the right definition, then all the rest of arithmetic should work out: 1/(-V) = -(1/V), -V = -\R \times V, (-V)\times W = -(V\times W), and, most importantly, --V = V (up to canonical isomorphism).
One can go further with such symbolic manipulation. You've certainly met the symmetric tensor algebra S^{.}(V) of multilinear symmetric tensors, and you've probably defined each graded component S^{n}(V) as V^{\tensor n} / S_n, where by "/ S_n" I mean mean "modulo the S_n action that permutes the components in the n-times tensor product." (If you are a physicists, you probably defined the symmetric tensors as a _subspace_ of all tensors, rather than a quotient space, but this is ok, because the S_n identification generates a projection operator Sym: \omega \to (1/n!)\sum_{\pi\in S_n} \pi(\omega), and so the subspace is equal to the quotient. At least when the characteristic of the ground field is 0.) Well, S_n looks an awful lot like n!, so the symmetric algebra really looks like 1 + V + V^2/2! + ... = e^V. Which is reasonable: we can naturally identify S^{.}(V+W) = S^{.}V\tensor S^{.}W.
It's not quite perfect, though. The dimension of S^{.}V, if dim V = n, is not e^n, but 1 + n + n(n+1)/2 + n(n+1)(n+2)/6 + ..., which is only correct in the limit n\to\infty. Well, so why is that the dimension? When we symmetrize v\tensor w to 1/2(vw+wv), we generically identify different tensors. But v^2 symmetrizes to itself. Baez, though, says how to think about this: when a group action does not act freely, we should think of points like v^2 as only going to "half" points. So, for example, the group 2 can act on the vector space \R in a trivial way; we should think of \R/2 as consisting of only "half a dimension".
Anyway, the point is that we can divide by groups, and this is similar to our division by (dual) vector spaces: in either case, we are identifying, in a linear way, equivalence classes (either orbits or preimages).
Now, though, it becomes very obvious that we need to extend what kinds of spaces we're considering. Groups can act linearly in lots of ways, and it's rare that the quotient space is in fact a vector space. Perhaps the physicists are smart to confuse fixed subspaces and quotients: it restricts them just to projection operators. But, for instance, if we mod out \C by 2 = complex conjugation (which is real-linear, although not complex-linear), do we get \R or some more complicated orbifold? Is there a sense in which \R/2 + \R/2 = \R, where 2 acts by negation? \R/2 is the ray, so perhaps the direct sum model works, but you don't naturally get \R, just a one-dim space? To give interesting physics, it would be nice if these operations really did act on the constituent parts of each space. And what about dividing by 3? Every field has a non-trivial square root of 1, but only \C has nontrivial nth roots. So perhaps we really should just work with Vect of \C-linear spaces. Then we can always mod out by cyclic actions, but we don't normally get vector spaces.
Of course, part of the fun of categorifying is that there are multiple categorical interpretations of any arithmetic object: 6 may be the cyclic group C_6 = C_2 \times C_3, but 3! is the symmetric group S_3, and the groups 4 and 2x2 are also unequal. But if we come up with a coherent-enough theory, we ought to be able to make interesting discussion of things like square roots: there's an important sense in which the square root of a Lorentz vector is a spinor, and we should be able to say (1+V)^{1/2} = 1 + (1/2)V + (1/2)(-1/2)V^2/2 + (1/2)(-1/2)(-3/2)V^3/3! + ....
Overall, the move from Set to Vect deserves to be called "quantization" — well, really quantization doesn't yield vector spaces but (complex) Hilbert spaces, so really it should be the forgetful functor Set \to Hilb. If we have a coherent theory of how to categorify from numbers to Set, then it should match our theory of how to categorify from numbers to Hilb. And, ultimately, we should be able to understand all of linear algebra as even more trivial than how we already understand it: linear algebra is simply properly-categorified arithmetic.
22 October 2006
10 October 2006
Tangent vectors and their fields
Voice-over: "Last time, on Blogging My Classes, Blogging My Fields,"
Screen flashes with images of surfaces and atlases. Main character says something cliche (but stunning because of the background music) about the definition of the manifold. Then screen switches to the final scene: The Scalar Bundle.
Voice-over: "And now, the continuation."
Classically, the tangent bundle T(M) to a manifold M was defined by taking equivalence classes of (parameterized) curves at each point, equivalent if they're tangent there. Slightly more universally, we can take our atlas of patches, and on each patch, consider the (locally trivial) bundle of tangent spaces to \R^n, then modding out by the transition functions between patches. But there is a better, more algebraic way to develop tangent vectors, directly from the sheaf of differentiable functions.
Within the space of linear functionals on C^\infty(M), consider those that are "derivations at the point p": l:C^\infty(M)\to\R should satisfy, for all f,g, l(fg) = f(p)l(g) + g(p)l(f). Of course, derivations at points of constant functions return zero, and one can check that derivations at points don't care about the value of the function outside a nbhd of the point, by considering bump functions. Given a coordinate patch x^i, the m derivations \d/\d x^i |_p (derivative in the i'th direction, evaluated at p) are examples, and it turns out that these form a basis for the (linear) space of derivations at p. (This is not entirely obvious. In coordinates, it comes from the fact that I can write any f(x):\R^n\to\R as f(x) = f(0) + \sum g_i(x) x^i (in a small nbhd of 0), by letting h_x(t) = f(xt) and thus g_i(x) = \int_0^t h_{x^i}(u) du.) So we have, given an n-dimensional manifold, n dimensions worth of derivations at each point.
Now, intuitively, any tangent vector gives a derivation at its basepoint, by differentiating the function "in the direction of the vector". And, intuitively, there are n dimensions worth of tangent vectors. So we can define a tangent vector at p to be a derivation at p.
