theory – Central extensions of torsion groups by $mathbb{R}^n$

Let $Gamma$ be a torsion group (i.e. ever element has finite order). I am interested in understanding central extensions of the form:
0 @>>> mathbb{R}^n @>exp>> G @>pi>> Gamma @>>> 1\

Equivalently, I want examples of groups $Gamma$ with non-trivial classes in $H^2(Gamma,mathbb{R}^n)$. When $Gamma$ is finite I’m aware that $H^2(Gamma,mathbb{R}^n) = 0$ but I have little intuition or examples for infinite torsion groups.

Interestingly, any central extension such a $Gamma$ by $mathbb{R}^n$ has a canonical set theoretic splitting $sigma colon Gamma to G$ with the property that:

$$ sigma(gamma) = g quad Leftrightarrow quad pi(g) = gamma text{ and } g text{ has finite order}$$
where $pi colon G to Gamma$ is the projection. It is not too hard to show that if any group-theoretic splitting exists then it must be $sigma$.

Using usual group cohomology arguments, this gives rise to a cocycle:
$$ alpha colon G times G to mathbb{R}^n $$
I’ve managed to prove a few interesting properties of $alpha$ (for example, $alpha$ must be symmetric) but have not been able to show it is zero and I suspect a counter example probably exists.

ct.category theory – Adjoining extensions in bicategories

Given a bicategory $mathcal K$, is there a universal construction of a bicategory $mathcal K’$ and faithful locally fully faithful pseudofunctor $mathcal K hookrightarrow mathcal K’$ such that for each pair of 1-cells $f : A to B$ and $g : A to C$ in $mathcal K$, the left extension $mathrm{lan}_f g : B to C$ exists in $mathcal K’$? That is, is it possible to freely adjoin left extensions for every pair of 1-cells with common codomain?

If there is not an explicit construction (in the literature or elsewhere), I would be happy for a demonstration that a construction exists. I suspect one can prove that a syntactic construction works, but I would rather avoid trying this if there is a more elegant approach.

A related question regards adjoining adjoints to 1-cells in bicategories, for which a partial solution can be found in Dawson–Paré–Pronk’s Adjoining adjoints.

audio – Can Mac (Big Sur) System extensions collect data?

I’ve been meaning to install Blackhole or Background Music or other third party audio driver in order to record audio output while screen recording through QuickTime.

What I’m not sure of is whether these third party audio drivers (system extensions) collect any data or compromise privacy in any manner?

Couldn’t find any policy on their GitHub docs about the same.

How to configure Magento 2 to have personal/private develpment extensions just for myself

I have several development tools which I personally use like:


Is it possible to somehow configure and use it for myself? Other team members are not using it.

The specific issues:

  • app/etc/config.php is in the repo, I cannot commit it
  • composer.json and composer.lock are in the repo, I cannot commit it

I would like to use both modules and enable them without changing any repo files. Would that be possible?

NullPointerException when manually loading required Vulkan instance extensions

I am currently developing a Win32 app with Vulkan via LWJGL, and when I run the following code,

