ct.category theory – Finitary monads on $Set$ are substitution monoids. Finitary monads on $Set_*$ are…?

It is well known that the category of functors $F : Fin to Set$ is equivalent to the category of finitary endofunctors $Set to Set$; in this equivalence, finitary monads correspond to what are called substitution monoids on $(Fin,Set)$, i.e. to monoids with respect to the monoidal structure
F diamond G = mmapsto int^n Fn times G^{ast n}m tag{$star$}
where $G^{*n}$ is th functor
m mapsto int^{p_1,dots, p_n} Gp_1 times dots times Gp_n times Fin(sum p_i, m).

More precisely, the equivalence $(Set,Set)_{omega} cong (Fin,Set)$ can be promoted to a monoidal equivalence, and composition of endofunctors corresponds to substitution of presheaves in the following sense: let $J : Fin to Set$ be the inclusion functor, then
Lan_J(Fdiamond G) cong Lan_JF circ Lan_JG tag{$heartsuit$}

(Scirc T) J cong SJ diamond TJtag{$clubsuit$}
for two finitary endofunctors $S,T : Set to Set$. (Kan extending along $J$ and precomposing an endofunctor of $Set$ with $J$ is what defines the equivalence.)

I would like to prove the same exact theorem, replacing everywhere the cartesian category of sets with the monoidal category of pointed sets and smash product, but I keep failing.

The equivalence of categories
(Fin_*,Set_*)cong (Set_*,Set_*)
remains true; and this equivalence must induce an equivalence between the category of finitary monads on pointed sets, and the category of suitable “pointed substitution” monoids, that are obtained from the iterated convolution on $(Fin_*,Set_*)$ as
Fdiamond’ G = m mapsto int^n Fn land G^{*n}m
where $land$ is the smash product, and $G^{*n}$ iterates the convolution on $(Fin_*, Set_*)$ induced by coproduct on domain, and smash on codomain:
G^{*n}m = int^{p_1,dots,p_n} Gp_1 land dotsland Gp_n land Fin_*(bigvee p_i, m)
where $bigvee p_i$ is the coproduct of pointed sets, joining all sets along their basepoint.

This shall be the perfect equivalent of $(star)$.

However, if to prove the isomorphisms $heartsuit, clubsuit$, I find that it is not true that $Lan_J(Fdiamond G) cong Lan_JF circ Lan_JG$. I am starting to suspect that the generalisation is false as I have stated it, or that it is true in a more fine-tuned sense.

So, I kindly ask for your help:

To what kind of monoids on $(Fin_*,Set_*)$ do finitary monads on pointed sets correspond?

On some optimal crontainers of a set of points on the 2D plane

Given a set of N points in general position on the plane, give efficient algorithms to find

  1. the smallest semicircular region (semidisk) that contains the points
  2. the smallest circular segment that contains the points (2 variants – ‘smallest’ could mean either of ‘least area’ or ‘least perimeter’).
  3. the smallest sector that contains the points (again, 2 variants)

I am not aware of previous works on these questions.

opengl – Can I set the unity camera to have the same position in a real world?

I have a problem about setting the camera in the unity to have the same extrinsic parameters as my real world camera to the origin. So let’s me describe my problem step by step.

  1. I have a camera in a real world and Its extrinsic to an aruco plane was calculated. So I use the Aruco plane(You can think as a plane reference) as an world origin(0, 0, 0).

  2. The in my unity, I want to set my camera to have the same extrinsic as my real world camera. So I take an translation to set the Position in unity transform toolbox. And set the rotation by convert my rotation matrix from extrinsic to the euler’s angle and set it to x, y, z rotation.

  3. So I test this by use the unity properties named “camera.cameraToWorldMatrix”. You can treat this as an extrinsic of the unity camera transform to the world coordinates(0, 0, 0). So this value should equal to my extrinsic since it’s use the same rotation euler’s angle. But unfortunately it’s not.

So how can i solve this problem or there’s a simpler way to set the cameraToWorldMatrix equal to my extrinsic parameters.

Thank you very much.

