I know the definition of a category and know several examples. But I have not studied the theory of categories.   Much of mathematics I studied can be written as a category and a functor.  But for me, a category is just an organizational language. What is the merit of studying category theory?  What can I obtain by reading Mac Lane's book?  Is it necessary to learn category theory to be a good mathematician, especially for number theorists, algebraists, and topologists?
Okay, so I understand what calculus, linear algebra, combinatorics and even topology try to answer (update: this is not the case in hindsight), but why invent category theory? In Wikipedia it says it is to formalize. As far as I can tell, it sort of generalizes a bunch of fields in mathematics, like in topology, graphs and groups we have isomorphisms and automorphisms and homomorphisms, it seems to me like it just finds a bunch of results that are true to these functions in a bunch of disciplines. But why would generalizing that be of use? I mean, wouldn't it be easier to prove them specifically in each of the important disciplines?
A similar question was asked , but it was specific to .NET 3.5. Specifically, I'm looking for the following:   What is the correct way to determine which .NET Framework versions and service packs are installed? Is there a list of registry keys that can be used? Are there any dependencies between Framework versions? Obligatory: This is not a duplicate of , which is concerning the much older duplicate closure process.  Have a gander at . Notice anything weird?  The Community♦ user was invoked by the question asker to self-close as a duplicate (presumably of the comment below, which links to ), but the list of duplicates is entirely empty. So it would seem the question asker was able to close their own question as a duplicate of... nothing?  Screenshot, in case the post is altered later:    Any idea what happened here? Making tags visible when using a Tumblr theme that doesn't show them Is it OK to require users to have a third party login account? Webform redirect I'd like to do something pretty straightforward: After user fills out a , I'd like a button that simply redirects them to a URL (in this case it's to a demo sandbox). Of course I'd want the user to fill in their info to capture for lead-generation purposes. My  is set up correctly in that it correctly captures all the user fields and submits them to me via email. However, upon completion "click" I'd like to have the user redirected to a determined internal URL www.mysite.com/foo  I can't figure out how this is done with the vanilla webform settings. Am I missing something obvious? How to prove that the radial basis function is a kernel? How to prove that the radial basis function $k(x, y) = \exp(-\frac{||x-y||^2)}{2\sigma^2})$ is a kernel? As far as I understand, in order to prove this we have to prove either of the following:   For any set of vectors $x_1, x_2, ..., x_n$ matrix $K(x_1, x_2, ..., x_n)$ = $(k(x_i, x_j))_{n \times n}$ is positive semidefinite. A mapping $\Phi$ can be presented such as $k(x, y)$ = $\langle\Phi(x), \Phi(y)\rangle$.   Any help? Proof that $(x-1)^2$ divides a $nx^{n+1} – (n+1)x^{n}+1$ I want to solve:  Show that, in $\mathbb Q[x]$ (rationals), the polynomial $(x-1)^2$ divides a $zx^{z+1} – (z+1)x^{z}+1$, where $z \geq 1$ is any integer  I am really lost with this one, don't know how to aproach it How can I prevent my Wi-Fi connection from glitching? Occassionally my Wi-Fi goes into this weird state where the phone "thinks" I'm connected, but the connection simply doesn't work.  Apps still seem to think I have connectivity, though they can't actually connect to anything, and the status bar still shows that I'm connected to Wi-Fi.  Turning Wi-Fi off and back on resolves the issue, and this occurs on multiple networks.  It seems sort of like whatever service manages the Wi-Fi connection gets killed occasionally.  It usually happens when I've been playing a game (that doesn't access the Internet) for a while or if my phone's been sitting for a while; it never happens while I'm actively using the connection.  I have my Wi-Fi timeout set to "Never" and the normal timeout isn't the issue since my phone regularly lasts several hours without the Wi-Fi dying.  Sometimes it dies a couple times in a day, and recently I went a week or so without it happening.  How can I fix this or prevent the service from being killed (if that's indeed what's happening)?  Shouldn't that be a system process that never gets killed?  (Does Android ever kill system processes other than when they crash?)  Could it be related to TouchWiz?  I have an SGS Vibrant running 2.2 (Bionix NextGen v1, but this has occurred on multiple ROMs). I'm making an app with a custom list in multiple languages.  Is there an easy way so that I only have to create the site columns, content type and list once? So I don't have to create them for each language?  My app will be in English, French and Dutch.  My lists elements.xml looks like this  &lt;?xml version="1.0" encoding="utf-8"?&gt; &lt;Elements xmlns="http://schemas.microsoft.com/sharepoint/"&gt;     &lt;!