Thus, it's clear that a vector field is exactly a derivation: a field worth of derivations, one at each point. Indeed, any derivation — an algebraically-defined object, exactly a linear operator L from C^\infty(M) to itself that satisfies the Leibniz rule L(fg) = L(f)g + fL(g) — gives us a derivation at each point: L_p(f) = L(f)(p). (And, by chasing definitions, two derivations agree iff they agree at each point.) More generally, we can talk about the sheaf of (tangent) vector fields, by asking about derivations of functions defined only on various open sets.
It's worth now mentioning cotangent vectors, and specifying some notation. Of course, to any vector space (e.g. T_p(M), the tangent vectors at p), we can define the dual space (of linear functionals). By linear algebra, if dim(V)<\infty, then the dual space has the same dimension; given a basis, we can construct a dual basis. Working now with manifolds, given any function f, I can get a cotangent field df defined by df(v) = v[f], where we think of v as a derivation. In particular, by the claim I made above about being able to write f in some local normal form, given a coordinate system x = {x^i} on a nbhd U, it's clear that the {dx^i} are a basis for the space of sections of T^*(U) as a module over C^\infty(U). (Similarly, the partials \d/\d x^i are a basis of {sections of T(U)} as a module over functions.)
Following the physicists' convention, I will usually just write p_i for the cotangent field p_i dx^i, and similarly I will usually just write q^i for the vector field q^i \d/\d x^i. (Continuing the summation convention.) This works, because dx^i \d/\d x^j = \delta^i_j, so (p_i dx^i)(q^j \d/\d x^j) = p_i q^j \delta^i_j = p_i q^i, so the dot-products work out right. This is only because I happen to be using a basis and its dual basis. Eventually, I may redefine the index conventions truly coordinate-independently, but for now let's maintain the convention that whenever we interpret our formulas in terms of coordinates, we always use dual bases for T and T^*.
Next time, I'd like to talk more about tensors, metrics, and similar structures. In particular, I'd like to define the Lorentz group and classify its representations.
Screen flashes with images of surfaces and atlases. Main character says something cliche (but stunning because of the background music) about the definition of the manifold. Then screen switches to the final scene: The Scalar Bundle.
Voice-over: "And now, the continuation."
Classically, the tangent bundle T(M) to a manifold M was defined by taking equivalence classes of (parameterized) curves at each point, equivalent if they're tangent there. Slightly more universally, we can take our atlas of patches, and on each patch, consider the (locally trivial) bundle of tangent spaces to \R^n, then modding out by the transition functions between patches. But there is a better, more algebraic way to develop tangent vectors, directly from the sheaf of differentiable functions.
Within the space of linear functionals on C^\infty(M), consider those that are "derivations at the point p": l:C^\infty(M)\to\R should satisfy, for all f,g, l(fg) = f(p)l(g) + g(p)l(f). Of course, derivations at points of constant functions return zero, and one can check that derivations at points don't care about the value of the function outside a nbhd of the point, by considering bump functions. Given a coordinate patch x^i, the m derivations \d/\d x^i |_p (derivative in the i'th direction, evaluated at p) are examples, and it turns out that these form a basis for the (linear) space of derivations at p. (This is not entirely obvious. In coordinates, it comes from the fact that I can write any f(x):\R^n\to\R as f(x) = f(0) + \sum g_i(x) x^i (in a small nbhd of 0), by letting h_x(t) = f(xt) and thus g_i(x) = \int_0^t h_{x^i}(u) du.) So we have, given an n-dimensional manifold, n dimensions worth of derivations at each point.
Now, intuitively, any tangent vector gives a derivation at its basepoint, by differentiating the function "in the direction of the vector". And, intuitively, there are n dimensions worth of tangent vectors. So we can define a tangent vector at p to be a derivation at p.
Thus, it's clear that a vector field is exactly a derivation: a field worth of derivations, one at each point. Indeed, any derivation — an algebraically-defined object, exactly a linear operator L from C^\infty(M) to itself that satisfies the Leibniz rule L(fg) = L(f)g + fL(g) — gives us a derivation at each point: L_p(f) = L(f)(p). (And, by chasing definitions, two derivations agree iff they agree at each point.) More generally, we can talk about the sheaf of (tangent) vector fields, by asking about derivations of functions defined only on various open sets.
It's worth now mentioning cotangent vectors, and specifying some notation. Of course, to any vector space (e.g. T_p(M), the tangent vectors at p), we can define the dual space (of linear functionals). By linear algebra, if dim(V)<\infty, then the dual space has the same dimension; given a basis, we can construct a dual basis. Working now with manifolds, given any function f, I can get a cotangent field df defined by df(v) = v[f], where we think of v as a derivation. In particular, by the claim I made above about being able to write f in some local normal form, given a coordinate system x = {x^i} on a nbhd U, it's clear that the {dx^i} are a basis for the space of sections of T^*(U) as a module over C^\infty(U). (Similarly, the partials \d/\d x^i are a basis of {sections of T(U)} as a module over functions.)
Following the physicists' convention, I will usually just write p_i for the cotangent field p_i dx^i, and similarly I will usually just write q^i for the vector field q^i \d/\d x^i. (Continuing the summation convention.) This works, because dx^i \d/\d x^j = \delta^i_j, so (p_i dx^i)(q^j \d/\d x^j) = p_i q^j \delta^i_j = p_i q^i, so the dot-products work out right. This is only because I happen to be using a basis and its dual basis. Eventually, I may redefine the index conventions truly coordinate-independently, but for now let's maintain the convention that whenever we interpret our formulas in terms of coordinates, we always use dual bases for T and T^*.
Next time, I'd like to talk more about tensors, metrics, and similar structures. In particular, I'd like to define the Lorentz group and classify its representations.
09 October 2006
A new class of entries
I think I might like to spend some time thinking about definitions in mathematical physics. What is a quantum field, for instance? Physicists usually give a slightly incoherent answer: a quantum field is a quantum particle at every point, just like a field is a number at every point. You ask them to unpack this a bit, and some might remember that there may be global — what the physicists call "topological" — issues with such a definition, but for now let's only be concerned with the local definition, where a field is a function. So what should a quantum field be?
