category theory – Products and coproducts in $mathsf{Poset}$

I was advised to explore products and coproducts in $mathsf{Poset}$ in order to improve my understanding of products and coproducts in general. Here I will type up my understanding.

Power set as category

Given set $X$ (which I suppose we should require to be non-empty), $(mathcal P(X), subseteq)$ is a partially ordered set. We can consider the partially ordered set to be a category wherein the objects are elements of $mathcal P(X)$, and there is a morphism $f colon A to B$ for $A, B in mathcal P(X)$ if and only if $A subseteq B$. This means that any hom set contains either zero or one morphism. The composition rule is satisfied because of transitivity of the order relation.

Composition of morphisms is associative because if $h colon X to Y, g colon Y to Z$, and $f colon Z to W$, then $f(gh) = (fg)h$ because both represent the true statement that $X subseteq W$. Identity morphisms exist because of the reflexivity of the order operation.

The product in $mathsf{Poset}$

Given category $mathsf C$, index set $mathsf I$ and indexed objects $X_i$ from the category, the product of ${ X_i ; i in mathsf I }$ is an object of $mathsf C$, denoted $prod X_i$, together with an indexed family of morphisms $pi_i colon prod X_i to X_i$ such that for any object $Y$ in $mathsf C$ and indexed family of morphisms $f_i colon Y to X_i$, there is a unique map $f colon Y to prod X_i$ such that $pi_i f = f_i$ for all $i in mathsf I$.

If we translate this into the more intuitive language of $mathsf{Poset}$, then $prod X_i$ is an object such that for any object $Y$ such that $Y subseteq X_i$ for all $i in mathsf I$, we have $Y subseteq prod X_i subseteq X_i$. The natural choice is $prod X_i := inf {X_i}$, because if we choose anything smaller then we might contradict our requirement that $Y subseteq prod X_i$. Uniqueness is not an issue because there is only one morphism from $Y$ to $prod X_i$.

The coproduct in $mathsf{Poset}$

Since the coproduct is dual to the product, I should be able to reverse arrows in the preceding abstract description in order to get the correct abstract description of a coproduct. Therefore, the coproduct of ${ X_i ; i in mathsf I }$ is an object of $mathsf C$, denoted $coprod X_i$, together with an indexed family of morphisms $pi_i colon X_i to coprod X_i$ such that for any object $Y$ in $mathsf C$ and indexed family of morphisms $f_i colon X_i to Y$, there is a unique map $f colon coprod X_i to Y$ such that $fpi_i = f_i$ for all $i in mathsf I$.

Translating into the language of $mathsf{Poset}$ again, $coprod X_i$ is an object such that for any object $Y$ such that $X_i subseteq Y$ for all $i in mathsf I$, we have $X_i subseteq coprod X_i subseteq Y$. The natural choice is $coprod X_i := sup { X_i ; i in mathsf I }$ because if we choose anything larger we might contradict our requirement that $coprod X_i subseteq Y$. Uniqueness is not an issue because there is only one morphism from $coprod X_i$ to $Y$.

Questions:

Is my work above correct? (It’s my understanding that $pi$ is not used for the coproduct but I didn’t feel like changing it since I’m emphasizing that the coproduct at least in the abstract is the reverse of the product.)

I’m under the impression that universal properties are worth learning about. How should I go about learning about universal properties in relation to what I’m doing here? Should I just look at examples?

I appreciate any help.

How to use an exclusive domain in a specific category page?

I’m using Magento 2.4 and I want to offer to my partners a page with products in a new domain and I think I can do it assigning a domain to some specific categories, instead of creating many stores.

How could I do it?

How to configure the category gridlines in Numbers?

Below are two charts with and without gridlines. Obviously the one with gridlines isn’t very useful. I would like to just have a gridline every 10 years or something like that. How do I do that?

without gridlines

With gridlines

plugins – Onchange the category dropdown display the feature post and blog list

I have three functions.

