dg.differential geometry – What is the appropriate notion of Weakly Equivalent or Morita Equivalent categories internal to a category of generalized Smooth Spaces?

Let $G$ and $H$ be Lie Groupoids. We know that there are two notions of equivalences of Lie Groupoids:

  1. Strongly Equivalent Lie Groupoids: (My terminology)

A homomorphism $phi:G rightarrow H$ of Lie Groupoids is called a strong equivalence if there is a Lie groupoid homomorphism $psi:H rightarrow G$ and natural transformation of Lie Groupoid homomorphism $T: phi circ psi Rightarrow id_H$ and $S: psi circ phi rightarrow id_G$. In this case $G$ and $H$ is said to be Strongly Equivalent Lie Groupoids.

  1. Weakly Equivalent or Morita Equivalent Lie Groupoids :

A homomorphism $phi:G rightarrow H$ of Lie Groupoids is called a weak equivalence if it satisfies the following two conditions

enter image descriptiobb

where $H_0$, $H_1$ are object set and morphism set of Lie Groupoid H respectively. Similar meaning holds for symbols $G_0$ and $G_1$. Here Symbols $s$ and $t$ are source and target maps respectively. The notation $pr_1$ is the projection to the first factor from the fibre product. from t. Here the condition (ES) says about Essential Surjectivity and the condition (FF) says about Fully Faithfulness.

One says that two Lie Groupoids $G$ and $H$ are weakly equivalent or Morita Equivalent if there exist weak equivalences $phi:P rightarrow G$ and $phi’:P rightarrow H$ for a third Lie groupoid $P$.

(According to https://ncatlab.org/nlab/show/Lie+groupoid#2CatOfGrpds one motivation for introducing Morita Equivalence is the failure of Axiom of Choice in the category of smooth manifolds )

WHAT I AM LOOKING FOR:

Now let we replace $G$ and $H$ by categories $G’$ and $H’$ which are categories internal to a category of Generalized smooth spaces (For example, category of Chen Spaces or Category of Difffeological spaces,..etc). For instance, our categories $G’$ , $H’$ can be Path groupoids.

Analogous to the case of Lie Groupoids I can easily define the notion of Strongly Equivalent Categories internal to a category of Generalized Smooth Spaces.

Now if I assume that the Axiom of choice fails also in the category of generalized smooth spaces then it seems reasonable to introduce a notion of Weakly Equivalent or some sort of Morita Equivalent categories internal to a category of generalized smooth spaces.

But it seems that we cannot directly define the notion of Weak Equivalent or Morita Equivalent Categories internal to a category of Generalized Smooth Spaces in an analogous way as we have done for Lie Groupoids. Precisely in the condition of Essential Surjectiveness (ES) we need a notion of surjective submersion but I don’t know the analogue of surjective submersion for generalized smooth spaces

I heard that Morita Equivalence of Lie Groupoids are actually something called “Anaequivalences” between Lie Groupoids.(Though I don’t have much idea about anafunctors and anaequivalences).

So my guess is that the appropriate notion of Weakly Equivalent or Morita Equivalent categories internal to a category of Generalized smooth spaces has something to do with anaequivalence between categories internal to a category of generalized smooth spaces. Is it correct?

My Questions is the following:

What is the appropriate notion of Weakly Equivalent or Morita Equivalent categories internal to a category of generalized Smooth Spaces?

plugins – Trying to add categories texts inside the container

I am trying to add categories texts, but I don’t why when I place the code, it doesn’t show up.

Here is the code for related posts box (related-posts.php)

function related_posts_after_post_content($content){

if ( !is_admin() && is_singular( 'post' ) ) {
    
    global $post;
    $options = get_option('related_posts');
    $related_posts_custom_css = $options('related_posts_custom_css') ?: '';
    $related_post_background_color = $options('related_post_background_color') ?: '#FFFFFF';
    $related_post_title = $options('related_post_title') ?: '#3365c3';
    $related_post_content = $options('related_post_content') ?: '#000';
    $related_posts_heading = $options('related_posts_heading') ?: 'Recommended For You';
    $related_posts_image = $options('related_posts_image') ?: '';
    $related_posts_excerpt = $options('related_posts_excerpt') ?: '';
    $related_posts_image_resolution = $options('related_posts_image_resolution') ?: 'thumbnail';
    $related_posts_value = $options('related_posts_value') ?: '6';
    
    // Custom CSS
    if(!empty($related_posts_custom_css)) { 
        $content .= '<style type="text/css">'. $related_posts_custom_css .'</style>'; 
    }
    
    $content .= '<style type="text/css">
        .visualmodo-related-post {background-color:'. $related_post_background_color('regular') .'; }
        .visualmodo-related-post:hover {background-color:'. $related_post_background_color('hover') .'; }
        .visualmodo-related-post:hover .visualmodo-related-post-body-title {color:'. $related_post_title('hover') .'; }
        .visualmodo-related-post:hover .visualmodo-related-post-body-content {color:'. $related_post_content('hover') .'; } 
    </style>';
    
    $content .= '<div class="visualmodo-related-posts">';
    