Conveniently, I'm taking three classes right now on related questions: Differential Geometry, Geometric Methods to ODEs, and Quantum Field Theory. I would like to start a series of entries blogging those classes, and relating it back to such foundational questions. I hope to get to answers involving infinitesimals: Robinson's "Non-standard Analysis", or Kock's "Synthetic Geometry". I don't have the answers yet.
What's most important about fields is their geometric nature. Like the physicists and the classical differential geometers, I may from time to time refer to coordinates, but ultimately I'd like a coordinate-invariant picture — indeed, one without coordinates at all. I also hope to ask and answer issues about how to regularize our fields, by which I mean "how continuous should they be?" This is an extremely non-trivial question: not only is it extremely unclear how to demand that two
"nearby" "quantum particles" be "similar" (we can demand as much of classical fields: for any epsilon, there should be a delta at each point so that within the delta ball at that point the fields don't vary more than epsilon; perhaps we should find the right metric on Schrodinger-quantized particles?), but the physicists don't even want to be stuck with, say, C^\infty fields. They want \delta functions to work within their formalism. And yet they adamantly refuse to consider "pathological" fields that are too "wildly varying".
Eventually, it would be nice also to understand the Lagrangian and Hamiltonian, and this almost-symmetry between position and momentum. For now, I'd like to end this entry with some basic definitions.
Manifolds: There are many equivalent definitions of a manifold. Since the physicists and classical geometers like to work with coordinates (replacing geometry-defined, invariant objects with coordinate-defined, covariant objects), I'll use the definition that mentions coordinates explicitly. A manifold is a (metrizable) topological space M with a maximal atlas — to each "small" open set U in M we assign a module of "coordinate patches" \phi:U\to\R^n, which should be homeomorphisms, subject to some regularity condition: if \phi:U\to\R^n and \psi:V\to\R^n, then \phi\psi^{-1} should be, say, smooth wherever it's defined. In general, modifying the word manifold modifies the condition on \phi\psi^{-1}: a C^\infty manifold has that all the \phi\psi^{-1}'s are C^\infty, for example. I will generally be interested only in C^\infty (aka "smooth") manifolds, although once we understand what kinds of functions the physicists are ok with, we may change that restriction. For a manifold, I demand that the atlas be maximal in the sense that it list all possible coordinatizations consistent with the smoothness condition. It is, of course, sufficient to simply cover our space with (coherent) patches, defining the rest as all other possibilities.
So that we can generalize this definition if we need to, it would be nice to reword this definition in the language of sheaves. The god-given structure on a smooth manifold is exactly enough to tell which functions are differentiable: a sheaf is a topological space along with a ring of "smooth functions" on each open set, so that the function rings align coherently (in full glory, a sheaf is a (contravariant) functor from the category of open sets in the space to the category of commutative\R-algebras whatever your sheaf is of, along with some "local" axioms, which ultimately say that to know a function I need exactly to know it on an open cover). I probably won't use this description, largely because I don't know what other conditions I would want to put on my sheaf in order to make it into something like a smooth manifold. Clearly every manifold generates a sheaf, and I have it on good authority that if two manifolds have the same sheaf, then they are the same manifold.
So what about our most-important of objects: a field? A field is a "section" of a "bundle".
Let's start with the latter of those undefined words. To each point p\in M, we associate a (for now) vector space V_p, called the "fiber at p". And let's (for now) demand some isotropy: V_p should be isomorphic to V_q for any given p and q in M, although not necessarily canonically so. (When we move to the realm of infinite-dimensional fibers, we may demand only that the fibers be somehow "smoothly varying" — I'm not sure yet how to define this. So long as everything is finite-dimensional, the isomorphism class of a fiber is determined by an integer, and integers cannot smoothly vary, so it suffices to consider bundles where the dimension of the fibers is constant.)
There should be some sort of association between nearby fibers: locally (on small neighborhoods U) the bundle should look like U\times V. So I ought to demand that the bundle be equipped with a manifold structure, which aligns coherently with M: a bundle E is a manifold along with a projection map \pi_E : E\to M, such that the inverse image of each point is a vector space. This is the same as saying that among the coordinate patches in E's atlas, there are some of the form \Phi: \pi^{-1}(U) \to \R^(n+k), (where, of course, n is the dimension of M and k is the dimension of each fiber) so that \Phi = (\phi,\alpha), where \phi is a coordinate patch on M and \alpha is linear on each fiber. We can naturally embed M\into E by identifying each point p\in M with (p,0) in E (where 0 is the origin of the fiber at p).
I will soon make like a physicist and forget about global issues, but I do want to provide one example of why global issues are important: the cylinder and the mobius strip are both one-dimensional ("line") bundles over the circle. The latter has a "twist" in it: as you go around the circle, you come back with an extra factor of -1.
So what's a section of a bundle? A (global) section is a map s:M\to E so that \pi s:M\to M is the identity, i.e. a section picks out one vector from each fiber. We will for now think of our sections as being C^\infty.
The most important kinds of fields are "scalar" fields, by which I exactly mean a function, i.e. a number at every point. I want to do this because I want to consider other spaces of fields as modules over the ring of scalar fields, so I need to be able to multiply. Of course, there are many times when I don't want a full-fledged scalar field. The potential energy, for instance, is only defined up to a constant: I will eventually need my formalism to accommodate objects that have fields as derivatives, but aren't fields themselves. Since potentials don't care about constants, we could imagine that after going around a circle we measure a different potential energy than we had to begin with, but that we never picked up any force. The string theorists, in fact, need similar objects: locally, string theory looks like (conformal) field theory on the string's worldsheet. But perhaps the string wraps around a small extra dimension? This is why in the previous paragraph I refer to "global" sections: I really ought to allow myself a whole sheaf of fields, understanding that sometimes I want to work with fields that are only defined in a local area. But the physicists are generally clever about this type of problem, so, at the risk of saying things that we might think generalize but actually don't, I'm going to restrict my attention to scalar fields.