            String className = "Hello Window App";
            ByteBuffer classNameBuffer = stack.UTF16(className);
            WindowProc defaultWndProc = new WindowProc() {
                public long invoke(long hwnd, int uMsg, long wParam, long lParam) {
                    if (uMsg == User32.WM_DESTROY) {
                    } else if (uMsg == User32.WM_PAINT) {
                    return User32.DefWindowProc(hwnd, uMsg, wParam, lParam);
            WNDCLASSEX wc = WNDCLASSEX.callocStack(stack);
            hwnd = User32.CreateWindowEx(User32.WS_EX_APPWINDOW, className, "Hello Windows App",
                    User32.WS_OVERLAPPEDWINDOW, 100, 100, 640, 480, 0, 0, WindowsLibrary.HINSTANCE, 0);
            if (hwnd == 0) {
                throw new RuntimeException("Unable to create Window");
            User32.ShowWindow(hwnd, User32.SW_SHOW);
            User32.SetCursorPos(0, 0);
            MSG msg = MSG.callocStack(stack);
            PointerBuffer ppEnabledLayerNames = null; // We'll add support for validation layers later
            // PointerBuffer windowExtensions = GLFWVulkan.glfwGetRequiredInstanceExtensions();
            PointerBuffer windowExtensions = MemoryUtil.memAllocPointer(2);
            PointerBuffer ppEnabledExtensionNames = windowExtensions;
            VkApplicationInfo value = VkApplicationInfo.mallocStack(stack);
            value.apiVersion(VK11.VK_MAKE_VERSION(1, 1, 0));
            VkInstanceCreateInfo pCreateInfo = VkInstanceCreateInfo.mallocStack(stack);
            PointerBuffer pInstance = stack.mallocPointer(1);
            if (VK11.vkCreateInstance(pCreateInfo, null, pInstance) != VK11.VK_SUCCESS) {
            instance = new VkInstance(pInstance.get(0), pCreateInfo);

            VkWin32SurfaceCreateInfoKHR sfc = VkWin32SurfaceCreateInfoKHR.calloc();
            LongBuffer pSurface = stack.callocLong(1);
            KHRWin32Surface.vkCreateWin32SurfaceKHR(instance, sfc, null, pSurface);
            surface = pSurface.get(0);
            while (User32.GetMessage(msg, 0L, 0, 0) == true) {

I get the exception below. It seems to be an error regarding loading the required Vulkan instance extensions, because when I uncomment the GLFW.glfwInit(); and PointerBuffer windowExtensions = GLFWVulkan.glfwGetRequiredInstanceExtensions(); lines and use those instead to load the required extensions, the error does not occur and the Window and Vulkan Instance and Surface create successfully, perhaps someone experienced can correct what I am doing wrong here?
Those two extensions I am trying to manually load are the only ones GLFW’s method does.

Exception in thread "main" java.lang.NullPointerException
        at org.lwjgl.system.Checks.check(
        at org.lwjgl.vulkan.KHRWin32Surface.nvkCreateWin32SurfaceKHR(
        at org.lwjgl.vulkan.KHRWin32Surface.vkCreateWin32SurfaceKHR(
        at win32_test.Win32Window.init(
        at win32_test.Win32Window.main(

uml – How use case scenario should look like, when use case have extensions points or include something

The first thing I’d do is throw away the use case diagram. In other answers, I’ve frequently quoted Martin Fowler, either from his site or his book, UML Distilled, where he expresses the lack of usefulness and value in UML Use Case diagrams. I’d only look at the two textual representations of the use case.

The next step is to clearly define what a use case is. A use case is a definition for how a user interacts with software (or a system) to achieve a useful result, considering inputs and responses to those inputs (or outputs). Based on this definition, the first example of a textual representation of the use case is the better of the two, since the main course actually ends with something valuable happening.

I’m not a fan of the extraction into child use cases, though, at least based on this example. It may make sense to promote reuse of the detailed textual descriptions, but unless the same steps related to “select repair type”, “enter home address” and “enter city” are reused in several use cases, I would consider those steps in the “enter repair order” use case. I’d also question why “enter home address” and “enter city” are separate use cases, since a city is a part of an address. It seems like there are more detailed design decisions being made here regarding how the domain is being modeled that wouldn’t be appropriate for consideration at the use case level.

Connection between the classifications of group extensions and group-graded algebras in terms of non-abelian cohomology

First, consider group extensions with non-abelian kernel
$$1to Kto G to Q to 1$$
It is well-known that these are classified by certain cohomological objects, specifically: Any such extension induces an outer action, i.e. a group homomorphism $Qtooperatorname{Out}(K)$ (which turns $Z(K)$ into a $mathbb{Z}Q$-module) and to any outer action we can associate an obstruction in $H^3(Q,Z(K))$ that vanishes iff any extension with the given outer action exists. And if they exist, they are in bijection with the non-abelian cohomology $H^2(Q,K)$.