-- Do not change the value of the Name attribute below. If it does not match the folder name of the List project item, an error will occur when the project is run. --&gt;     &lt;ListTemplate         Name="List1"         Type="10000"         BaseType="0"         OnQuickLaunch="TRUE"         SecurityBits="11"         Sequence="410"         DisplayName="List1"         Description="My List Definition"         Image="/_layouts/15/images/itgen.png"/&gt;   &lt;Field       Name="Adres"       ID="{2a32d9eb-ad3b-4dbb-abba-66dd01ca7c6e}"       DisplayName="$Resources:Adres"       Type="Text"       Required="TRUE" /&gt;   &lt;Field       Name="Naampje"       ID="{ac47be45-2f43-4d16-9a06-162c85945ec9}"       DisplayName="$Resources:Naampje"       Type="Text"       Required="FALSE" /&gt;   &lt;Field     Name="Nickname"     ID="{6B0A2CD7-A7F9-41ca-B932-F3BEBB603793}"     DisplayName="$Resources:Bijnaam"     Type="Text"     Required="FALSE" /&gt;  &lt;/Elements&gt;   Well if i change my language in Sharepoint (Office 365) it's not changing the titles from dutch to the other languages i have added. Can a self-complementary graph on 100 vertices have precisely one vertex of degree 50? I found a feature request at  but it doesn't seem easy to get a column view with Nautilus while with Mac OS X it's a default feature.  Have I missed something or is this not currently possible? Are the rules regarding absolute modifiers too absolute? A common grammar lesson that was taught to me in the US and that I've had to teach abroad in EFL classrooms is that we're not to use adverbs of emphasis with absolute modifiers, just as we're not supposed to use them as comparatives or superlatives.  Classic egregious examples of this mistake include very unique and more perfect, which seem obviously flawed to me. Other instances of further modified absolute modifiers are similarly meaningless or contradictory in nature.  I wonder, though, if the rule that I learned isn't overly broad, and whether it's taught similarly in the UK and elsewhere. I find myself modifying absolute modifiers quite regularly, and in some circumstances I think it's not only logical and correct, but quite meaningful. Phrases like almost exactly or virtually all or practically infinite possess a specific meaning that can't really be replicated using nonabsolutes like very nearly or most or quite long.  In my own speech and writing, I use such phrases exactly as I see fit, so this isn't a request for permission, per se. I'm more curious about whether there's a more appropriate rule regarding absolutes that you rely on (and might be used in a classroom). Is AES in CBC mode secure if a known and/or fixed IV is used? I have a need to encrypt credentials for a third-party app used by a secured internal app. Over on ITSec.SE, I was helpfully shown a scheme to encrypt the third-party credentials based on a hash of the credentials for the internal app.   I picked AES as the encryption algorithm, but the problem is that the password-based scheme doesn't produce a "secret" IV. So, the IV must at least be known to an attacker (stored alongside the encrypted data). A hash value used for password verification could work, or I could just generate a pseudorandom byte array and drop it in the DB as a new column. I was further considering, for simplicity, to use a constant IV.   What adverse effect will either of these choices have on the security of the encryption? Does AES depend, as many block ciphers do, on an unpredictable IV? Does it matter if the IV is stored as unencrypted plaintext? Can Ubuntu 32-bit access 8GB RAM? I need to use a large software package (Xojo) that requires the 32-bit architecture to run. Try as I might, I can't seem to install the 32-bit libraries on my 64-bit installation of 15.04.  I don't have a problem running a 32-bit version of Ubuntu (I'm only using this computer for Xojo) but the computer has 8GB of RAM and I'd like to be able to use it all. Is there a way to utilise all 8GB of RAM with a 32-bit Ubuntu install? Does Ubuntu do this out of the box? What geometric object is given by this equation? Monos in $\mathsf{Mon}$ are injective homomorphisms. Let $f: M \to N$ be a mono in $\mathsf{Mon}$. I want to prove that the underlying function $U(f): U(M) \to U(N)$ is injective.   How can this be done ? Thanks in advance. Seriously, it was fun and all, but I can't and I don't want to stay married to the same woman for the rest of my life - one real life marriage is enough..  Is there a lawyer or process of some kind in the world of Skyrim, and if not, who could help me divorce her?  If all else should fail, (sigh), do I have to take her to the backyard, thwack her head with the shovel and dig a shallow grave? Book: Nanobot apocalypse on Earth, originally used for weather control We've heard it many times, nothing can escape the gravity of a black hole, even light once it's past the event horizon.  If this is true, how can a black hole emit anything?  are massive black holes that have consumed enormous amounts of matter and energy, and then blasted it back out into the universe.  How is this possible?  Time stands still in a black hole, that in itself should stop anything from moving. A new environment based on enumerate I have defined the following simple environment:  \newenvironment{renumerate}{\begin{enumerate}\raggedright}{\end{enumerate}}   I wonder how the above can be modified to use with, say, the following:  \begin{renumerate}[label=\color{blue}10.\theenumi] \item blah \item blah \end{renumerate}   The options given in [] are not known in advance.