In which case, yes, by "scalar field" I mean "function from M \to \R". "A section of M\times\R". "A number at each point". Those who prefer to start with the sheaf of scalar fields will be happy to know that, when I define tangent vectors and their relatives in the next entry, I will start with these scalar fields.
Conveniently, I'm taking three classes right now on related questions: Differential Geometry, Geometric Methods to ODEs, and Quantum Field Theory. I would like to start a series of entries blogging those classes, and relating it back to such foundational questions. I hope to get to answers involving infinitesimals: Robinson's "Non-standard Analysis", or Kock's "Synthetic Geometry". I don't have the answers yet.
What's most important about fields is their geometric nature. Like the physicists and the classical differential geometers, I may from time to time refer to coordinates, but ultimately I'd like a coordinate-invariant picture — indeed, one without coordinates at all. I also hope to ask and answer issues about how to regularize our fields, by which I mean "how continuous should they be?" This is an extremely non-trivial question: not only is it extremely unclear how to demand that two
"nearby" "quantum particles" be "similar" (we can demand as much of classical fields: for any epsilon, there should be a delta at each point so that within the delta ball at that point the fields don't vary more than epsilon; perhaps we should find the right metric on Schrodinger-quantized particles?), but the physicists don't even want to be stuck with, say, C^\infty fields. They want \delta functions to work within their formalism. And yet they adamantly refuse to consider "pathological" fields that are too "wildly varying".
Eventually, it would be nice also to understand the Lagrangian and Hamiltonian, and this almost-symmetry between position and momentum. For now, I'd like to end this entry with some basic definitions.
Manifolds: There are many equivalent definitions of a manifold. Since the physicists and classical geometers like to work with coordinates (replacing geometry-defined, invariant objects with coordinate-defined, covariant objects), I'll use the definition that mentions coordinates explicitly. A manifold is a (metrizable) topological space M with a maximal atlas — to each "small" open set U in M we assign a module of "coordinate patches" \phi:U\to\R^n, which should be homeomorphisms, subject to some regularity condition: if \phi:U\to\R^n and \psi:V\to\R^n, then \phi\psi^{-1} should be, say, smooth wherever it's defined. In general, modifying the word manifold modifies the condition on \phi\psi^{-1}: a C^\infty manifold has that all the \phi\psi^{-1}'s are C^\infty, for example. I will generally be interested only in C^\infty (aka "smooth") manifolds, although once we understand what kinds of functions the physicists are ok with, we may change that restriction. For a manifold, I demand that the atlas be maximal in the sense that it list all possible coordinatizations consistent with the smoothness condition. It is, of course, sufficient to simply cover our space with (coherent) patches, defining the rest as all other possibilities.
So that we can generalize this definition if we need to, it would be nice to reword this definition in the language of sheaves. The god-given structure on a smooth manifold is exactly enough to tell which functions are differentiable: a sheaf is a topological space along with a ring of "smooth functions" on each open set, so that the function rings align coherently (in full glory, a sheaf is a (contravariant) functor from the category of open sets in the space to the category of commutative
So what about our most-important of objects: a field? A field is a "section" of a "bundle".
Let's start with the latter of those undefined words. To each point p\in M, we associate a (for now) vector space V_p, called the "fiber at p". And let's (for now) demand some isotropy: V_p should be isomorphic to V_q for any given p and q in M, although not necessarily canonically so. (When we move to the realm of infinite-dimensional fibers, we may demand only that the fibers be somehow "smoothly varying" — I'm not sure yet how to define this. So long as everything is finite-dimensional, the isomorphism class of a fiber is determined by an integer, and integers cannot smoothly vary, so it suffices to consider bundles where the dimension of the fibers is constant.)
There should be some sort of association between nearby fibers: locally (on small neighborhoods U) the bundle should look like U\times V. So I ought to demand that the bundle be equipped with a manifold structure, which aligns coherently with M: a bundle E is a manifold along with a projection map \pi_E : E\to M, such that the inverse image of each point is a vector space. This is the same as saying that among the coordinate patches in E's atlas, there are some of the form \Phi: \pi^{-1}(U) \to \R^(n+k), (where, of course, n is the dimension of M and k is the dimension of each fiber) so that \Phi = (\phi,\alpha), where \phi is a coordinate patch on M and \alpha is linear on each fiber. We can naturally embed M\into E by identifying each point p\in M with (p,0) in E (where 0 is the origin of the fiber at p).
I will soon make like a physicist and forget about global issues, but I do want to provide one example of why global issues are important: the cylinder and the mobius strip are both one-dimensional ("line") bundles over the circle. The latter has a "twist" in it: as you go around the circle, you come back with an extra factor of -1.
So what's a section of a bundle? A (global) section is a map s:M\to E so that \pi s:M\to M is the identity, i.e. a section picks out one vector from each fiber. We will for now think of our sections as being C^\infty.
The most important kinds of fields are "scalar" fields, by which I exactly mean a function, i.e. a number at every point. I want to do this because I want to consider other spaces of fields as modules over the ring of scalar fields, so I need to be able to multiply. Of course, there are many times when I don't want a full-fledged scalar field. The potential energy, for instance, is only defined up to a constant: I will eventually need my formalism to accommodate objects that have fields as derivatives, but aren't fields themselves. Since potentials don't care about constants, we could imagine that after going around a circle we measure a different potential energy than we had to begin with, but that we never picked up any force. The string theorists, in fact, need similar objects: locally, string theory looks like (conformal) field theory on the string's worldsheet. But perhaps the string wraps around a small extra dimension? This is why in the previous paragraph I refer to "global" sections: I really ought to allow myself a whole sheaf of fields, understanding that sometimes I want to work with fields that are only defined in a local area. But the physicists are generally clever about this type of problem, so, at the risk of saying things that we might think generalize but actually don't, I'm going to restrict my attention to scalar fields.