Second, consider $Q$-graded, crossed $k$-algebras, i.e. $k$-algebras with a decomposition $A=bigoplus_{qin Q} A_q$ such that $1in A_1$, $A_{q_1} cdot A_{q_2} subseteq A_{q_1 q_2}$, and all $A_q$ contain a unit. Again, this situation induces an outer action $Qtooperatorname{Out}(A_1)$ (and $Z(A_1)$ becomes a $kQ$-module), for any such outer action there is an obstruction in $H^3(Q,Z(A_1)^times)$ that vanishes iff any $Q$-graded, crossed $k$-algebra with the given outer action exists. And if they exists, they are classified by non-abelian cohomology $H^2(Q,A_1^times)$.

These statements are so similar that it is natural to ask:

Question: What is a natural, common generalization of both statements?

There are natural constructions relating the two: Given a group extension, $A:=k(G)$ is $Q$-graded and crossed with $A_q:=k(qN)$. Conversely, given a crossed algebra, the group of homogeneous units fits into a natural extension $1to A_1^times to (A^times)_{homog.} to Gto 1$

But these constructions do not immediately give implications from one theorem to the other: The group extension version does not give you the algebra version, because $operatorname{Out}(A_1)$ may be different from $operatorname{Out}(A_1^times)$ and $Z(A_1)^times$ may have little to do with $Z(A_1^times)$.

Conversely, the algebra version does not give you the group version, $A_1=k(K)$, again because $operatorname{Out}(K)$ and $operatorname{Out}(k(K))$ can be very different.

magento2 – How could I check outdated extensions installed via Composer?

I have 30 extensions installed via Composer but when I need to check all the outdated extensions running the command below a huge list of non-extensions shows up, containing all the dependencies and etc.