How to Set Up Plex Media Server on Your Cheap VPS

With a huge portion of the world’s population told to stay home, many people are looking for ways to fill the time. Setting up a home media server might be a great idea and help keep you and your loved ones entertained. When it comes to software for home media systems, Plex is by far the best option.

Plex is a brilliant software that lets you access your media library on all your devices. It makes the process of setting up a media server a breeze. Plex will also automatically organize your media files, such as movies and songs, for a fantastic experience. It is available on almost every platform you can think of. This includes Windows, macOS, Android, iOS, and Linux. Plex is also available to install on Smart TV’s, such as Roku-enabled or Amazon Fire Stick platforms.

Today, we are going to guide you through the process of setting up a Plex Media Server on your remote Linux server, specifically on a server running Ubuntu 18.04 (this tutorial will also work with Ubuntu 16.04 servers).

We will divide the process into two parts: installation on your remote server (VPS) and configuration (local).

Let’s start with the installation process.


Step 1. Update your system

Before you download and start installing the Plex media server, you need to make sure that your system is updated. For this, enter this command in the terminal:

Step 2. Download Plex Media Server

Now you are ready to download and install Plex. For that, head over to the download page on Plex’s website. Now choose “Plex Media Server,” select Linux as your OS, and then select Ubuntu as your distribution. Now at this stage, there are two ways to download the deb package. Copy the direct URL so we can download it on the server using wget. For that, copy the download link and run this command, make sure that you download the latest version though:

Step 3. Install the downloaded file

Now we will install the deb file using dkpg. So, once you are done downloading the file, go to that directory and run the following command:

  • sudo dpkg -i plexmediaserver*.deb

Step 4. Run the Plex server

Now, you are done with the installation process! To start the Plex Server, run these two commands:

  • sudo systemctl enable plexmediaserver.service
  • sudo systemctl start plexmediaserver.service

Let’s now move on to the configuration of your Plex Media Server.


IMPORTANT NOTE: These steps will be performed on your computer (assuming you are running on MacOS or Linux) — not on your VPS.

Step 1. Open Plex web interface

To start configuring Plex, you first need to connect to your remote server via SSH. To do this, perform the command:

  • ssh root@VPSIP -L 8888:localhost:32400

NOTE: Replace “VPSIP” with the IP of your server.

Now that you are connected, open your web browser and go to http://localhost:8888/web

 Step 2. Sign in/Sign up on Plex

When you open the web interface, you would be asked to sign in or register a Plex account.

Step 3. Add directories to the server

Perform this step on the remote server.

Create directories for your media files, such as movies, TV shows, and music.

Step 4. Add libraries

Now on your computer, choose the “Add Library” option. This will prompt you to choose the type of library. You can select the relevant option (Movies, TV shows, Music, and Photos).

After that, you would be asked to choose a directory for the library. Choose the one that you made in step 3 (/home/plex/movies, for example.)


Step 5. Add media

Now you are all set up to add the media to your libraries. When doing that, make sure that you are following Plex’s naming conventions.

Congratulations! You are all done with the setup and installation of your Plex Media Server. All that is left now is to download the Plex client for your devices (this is supported on every platform, including Smart TV’s as mentioned earlier). Once you log in, you would be able to access all your media that is hosted upon your remote server/VPS with ease!

Editors note: This article was contributed to the LowEndBox community by RackNerd – a provider of VPS and other hosting solutions. If you are looking for a VPS server to run a Plex Media Server on without breaking the bank, be sure to take a look at their exclusive LEB offers by clicking here!

Jon Biloh

I’m Jon Biloh and I own LowEndBox and LowEndTalk. I’ve spent my nearly 20 year career in IT building companies and now I’m excited to focus on building and enhancing the community at LowEndBox and LowEndTalk.

networking – How to set static ip on Hyper-V vm when host has one network card?

I am running Hyper-V on Windows 10 and created an Ubuntu server vm running postgresql. My pc has a single nic card.
There is a single network adapter on the VM that is connected to the Default switch in Hyper-V. I am able to connect to the internet from the vm and can connect to the instance of Postgresql on the VM from the host and everything works fine.
But each time I reboot the VM it gets a new IP address so everyday I have to reconfigure the connection string on my app that makes calls to the Postgresql instance on the vm.
How can I configure the ip addresses that are assigned to the devices on the vlan (not sure if that is the correct term) that are sitting behind the Default Switch? I am guessing there is a DHCP server somewhere but I can’t find anything.