In which case, yes, by "scalar field" I mean "function from M \to \R". "A section of M\times\R". "A number at each point". Those who prefer to start with the sheaf of scalar fields will be happy to know that, when I define tangent vectors and their relatives in the next entry, I will start with these scalar fields.
16 September 2006
a statement of belief
Pacifism is something I've struggled with since at least mid high school. When the President started making waves about Iraq, the American Left moved strongly towards an isolationist/pacifist stance, and although I was nervous about the occasional paleoconservative philosophy, I was already on the bandwagon, having felt that the President's hasty response in Afghanistan was poorly executed, hasty, and morally questionable. At the same time, however, I was reading A Problem From Hell: America and the Age of Genocide by Samantha Power, a fantastic book by a New York Times writer that lays the blame for the 20th Century's genocides squarely at the feet of this country and its reluctance to involve itself militarily in foreign affairs.
I did, at the time, describe myself as "trying to move towards pacifism". In my case, it wasn't a question of will power, but of wrestling with the morally ambiguous issue of military humanitarian intervention.
Having grown up in a Christian society, immersed in "turn the other cheek" rhetoric, I definitely understand the appeal. It is the noble thing to do for the resource-rich. For the resource-poor, "turning the other cheek" effectively means not responding to oppression, and it is totally not clear to me, in instances of direct physical threat, when the switch from resource rich to resource poor happens.
Were I attacked, would I be able to kill someone? No. Of course not. Do I think it would be moral to do so? Probably not. Were I to watch someone rape and murder my sister, I still would probably be unable to kill them; were the choice between killing them or having them rape and murder my sister, I think that I would not be able to bring myself to killing someone. But the moral action? Probably, yes, to prevent imminent harm murder might be valid.
More generally, I simply do not believe that retributive justice is ethical. And since it is unethical to deprive you of the right to make personal decisions about life and death (you, for instance, have the right, in my mind, to suicide), it is certainly unethical to do so as punishment. But incarceration has four uses (and, since I don't value life per se the way many people do, I see murder as essentially a complete and violent form of incarceration) --- as retributive justice, as a way of bettering people, as deterrent, and to prevent other harm --- and the last is potentially ethical (the second would be if it were effective, but it is not). I can morally justify murder for, and only for, the purpose of preventing future harm, only as a last resort, and only when "turn the other cheek" is not the correct response. Ultimately, ethical decisions do involve balancing acts.
So what about the utilitarian test of the tourist, who may either kill one captured Indian (setting the rest free), or allow all twenty to be killed? I think that either choice must be allowed as an ethical choice; I myself would be entirely unable to fire the gun. But ultimately the answer is not really either: the completely ethical action is to consult first with the Indians and ask what they want. What's unethical about the situation is that the tourist is ultimately one of the oppressors, making life and death decisions for the oppressed people. Perhaps one Indian is willing to sacrifice themselves. Perhaps they decide to draw straws. Or perhaps they decide that they would all be happier dying than knowing that they lived only because someone else died for them. It should be their decision to make.
Similarly in international affairs, if we see endemic oppression, we may, and indeed we must, involve ourselves to help the oppressed. We must be careful to do the most effective things, and this is rarely, I believe, militaristic, and we must base our decisions strongly on what the oppressed people would like us to do. (This is, of course, hard. The most oppressed people are often sub-altern.) But, when we have the resources to just stand in the way of oppression, and absorb the onslaught of violent attempts to maintain the oppression while "turning the other cheek", then we cannot justify engaging ourselves in violence.
And, yet, we often find that we do not have such resources. And then is humanitarian military aid ethical? It helped, Power says, in Kosovo. I don't know.
I did, at the time, describe myself as "trying to move towards pacifism". In my case, it wasn't a question of will power, but of wrestling with the morally ambiguous issue of military humanitarian intervention.
Having grown up in a Christian society, immersed in "turn the other cheek" rhetoric, I definitely understand the appeal. It is the noble thing to do for the resource-rich. For the resource-poor, "turning the other cheek" effectively means not responding to oppression, and it is totally not clear to me, in instances of direct physical threat, when the switch from resource rich to resource poor happens.
Were I attacked, would I be able to kill someone? No. Of course not. Do I think it would be moral to do so? Probably not. Were I to watch someone rape and murder my sister, I still would probably be unable to kill them; were the choice between killing them or having them rape and murder my sister, I think that I would not be able to bring myself to killing someone. But the moral action? Probably, yes, to prevent imminent harm murder might be valid.
More generally, I simply do not believe that retributive justice is ethical. And since it is unethical to deprive you of the right to make personal decisions about life and death (you, for instance, have the right, in my mind, to suicide), it is certainly unethical to do so as punishment. But incarceration has four uses (and, since I don't value life per se the way many people do, I see murder as essentially a complete and violent form of incarceration) --- as retributive justice, as a way of bettering people, as deterrent, and to prevent other harm --- and the last is potentially ethical (the second would be if it were effective, but it is not). I can morally justify murder for, and only for, the purpose of preventing future harm, only as a last resort, and only when "turn the other cheek" is not the correct response. Ultimately, ethical decisions do involve balancing acts.
So what about the utilitarian test of the tourist, who may either kill one captured Indian (setting the rest free), or allow all twenty to be killed? I think that either choice must be allowed as an ethical choice; I myself would be entirely unable to fire the gun. But ultimately the answer is not really either: the completely ethical action is to consult first with the Indians and ask what they want. What's unethical about the situation is that the tourist is ultimately one of the oppressors, making life and death decisions for the oppressed people. Perhaps one Indian is willing to sacrifice themselves. Perhaps they decide to draw straws. Or perhaps they decide that they would all be happier dying than knowing that they lived only because someone else died for them. It should be their decision to make.