  1. The first one is displaying my categories dropdown from the custom post.
  2. The second is displaying the latest post(I have added the checkbox in each post if the user checked that then that will display in the latest post)
    3)And the third will display my all the post.

Below is the code I am using it.

//category dropdown
    function categoriesDropdown(){
    $categories = get_categories( array(
        'orderby' => 'name',
        'order'   => 'ASC',
        'taxonomy' => 'blogs_cat',
    ) );
     $output='';
     $output.='<select>';
    foreach( $categories as $category ) {
        $output.='<option value="'.$category->term_id.'">'.$category->name.'</option>'; 
    }
        $output.='</select>';
        return $output;
    }
    add_shortcode( 'showCategoryList', 'categoriesDropdown');
    
    // Feature blog if check box selected.
    function latestBlogView( $atts ){
          $the_query =array(
          'post_type' => 'blog',
          'post_status' => 'publish',
          'posts_per_page' => 3,
          'meta_key' => 'latestblog',
          'meta_value' => 1,
          'order'      => 'DESC'
        );
        
        $postData = '';
        // The Loop
         $featured = new WP_Query($the_query);
        $postData.='<div class="latestBlogsWrapper articlesWrapper"><ul>';
        if ($featured->have_posts()): while($featured->have_posts()): $featured->the_post();
            
        $postData.= '<li><div class="grid-item"><a href="'.get_permalink($post->ID).'">
        <div class="blogBoxwrapper"> 
        <img src="'. get_the_post_thumbnail_url($post->ID, "full").'"> 
        <div class="blogCatname"><h5>'.get_the_title($post->ID).'</h5></div>
        </div></div></a></li>'; 
        
        endwhile; else:
        $postData.="Please select the feature post check box";
    
        endif;
         $postData .= '</ul></div>';
        wp_reset_postdata();
        
        return $postData; 
    }
    add_shortcode( 'latestblogs', 'latestBlogView');
    
    // Blog list
    function BlogView( $atts ){
        $args = array(  
            'post_type' => 'blog',
            'post_status' => 'publish',
            'posts_per_page' => 30, 
            'orderby' => 'title', 
            'order' => 'DESC', 
        );    $loop = new WP_Query( $args ); 
        $data ='';
        $data.='<div class="articlesWrapper"><ul>';
        while ( $loop->have_posts() ) : $loop->the_post(); 
            $tid = $loop->ID;
            $data.= ' 
              <li>
                 <a href="'.get_permalink($tid).'">
                      <div class="blogBoxwrapper">
                         <img src="'.get_the_post_thumbnail_url($tid).'">
                            <div class="blogCatname">
                              <h5>'.get_the_title($id).'</h5>
                            </div>
                     </div>
                    </a>
            </li>'; 
        endwhile;  
        $data.='</ul>
        
        <div class="pt-5 text-center btnLoadmore"><a class="blogbtn blogbtnred loadMore" href="javascript:void(0);">Read More Blog Posts</a></div>
        </div>';
        wp_reset_postdata(); 
        return $data; 
    }
    add_shortcode( 'blogandarticles', 'BlogView');

Now what I am doing is, When the user changes the category from the dropdown then I have to display the latest blog and blog list related to that category.

For example. On page load, I am displaying all the posts by default. Now I have a category called Movie in the dropdown. Once the user selects the Movie from the dropdown then I have to show the Movie related post in the latest blog and blog list.

Would you help me out with this issue?

ag.algebraic geometry – embedding of derived category into another derived category

I am considering the following two cases:

  1. Assume that there is an embedding: $D^b(mathcal{A})xrightarrow{Phi} D^b(mathbb{P}^2)$and the homological dimension of $mathcal{A}$ is equal to $1$($mathcal{A}$ is an abelian category), for simiplicity, maybe first I assume that $mathcal{A}$ is a module catgeory over a finite dimensional $A$, then $A$ is a hereditary algebra. Assume that $Phi$ is a Fourier-Mukai functor, in addition, $A$ is $textbf{not}$ fractional Calabi-Yau algebra. I was wondering, what kind of condition I should impose on $A$, to conculde that $Acong KQ$(path algbera) such that $Q$ is a Kronecker quiver with three vertices and three arrows.