I just want to have the same ip always assigned to the VM or be able to set the ip manually and know that it is not going to be assigned to another VM, but am at a loss as to how to do what seems like should be a super simple thing.

graphs – Minimum spanning tree with small set of possible edge weights

Given a undirected graph which only has two different edge weights $x$ and $y$ is it possible to make a faster algorithm than Prim’s algorithm or Kruskal’s algorithm?

I saw on Kruskal’s algorithm that it already assumes that the edges can be sorted in linear time with count sort, so then can we gain any benefit by knowing the weights of all of the edges ahead of time?

I don’t think there is any benefit, but I am not sure…

8 – Upload an image to filesystem, rename it and set it as Profile Picture for current user

I have the external link of an image where I am trying to:

  1. Upload the image to filesystem.
  2. Rename the image.
  3. Set the owner of the image to current user.
  4. Set the image as Profile Picture for current user.

So far I have:

$full_name = "John Smith";
$picture_url = "https://scontent.fbey14-1.fna.fbcdn.net/v/t31.0-8/14054356_124897527960923_50194821455451882_o.jpg?_nc_cat=106&_nc_sid=09cbfe&_nc_ohc=P4i5oscJf0IAX_KFCfR&_nc_ht=scontent.fbey14-1.fna&oh=c13b6b2d8712db359f3be5e1ec835340&oe=5EF2B1A4";

// Download the picture to local filesystem.
$profile_pic = system_retrieve_file($picture_url, "public://profile-pictures", TRUE, FILE_EXISTS_REPLACE);

// Get the Profile Pic ID.
$profile_pic_id = $profile_pic->id();

// Load the image.
$picture = DrupalfileEntityFile::load($profile_pic_id);

// Rename the image.
// Set the owner of the profile picture file.

// Save the image.

$current_user->set('user_picture', $profile_pic_id);

The image is indeed uploaded to filesystem under public://profile-pictures, also is successfully set as profile picture of the current user.

and the filename of the image at /user/*/edit edit page is set to: john-smith.jpg, however, the filename of the image at public://profile-pictures is 14054356_124897527960923_50194821455451882_o.jpg while I am expecting it to be john-smith.png !

How to rename the image at the filesystem ?

and does $picture->setOwner($current_user) will set the owner of the image to the current user ?

Help needed: How to set up a campaign/project for non-English speaking regions properly?


I’ve been trying to to find my way around the Search Engine Ranker lately. I’m pretty sure that you can do some things wrong when you want to start a project for a language area outside of English.

Does anyone have experience with campaigns for other languages like German, French or Spanish. What to look for and what data to translate into the target language.

Your help is highly appreciated.

Thanks in advance

air travel – What is the name for the set of the outbound segment and inbound segment together?

I have a technical naming question.

Image I have this set of flights for a passegner flying from Paris to Thailand:

International Outbound:

- From Paris (France), Airport Charles De Gaulle (CDG)
- To Bangkok (Thailand), Airport Suvarnabhumi (BKK)
- 1 stop at Istambul (IST)


- From the city (Bangkok/Thailand) (BKK)
- To the beaches of Phuket (Phuket/(Thailand) (HKT)
- No stops

International Inbound:

- From Phuket (Thailand), aiport Phuket (HKT)
- To Paris (France), Airport Charles De Gaulle (CDG)
- 2 stops, one at BKK, one at IST.

With all this, the terminology is:


  • Any “piece” of flight: Each piece you takeoff and land.
  • For example the outbound flight has 2 legs, the second one being IST-BKK


  • The set of legs that connect the origin to the destination.
  • For example when returning from Thailand to Paris, there’s one segment HKT-CDG which is made up of 3 legs.


What is the “name” for the full set of segments in the tourism industry?

Ie: The same way { leg CDG-IST + leg IST-BKK } is named “segment”

what is the name for { segment CDG-BKK + segment BKK-HKT + segment HKT-CDG }?