Similarly in international affairs, if we see endemic oppression, we may, and indeed we must, involve ourselves to help the oppressed. We must be careful to do the most effective things, and this is rarely, I believe, militaristic, and we must base our decisions strongly on what the oppressed people would like us to do. (This is, of course, hard. The most oppressed people are often sub-altern.) But, when we have the resources to just stand in the way of oppression, and absorb the onslaught of violent attempts to maintain the oppression while "turning the other cheek", then we cannot justify engaging ourselves in violence.
And, yet, we often find that we do not have such resources. And then is humanitarian military aid ethical? It helped, Power says, in Kosovo. I don't know.
02 September 2006
A Categorical Definition
Categories, best described with commutative diagrams, allow for truly non-linear thinking, and yet they are usually defined linearly, similar to the way groups are usually introduced. This almost makes sense: morphisms, as one-dimensional objects, are about the most natural thing to compose linearly. And yet the power of category theory comes from the non-linear diagrams people draw, showing non-linear relationships between objects. The snake lemma, for instance, is poorly expressed and even more poorly understood if you are limited to writing words on a page.
I am, on this blog, constrained to poor, linear writing. Nevertheless, I would like to provide a definition of "category" using only visual, diagramatic ideas. Perhaps I will succeed in ASCIIing the diagrams. This definition, I hope, will ultimately be seen as providing a more basic understanding of these powerful creatures.
To begin with, a diagram is a (labeled) directed graph: it's composed of (labeled) vertices ("objects") connected by (labeled) directed edges ("arrows"). There is a natural notion of "subdiagram", formed by deleting any collection of edges, and any collection of vertices and all their edges. A category is a collection of diagrams, subject to certain rules, to be enunciated lower down. The diagrams in a category are said to commute. I will sometimes leave off labels, in which case I generally mean to refer to all (commutative) diagrams of that "shape".
Rule 0: A diagram commutes if and only if all its finite subdiagrams commute. (Thus any subdiagram of a commutative diagram commutes. In any category, the empty diagram commutes.)
One advantage of this construction is that I don't need any of that junk about "a collection of objects, and between each pair A and B of objects a set Hom(A,B) of arrows...". Instead, I can say simply that a morphism is a commutative diagram A ---> B, where I have left of the label on the arrow: I will write "f:A->B" for the very simple diagram of an arrow from A to B labeled by f, but only because I have to make it fit in this constrained formatting.
Rule 1: If two diagrams commute, then their disjoint union commutes. If diagram D contains an object label A, and E contains B, and if A ---> B commutes (morphism called f), then in the disjoint union we can connect A and B via f so that the diagram still commutes:
Often we will draw a dotted arrow in a diagram. In these notes, creating dotted arrows is too hard; I will use equal signs instead, as in ===>. A diagram with a dotted arrow means "If the diagram without the dotted arrow commutes, then there is exactly one diagram with an (labeled) arrow in place of the dotted one." Sometimes folks will write a label on the arrow, in which they mean the name (label) that the (unique) arrow which replaces it should have.
Rule 2: For any diagram E containing a finite chain (in the picture, I draw E "surrounding" the chain, to suggest that various objects in the chain might have other arrows to and from the rest of E), we have
So, in particular, some corollaries:
However, one more rule concerning the various 1_A morphisms is necessary.
Rule 3If a commutative diagram contains 1_A:A->A for some A, then we can maintain commutativity by replacing this diagram with A: all arrows into and out of either A in the original diagram now go into and out of the single A in the new diagram. Conversely, any object A in a diagram may be replaced by 1_A:A->A, where all arrows on A are now duplicated, and placed once on each A. I will not try to draw this, but you should.
And that, folks, is it. Rules 0 through 3 suffice to define a category.
Of course, we should say some more, to convince you that pure diagrammatic thinking is useful. For instance, an isomorphism is a commutative diagram of the form
With our "dotted arrow" (except I'm using equal signs) notation, we can go on to define "universal properties". It seems to me that there should really be "universal properties" and "co-universal properties", depending on which direction the arrows go. To define these, we introduce another new symbol [], which is kindof like ==>; whereas ==> defined an arrow uniquely, [] defines objects up to isomorphism. How? Well, say we have a diagram with a box. Then we're saying that (if the rest of the diagram commutes), then there's some object X which can fill the box (i.e. a labeling for that object that makes the diagram commute) s.t. if any other Y can also fill the box, then there's a dotted arrow from Y to X. I.e.:
where the idea in the last picture is that Y and X each have all the arrows going from and to them and D that the original diagram says they should have. The dual notion to a universal property is a "co-universal property", in which Y => X is replaced by "Y <= X" in the definition. If needed, I will write that as {}.
Now, I don't actually know of any useful (co-)universal properties that are not (co-)limits — well, I think tensor products might be one, but I don't remember how to define that — so I really ought to just define limit. And maybe I should have just started with them, I dunno.
Anyhoo, given a (commutative) diagram D, the limit of the diagram lim D is the (universal property) diagram given by D with a box added, and arrows from the added box to each object in D. Co-limits are the dual notion. Limits, like anything defined via universal property, need not exist. Some examples:
I will stop here. Many category theory books from around here and onward start using diagrammatic reasoning and definitions more frequently, so I refer you to them for other definitions of other objects. My goal was to give diagrammatic definitions of the most basic elements of category theory, and to suggest that categories are best thought of not as collections of objects and morphisms, but simply as collections of diagrams.
I am, on this blog, constrained to poor, linear writing. Nevertheless, I would like to provide a definition of "category" using only visual, diagramatic ideas. Perhaps I will succeed in ASCIIing the diagrams. This definition, I hope, will ultimately be seen as providing a more basic understanding of these powerful creatures.