composer outdated

amzn/amazon-pay-and-login-magento-2-module    4.2.1      5.6.0      Official Magento2 Plugin to integrate with Amazon Pay and Login with Amazon
Package amzn/amazon-pay-and-login-magento-2-module is abandoned, you should avoid using it. Use amzn/amazon-pay-magento-2-module instead.
braintree/braintree_php                       5.2.0      6.4.1      Braintree PHP Client Library
colinmollenhour/cache-backend-redis           1.11.0     1.14.2     Zend_Cache backend using Redis with full support for tags.
colinmollenhour/credis                        1.11.1     v1.12.1    Credis is a lightweight interface to the Redis key-value store which wraps ...
composer/composer                             1.10.22    2.1.5      Composer helps you declare, manage and install dependencies of PHP projects...
composer/semver                               1.7.2      3.2.5      Semver library that offers utilities, version constraint parsing and valida...
composer/xdebug-handler                       1.4.6      2.0.2      Restarts a process without Xdebug.
container-interop/container-interop           1.2.0      1.2.0      Promoting the interoperability of container objects (DIC, SL, etc.)
Package container-interop/container-interop is abandoned, you should avoid using it. Use psr/container instead.
csharpru/vault-php-guzzle6-transport          2.0.4      2.0.4      Guzzle6 transport for Vault PHP client
Package csharpru/vault-php-guzzle6-transport is abandoned, you should avoid using it. No replacement was suggested.
donatj/phpuseragentparser                     v0.20.0    v1.4.0     Lightning fast, minimalist PHP UserAgent string parser.
dotmailer/dotmailer-magento2-extension        4.12.0     4.13.0     dotmailer integration for magento 2
elasticsearch/elasticsearch                   v7.11.0    v7.14.0    PHP Client for Elasticsearch
endroid/qr-code                               3.9.7      4.2.1      Endroid QR Code
friendsofphp/php-cs-fixer                     v2.18.7    v3.0.2     A tool to automatically fix PHP code style
guzzlehttp/guzzle                             6.5.5      7.3.0      Guzzle is a PHP HTTP client library
guzzlehttp/psr7                               1.8.2      2.0.0      PSR-7 message implementation that also provides common utility methods
hoa/consistency                      The HoaConsistency library.
hoa/event                            The HoaEvent library.
hoa/exception                        The HoaException library.
laminas/laminas-code                          3.5.1      4.4.2      Extensions to the PHP Reflection API, static code scanning, and code genera...
laminas/laminas-escaper                       2.7.0      2.8.0      Securely and safely escape HTML, HTML attributes, JavaScript, CSS, and URLs
league/iso3166                                2.1.5      3.0.0      ISO 3166-1 PHP Library
magento/composer                              1.6.0      1.7.0      Magento composer library helps to instantiate Composer application and run ...
magento/magento2-functional-testing-framework 3.5.1      3.6.0      Magento2 Functional Testing Framework
monolog/monolog                               1.26.1     2.3.2      Sends your logs to files, sockets, inboxes, databases and various web services
nikic/php-parser                              v4.4.0     v4.12.0    A PHP parser written in PHP
pdepend/pdepend                               2.7.1      2.10.0     Official version of pdepend to be handled with Composer
phar-io/manifest                              1.0.3      2.0.3      Component for reading manifest information from a PHP Archive (PHAR)
phar-io/version                               2.0.1      3.1.0      Library for handling version information and constraints
php-amqplib/php-amqplib                       v2.10.1    v3.0.0     Formerly videlalvaro/php-amqplib.  This library is a pure PHP implementatio...
php-cs-fixer/diff                             v1.3.1     v2.0.2     sebastian/diff v2 backport support for PHP5.6
php-webdriver/webdriver                       1.8.2      1.11.1     A PHP client for Selenium WebDriver. Previously facebook/webdriver.
phpgt/dom                                     v2.1.6     v2.2.1     The modern DOM API for PHP 7 projects.
phpmd/phpmd                                   2.9.1      2.10.2     PHPMD is a spin-off project of PHP Depend and aims to be a PHP equivalent o...
phpseclib/mcrypt_compat                       1.0.8      2.0.0      PHP 7.1 polyfill for the mcrypt extension from PHP <= 7.0
phpseclib/phpseclib                           2.0.32     3.0.9      PHP Secure Communications Library - Pure-PHP implementations of RSA, AES, S...
phpunit/php-code-coverage                     8.0.2      9.2.6      Library that provides collection, processing, and rendering functionality f...
phpunit/php-token-stream                      4.0.4      4.0.4      Wrapper around PHP's tokenizer extension.
Package phpunit/php-token-stream is abandoned, you should avoid using it. No replacement was suggested.
phpunit/phpunit                               9.2.6      9.5.8      The PHP Unit Testing framework.
psr/container                                 1.1.1      2.0.1      Common Container Interface (PHP FIG PSR-11)
ramsey/uuid                                   4.1.1      4.2.1      A PHP library for generating and working with universally unique identifier...
sebastian/global-state                        4.0.0      5.0.3      Snapshotting of global state
squizlabs/php_codesniffer                     3.5.8      3.6.0      PHP_CodeSniffer tokenizes PHP, JavaScript and CSS files and detects violati...
symfony/console                               v4.4.29    v5.3.6     Eases the creation of beautiful and testable command line interfaces
symfony/error-handler                         v4.4.27    v5.3.4     Provides tools to manage errors and ease debugging PHP code
symfony/event-dispatcher                      v4.4.27    v5.3.4     Provides tools that allow your application components to communicate with e...
symfony/event-dispatcher-contracts            v1.1.9     v2.4.0     Generic abstractions related to dispatching event
symfony/http-kernel                           v4.4.29    v5.3.6     Provides a structured process for converting a Request into a Response
symfony/process                               v4.4.27    v5.3.4     Executes commands in sub-processes
vlucas/phpdotenv                              v2.6.7     v5.3.0     Loads environment variables from `.env` to `getenv()`, `$_ENV` and `$_SERVE...
webonyx/graphql-php                           v0.13.9    v14.9.0    A PHP port of GraphQL reference implementation

How could I have a list of my outdated Composer extensions only?