    $content .= '<h3 class="visualmodo-related-posts-title">'.$related_posts_heading.'</h3>';
    
    function get_excerpt(){
        $options = get_option('related_posts');
        $related_posts_excerpt_value = $options('related_posts_excerpt_value') ?: '60';
        $excerpt = get_the_content();
        $excerpt = preg_replace(" ((.*?))",'',$excerpt);
        $excerpt = strip_shortcodes($excerpt);
        $excerpt = strip_tags($excerpt);
        $excerpt = substr($excerpt, 0, $related_posts_excerpt_value);
        return $excerpt;
    }
    
    $related = get_posts( 
        array( 
            'category__in' => wp_get_post_categories(get_the_ID()), 
            'numberposts' => $related_posts_value, 
            'post__not_in' => array(get_the_ID()) 
            ) 
        );
        
        if( $related )
        
        $content .= '<div class="visualmodo-related-posts-grid">';
        
        foreach( $related as $post ) {
            
            setup_postdata($post);
            
            if ($related_posts_image == true && has_post_thumbnail()) { 
                $img = get_the_post_thumbnail( get_the_ID(), $related_posts_image_resolution, array( 'class' => 'visualmodo-related-post-body-image' ) ); 
            }
            
            $content .= '<div class="visualmodo-related-post">';
            $content .= '<a rel="bookmark" href="'.get_the_permalink().'">';
            
            $content .= $img;
            
            $content .= '<div class="visualmodo-related-post-body">';
            $content .= '<h6 class="visualmodo-related-post-body-title">'. get_the_title() .'</h6>';
            if ($related_posts_excerpt == true) { 
                $content .= '<p class="visualmodo-related-post-body-content">'. get_excerpt() .'</p>'; 
            }
            $content .= '</div>';
            
            $content .= '</a>';
            $content .= '</div>';
        }
        
        $content .= '</div>';

I just want to place the categories on the top right corner of every box (Above image), but don’t know the exact code to put in work.

enter image description here

And, this is the code I’m using to display categories texts.

<?php

foreach((get_the_category()) as $category) {
echo $category->cat_name . ‘ ‘;
}

  • Reference – theappflow.com/best-anxiety-apps-2020

category theory – Non-Examples of Functors and Categories

I’m preparing to deliver some lectures on homological algebra and category theory, and have found lots of nice long lists of examples of functors and categories arising in every-day mathematical practice. I am interested in a similar list, but for non-examples.

I know, for instance, that the center $Z(G)={gin G,|, hg=gh text{ for all } hin G}$ of a group/ring/etc. fails to be a functor, and that the association of a Cayley graph to a group fails to be a functor from Groups to Graphs.

There was an earlier thread about this, but with the restriction that non-examples must be functions on objects and on morphisms but fail to respect morphism composition. I felt like the examples in this thread were somewhat artificial as well. I’m interested in examples where a student may expect there to be a category or functor involved, but there is not.

categorization – Nested categories in sidebar

I want to build a sidebar with categories and sub-categories in it, so to say nested categories, but without a limitation on depth, so you can have as many sub categories of unlimited number of parent categories.

The thing and the problem is, that the sidebar is limited by width and when sub-category is present, it is nested with a bit of right padding(to illustrate it is a sub-category), so if you have enough depth in sub-categories it can cause problems, as there is limitation of width and therefor there won’t be any space left to add right padding. I can’t find a good solution for this, please share your thoughts with me, perhaps we could find a better way around.

Here’s a picture to illustrate how it might look:

enter image description here

ct.category theory – Are compact objects in presheaf categories finite colimits of representables?

An object $x$ in a category $mathsf{C}$ is called compact or finitely presentable if $$mathrm{hom}(x,-) : mathsf{C} to mathsf{Set}$$ preserves filtered colimits. This concept behaves best when $mathsf{C}$ has all filtered colimits, e.g. when it is the category of presheaves on some small category $mathsf{X}$:

$$ mathsf{C} = mathsf{Set}^{mathsf{X}^{mathrm{op}}} $$

Every representable presheaf is compact. In general, any finite colimit of compact objects is compact. Thus, any fimite colimit of representables is compact.

My question is about the converse: in the category of presheaves on a small category, is every compact object a finite colimit of representables?

custom post types – The Difference Between Categories and Tags and Taxonomies and Terms

I’ve got a custom post type ‘articles’ and I’ve added a custom taxonomy ‘articles_categories’ which has its own terms.

I was under the impression you can’t have categories and tags on custom post types? But I can’t see how Taxonomies and Terms are the same? And if you can have categories and tags on custom post types what is the reasoning for using custom Taxonomies? It’s all very confusing.

Could someone explain the context of taxonomies and why can’t you use categories and tags on custom post types?

Many thanks in advance

How to list all subcategories from all categories but not from a certain category

I know how to list all subcategories from a certain category, but don’t know how to list all subcategories from all categories.

For example, here is category:

Category 1
    -Child Category1
    -Child Category2
Category 2
    -Child Category3
    -Child Category4
Category 3
    -Child Category5
    -Child Category6

The list i want is :

-Child Category1
-Child Category2
-Child Category3
-Child Category4
-Child Category5
-Child Category6

This is code i have now

<?php wp_list_categories( array(
    'orderby'=> 'id',
    'order' => 'DESC',
    'show_count' => false,
    'use_desc_for_title' => false,
     'title_li' => '',
) ); ?>