To begin with, a diagram is a (labeled) directed graph: it's composed of (labeled) vertices ("objects") connected by (labeled) directed edges ("arrows"). There is a natural notion of "subdiagram", formed by deleting any collection of edges, and any collection of vertices and all their edges. A category is a collection of diagrams, subject to certain rules, to be enunciated lower down. The diagrams in a category are said to commute. I will sometimes leave off labels, in which case I generally mean to refer to all (commutative) diagrams of that "shape".
Rule 0: A diagram commutes if and only if all its finite subdiagrams commute. (Thus any subdiagram of a commutative diagram commutes. In any category, the empty diagram commutes.)
One advantage of this construction is that I don't need any of that junk about "a collection of objects, and between each pair A and B of objects a set Hom(A,B) of arrows...". Instead, I can say simply that a morphism is a commutative diagram A ---> B, where I have left of the label on the arrow: I will write "f:A->B" for the very simple diagram of an arrow from A to B labeled by f, but only because I have to make it fit in this constrained formatting.
Rule 1: If two diagrams commute, then their disjoint union commutes. If diagram D contains an object label A, and E contains B, and if A ---> B commutes (morphism called f), then in the disjoint union we can connect A and B via f so that the diagram still commutes:
DDD EEE
D f E
D A ---> B E
Often we will draw a dotted arrow in a diagram. In these notes, creating dotted arrows is too hard; I will use equal signs instead, as in ===>. A diagram with a dotted arrow means "If the diagram without the dotted arrow commutes, then there is exactly one diagram with an (labeled) arrow in place of the dotted one." Sometimes folks will write a label on the arrow, in which they mean the name (label) that the (unique) arrow which replaces it should have.
Rule 2: For any diagram E containing a finite chain (in the picture, I draw E "surrounding" the chain, to suggest that various objects in the chain might have other arrows to and from the rest of E), we have
EEEEEEEEEEEEEEEEEEEEEEEEE
E __ B -...-> C E
E /| \ E
E / _| E
E A =============> D E
So, in particular, some corollaries:
- "composition"
A -> B -> C
========> - "associativitiy"
if B -> C and B -> C commute
^ > \ |
| / > v
A D
then so do C and B
> | ^ / v | >
A -> D A -> D
(of course, with all edges (consistently) labeled.) - "identity"
A <==|| \\\ //
normally called "\mathbb{1}_A", I will just call this morphism "1_A". I will let you write out the "left and right identity laws" in this language; they follow from Rule 2.
\====/
However, one more rule concerning the various 1_A morphisms is necessary.
Rule 3If a commutative diagram contains 1_A:A->A for some A, then we can maintain commutativity by replacing this diagram with A: all arrows into and out of either A in the original diagram now go into and out of the single A in the new diagram. Conversely, any object A in a diagram may be replaced by 1_A:A->A, where all arrows on A are now duplicated, and placed once on each A. I will not try to draw this, but you should.
And that, folks, is it. Rules 0 through 3 suffice to define a category.
Of course, we should say some more, to convince you that pure diagrammatic thinking is useful. For instance, an isomorphism is a commutative diagram of the form
--->
A B.
<---
With our "dotted arrow" (except I'm using equal signs) notation, we can go on to define "universal properties". It seems to me that there should really be "universal properties" and "co-universal properties", depending on which direction the arrows go. To define these, we introduce another new symbol [], which is kindof like ==>; whereas ==> defined an arrow uniquely, [] defines objects up to isomorphism. How? Well, say we have a diagram with a box. Then we're saying that (if the rest of the diagram commutes), then there's some object X which can fill the box (i.e. a labeling for that object that makes the diagram commute) s.t. if any other Y can also fill the box, then there's a dotted arrow from Y to X. I.e.:
DD DD
DDDDDD DDDDDD
[] DD means that X DD
DDDDDD DDDDDD
DD DD
DD
DDDDDD
and if any other Y DD, then
DDDDDD
DD
DDDDDDDD
DD DD DD
Y => X DD
DD DD DD
DDDDDDDD
where the idea in the last picture is that Y and X each have all the arrows going from and to them and D that the original diagram says they should have. The dual notion to a universal property is a "co-universal property", in which Y => X is replaced by "Y <= X" in the definition. If needed, I will write that as {}.
Now, I don't actually know of any useful (co-)universal properties that are not (co-)limits — well, I think tensor products might be one, but I don't remember how to define that — so I really ought to just define limit. And maybe I should have just started with them, I dunno.
Anyhoo, given a (commutative) diagram D, the limit of the diagram lim D is the (universal property) diagram given by D with a box added, and arrows from the added box to each object in D. Co-limits are the dual notion. Limits, like anything defined via universal property, need not exist. Some examples:
A x B = lim A B
B
A x_C B = lim |
v
A -> C
terminal object = lim (empty diagram)
1_A
lim A = A ---> A, which we're considering to be equivalent to A.
I will stop here. Many category theory books from around here and onward start using diagrammatic reasoning and definitions more frequently, so I refer you to them for other definitions of other objects. My goal was to give diagrammatic definitions of the most basic elements of category theory, and to suggest that categories are best thought of not as collections of objects and morphisms, but simply as collections of diagrams.
31 August 2006
Today's News
Today's headlines in the New York Times:
No news is good news?
- Lockheed Martin got another government contract.
- Bush said something he said last week too.
- Folks post stuff online.
- Someone in Chicago wants to be mayor.
No news is good news?
28 August 2006
Young boys and a man
While looking out the window at a rainy Newark Airport and waiting for a very delayed flight, I found myself standing next to a young boy — perhaps five or six — eating a large roll of bread. I struck up a conversation, and we were soon joined by his older brother — six or seven. I let the conversation go wherever it wandered, and learned quite a lot: that their father is a pilot; that the Yankees are the best baseball team, pitching is the best position, and next year they won't use the tee until you get six strikes; that the bushes below the hotel in Hawaii with the big rooms (three balconies in the suite!) now house a favorite action figure; that the police climbing the stairs into the jet-way were probably entering the airplane, because if there were a bad guy in the terminal, the security would have caught him in the initial screening (in fact, they were there to escort a very drunk passenger, who had repeatedly opened an alarmed door, from the terminal to the hospital).