  2. Assume that there is an embedding: $D^b(mathcal{A}’)xrightarrow{Psi} D^b(J(Gamma))$, where $Gamma$ is a genus 2 degree 7 curve and $J(Gamma)$ is its Jacobian, which is an abelian surface. Also $mathcal{A}’$ has homological dimension 1 and $Psi$ is also Fourier-Mukai functor. What condition I should impose to conclude that $mathcal{A}’congmathrm{Coh}(Gamma)$? Note that in this case, $J(Gamma)$ is an abelian surface and there is no non-trivial SOD for its derived category, which means that $Psi(D^b(mathcal{A}’))$ is not a left or right admissble subcategory of $D^b(J(Gamma))$.

Motivation
I am considering $mathbb{P}^2$ as certain moduli space of stable objects in $mathcal{A}$ and $J(Gamma)$ as certain moduli space of stable objects in $mathcal{A}’$ and the embedding functor $Phi$ and $Psi$ are induced by Fourier-Mukai functor with the kernel given by universal family.

custom taxonomy – Code html in a product category

im working in a drinks web and one of the attributes is the location where the drink is originally, i want that not only shows the flag

I create a taxonomy call “Country” and tried the classic <img>name but only show me the text

I have a filter and there i use the propiety background to show the flag but for obvious reasons only show there and when i click in the product only show the text

I’ve been reading and searching and is a propiety of wordpress that prevents to insert html code in those thing

Is possible deactivate that function or i have to create a custom filter?

magento2 – Magento 2 : How to set different list.phtml for each category

I want to do like this list.phtml should be different different call in category page. I created custom category layout and assign phtml file for that. But, layered navigation at bottom after product grid.

It should be display same as like 2-column-left layout. How to solve it?

custom layout code :

app/design/frontend/Vendor/CustomTheme/Magento_Theme/page_layout/category_my_custom.xml

<?xml version="1.0" encoding="UTF-8"?>
<!--
/**
 * Copyright © Magento, Inc. All rights reserved.
 * See COPYING.txt for license details.
 */
-->
<layout xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" xsi:noNamespaceSchemaLocation="urn:magento:framework:View/Layout/etc/page_layout.xsd">
  <update handle="2columns-left"/>
  <body>
    <referenceContainer name="content">
    <referenceBlock name="category.products">
        <block name="product_list" class="MagentoCatalogBlockProductListProduct">
            <action method="setTemplate">
                <argument name="template" xsi:type="string">Magento_Catalog::product/book_list.phtml</argument>
            </action>
        </block>
    </referenceBlock>
    </referenceContainer>
</body>
</layout>

Please help me.

Thanks.

magento2.3.5 p1 – Something went wrong while saving the category

I get an error on Magento 2.3.5-p1 when trying to add products to an existing category.
The products won’t save and I get this message:

Argument 1 passed to MagentoCatalogModelCategoryFileInfo::removeStorePath() must be of the type string, array given, called in /cache/vendor/magento/module-catalog/Model/Category/FileInfo.php on line 167

If I navigate to the product page to add to a category i get this error:

Unable to unserialize value. Error: Syntax error

and this in the log:

(2020-09-18 19:44:02) main.CRITICAL: Unable to unserialize value. Error: Syntax error {“exception”:”(object) (InvalidArgumentException(code: 0): Unable to unserialize value. Error: Syntax error at /cache/vendor/magento/framework/Serialize/Serializer/Json.php:39)”} ()

How can I fix this?

How to get product category ID based on post?

I want to show certain images on product pages based on the category ID. How can I achieve it on the product page?

Display category name , category posts and create pagination using ajax [closed]

Display category name , category posts and create pagination by clicking on category name the related category posts need to show using ajax