After a while, their farther joined us at the window. "Tell the man next to you" — me — "what the kind of plane with the bump on top is," he asked his younger son. "I'll give you a hint: it starts Seven...."
"Um, Seven Seven?"
"No, Seven Forty-Seven."
"Seven Forty-Seven."
"And if there are [a particular kind of wing flaps]" — here my memory of the technical terms, which he used, has gone — "then it's a 747-400."
What I found most memorable about this discussion was not the ease with which we changed topics — an ease I normally associate with the uniformly brilliant kids at Mathcamp; an ease often pathologized as ADHD and ruined with drugs such asspeed ritalin — nor the freedom with which these kids would talk to a complete stranger. What stuck with me was one particular piece of language: "Tell the man next to you..."
Those who've known me for a while may remember previous discussions I've had (though I think not here) about the different words "boy", "man", "kid", etc., which I find fascinating. I've intentionally used some throughout this entry: Mathcamp students and five-year-olds I've both described as "kids," for instance, whereas my first companion was a "young boy." I generally insist that periodicals refer to high school, and certainly college, students as "men" and "women": my freshman roommate was on the men's swim team, and in my brother's CS class there are only six women, as opposed to "boys'" and "girls." Mathcampers, on the other hand, and even my housemates, I often think of as "boys and girls". Not "children," perhaps, but "kids."
What's hardest, though, is self-identity — I'm good at holding multiple contradictory beliefs about the external realty — I had never before defined myself as someone who could be a "man [standing] next to you." Perhaps, when discussing sexual and gender politics, I've identified myself as a "(suitably adjectived) man," but more often as a "male." Categories like "men who have sex with men" are so entirely foreign and don't seem to apply to me or any of my peers. People in my socioeconomic class don't become "adults" until closer to 26, but I'm definitely no longer a "young adult." I'm a "student" or a "guy," not a "man."
One reason for my sojourn to New York was to attend a ninetieth birthday party and family reunion, where I spent some time chatting with various second cousins whom I haven't seen in ten years. My father, an older brother, is younger than his cousins, so while I played cards and board games with my fourteen-year-old cousin, the majority of "my generation" were three to ten years older than me. One announced the wonderful news of her pregnancy, making the matriarch whose birthday we were celebrating extremely happy. I'm used to my peers consisting of younger siblings and students exactly my age; I'm used to understanding those classmates only a few years older than me as significantly closer to adult, since they tend to be grad students when I'm an undergrad, or undergrads when I'm in high school.
But I'll be graduating in four months, and dreaming of my own apartment, and, eventually, house and family. I watch my fresh-out-of-college friends with their jobs in Silicon Valley, and can't help but think how similar that life is to college — they have roommates, come to campus, go on dates. They're no more "adults" than I am.
I have no trouble being "mature", or "old", or even relatively "grown up". But I'm twenty-one years old, and have a hard time thinking of myself as an "adult". Identifying as a "man" is impossible, and it is my current self-descriptor.
After a while, their farther joined us at the window. "Tell the man next to you" — me — "what the kind of plane with the bump on top is," he asked his younger son. "I'll give you a hint: it starts Seven...."
"Um, Seven Seven?"
"No, Seven Forty-Seven."
"Seven Forty-Seven."
"And if there are [a particular kind of wing flaps]" — here my memory of the technical terms, which he used, has gone — "then it's a 747-400."
What I found most memorable about this discussion was not the ease with which we changed topics — an ease I normally associate with the uniformly brilliant kids at Mathcamp; an ease often pathologized as ADHD and ruined with drugs such as
Those who've known me for a while may remember previous discussions I've had (though I think not here) about the different words "boy", "man", "kid", etc., which I find fascinating. I've intentionally used some throughout this entry: Mathcamp students and five-year-olds I've both described as "kids," for instance, whereas my first companion was a "young boy." I generally insist that periodicals refer to high school, and certainly college, students as "men" and "women": my freshman roommate was on the men's swim team, and in my brother's CS class there are only six women, as opposed to "boys'" and "girls." Mathcampers, on the other hand, and even my housemates, I often think of as "boys and girls". Not "children," perhaps, but "kids."
What's hardest, though, is self-identity — I'm good at holding multiple contradictory beliefs about the external realty — I had never before defined myself as someone who could be a "man [standing] next to you." Perhaps, when discussing sexual and gender politics, I've identified myself as a "(suitably adjectived) man," but more often as a "male." Categories like "men who have sex with men" are so entirely foreign and don't seem to apply to me or any of my peers. People in my socioeconomic class don't become "adults" until closer to 26, but I'm definitely no longer a "young adult." I'm a "student" or a "guy," not a "man."
One reason for my sojourn to New York was to attend a ninetieth birthday party and family reunion, where I spent some time chatting with various second cousins whom I haven't seen in ten years. My father, an older brother, is younger than his cousins, so while I played cards and board games with my fourteen-year-old cousin, the majority of "my generation" were three to ten years older than me. One announced the wonderful news of her pregnancy, making the matriarch whose birthday we were celebrating extremely happy. I'm used to my peers consisting of younger siblings and students exactly my age; I'm used to understanding those classmates only a few years older than me as significantly closer to adult, since they tend to be grad students when I'm an undergrad, or undergrads when I'm in high school.
But I'll be graduating in four months, and dreaming of my own apartment, and, eventually, house and family. I watch my fresh-out-of-college friends with their jobs in Silicon Valley, and can't help but think how similar that life is to college — they have roommates, come to campus, go on dates. They're no more "adults" than I am.
I have no trouble being "mature", or "old", or even relatively "grown up". But I'm twenty-one years old, and have a hard time thinking of myself as an "adult". Identifying as a "man" is impossible, and it is my current self-descriptor.
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