Filtered triangulated category examples

I am reading Beilison Ginsburg Schechtman’s “Koszul duality”. In the section 1.3, they introduced the notion filtered triangulated categories with only one example, considering an abelian category with filtration and take its corresponding derived category.

I would like to see other examples of filtered triangulated category other than that. Any references or explicit examples are welcome.

Weak generators of the right-bounded derived category of a finite-dimensional algebra

The setup:

  • Let $ A$ be a finite-dimensional $ k$ -algebra over some field $ k$ .
  • Let $ \mathcal{B} = Hot^-(Proj \, A)$ denote the homotopy category of cochain complexes of (possibly infinitely generated) projective right $ A$ -modules which are bounded from the right. This category can be identified with the right-bounded derived category $ D^-(Mod \, A)$ of $ A$ -modules.
  • Let $ \mathcal{C} = Hot^-(proj \, A)$ denote the full subcategory of $ \mathcal{B}$ given by right-bounded complexes of finitely generated projective $ A$ -modules. In different terms, this category corresponds to the right-bounded derived category $ D^-(mod \, A)$ of finitely generated $ A$ -modules.
  • Let $ P$ be a perfect object of $ \mathcal{B}$ , that is, a bounded complex of finitely generated projective right $ A$ -modules. Assume also that $ P$ is a weak generator of the subcategory $ \mathcal{C}$ , so for any object $ X \in \mathcal{C}$ there is some integer $ m$ and some non-zero morphism $ P \to X[m]$ in $ \mathcal{C}$ .

My question:

Is $ P$ already a weak generator of the big category $ \mathcal{B}$ ?

Some background:

  1. By a result of Jeremy Rickard, the answer is affirmative if $ P$ is a partial tilting complex, that is, if $ Hom_{\mathcal{B}}(P,P[n])=0$ for any non-zero integer $ n$ .

Reference: Proposition 5.4 in Rickard, Jeremy, Morita theory for derived categories, J. Lond. Math. Soc., II. Ser. 39, No. 3, 436-456 (1989). ZBL0642.16034.

  1. By a result of Bruno J. Müller, any projective $ A$ -module is a (possibly infinite) direct sum of finitely generated projective $ A$ -modules.

Reference: Müller, Bruno J., On semi-perfect rings, Mathematical Report No. 19, Vol. 1. Hamilton, Ont.: McMaster University, Department of Mathematics. 11 p. (1969). ZBL0226.16026.

  1. $ P$ is a weak generator of the category $ \mathcal{B}$ if and only if $ P$ is a classical generator of the category $ Hot^b(proj \,A)$ , that is, the smallest triangulated category containing $ P$ (which is closed under cones, shifts, isomorphisms and direct summands) is given by the homotopy category of bounded complexes of finitely generated projective $ A$ -modules.

Reference: Stacks-Project, https://stacks.math.columbia.edu/tag/09SR. Please write me if you know the original reference of this fact.

  1. At least, $ P$ is a weak generator of the homotopy category of complexes of projective $ A$ -modules which have finitely generated cohomology at each degree.

Any comments and any input will be very appreciated.

how can i get all product of a category in a custom page in magento 1.9

i have created a custom sub category page, i which i’m calling all the sub-categories, now i want to show product of each sub-categories in the same page how can i do that. This is what i do to get my sub-categories

1- create a custom block “sub-category-listing”

2-then i created a subcategorylist.phtml page

<div class="category-products sub-cat"> <ul class="products-grid" style="margin-left:0px;margin-right:0px;"> <?php $  _categories=$  this->getCurrentChildCategories(); if($  _categories->count()): $  categorycount = 0; foreach ($  _categories as $  _category): if($  _category->getIsActive()): $  cur_category=Mage::getModel('catalog/category')->load($  _category-  >getId()); $  layer = Mage::getSingleton('catalog/layer'); $  layer->setCurrentCategory($  cur_category); $  catName = $  this->getCurrentCategory()->getName();  ?>  <li class="item <?=$  class?>"> <h2><a href="<?php echo $  _category->getURL() ?>" title="<?php echo  $  this->htmlEscape($  _category->getName()) ?>"><?php echo $  this-  >htmlEscape($  _category->getName()) ?></a></h2> </li> <?php endif; if($  categorycount == 100000){ $  categorycount = 0; echo "</ul>\n\n<ul class=\"products-grid\">"; } else{ $  categorycount++; } endforeach; endif; ?> </ul> </div> 

how can i get all product of a category in a custom page in magento 1.9

i have created a custom sub category page, i which i’m calling all the sub-categories, now i want to show product of each sub-categories in the same page how can i do that. This is what i do to get my sub-categories

1- create a custom block “sub-category-listing”

2-then i created a subcategorylist.phtml page

<div class="category-products sub-cat"> <ul class="products-grid" style="margin-left:0px;margin-right:0px;"> <?php $  _categories=$  this->getCurrentChildCategories(); if($  _categories->count()): $  categorycount = 0; foreach ($  _categories as $  _category): if($  _category->getIsActive()): $  cur_category=Mage::getModel('catalog/category')->load($  _category-  >getId()); $  layer = Mage::getSingleton('catalog/layer'); $  layer->setCurrentCategory($  cur_category); $  catName = $  this->getCurrentCategory()->getName();  ?>  <li class="item <?=$  class?>"> <h2><a href="<?php echo $  _category->getURL() ?>" title="<?php echo  $  this->htmlEscape($  _category->getName()) ?>"><?php echo $  this-  >htmlEscape($  _category->getName()) ?></a></h2> </li> <?php endif; if($  categorycount == 100000){ $  categorycount = 0; echo "</ul>\n\n<ul class=\"products-grid\">"; } else{ $  categorycount++; } endforeach; endif; ?> </ul> </div> 

Do non-existent ‘category’ pages present a problem for search engines?

Say for example if I have a news site and my url structure is

site.com/news/random-article

If there is no page at site.com/news/ in other words it’s a 404, is that problematic from an SEO perspective or problematic for the search engine crawlers?

The reason I ask is that the homepage is the actual “news” landing page so I was thinking of using a permanent redirect to the homepage if the nonexistent page is a problem.

But I also want to preserve the possibility of eventually moving the “news” landing page to site.com/news/. Would a permanent redirect mean that would become impossible from the perspective of being indexable in the future?

How to skip any category in seo meta title template?

We have magento 2 on website we have 4 categories, we sell photocopier parts, let me give an example what issue we are facing and what we want.

Konica Minolta – Bizhub Color Series- Bizhub 203 – Toner Cartridges Category 1 – Category 2 – Category 3 – Category 4

We have magewrox seo suit installed on website , we made a template for meta titles when we call categories in this template the meta title goes like this

“Konica Minolta Bizhub Color Series Bizhub 203 Toner Cartridges”

We want to skip series category in seo templates , but when we change the sequence like Category 1 – Category 3 – Category 4 on different pages and stages it catches the sequence at current stage of ccategory breadcrumbs and does not skip. We want to show this category to customers but want to hide and skip it in meta titles. We talked with magewrox they said that it is not possible in their extension. Do we have any customization in which we can show category on front end but it should be hidden in seo templates. Like on any page when we call category 2 in any template it should not consider the series category it should consider next one as category 2. Hope it make sense if you dont understand I will try to elaborate it more

Category Custom Attribute in rest API in magento 2

I am using magento 2.2.7 and i am creating rest API for Category. From swagger i am getting value like:

{ "id": 28, "parent_id": 2, "name": "Smart Phones", "is_active": true,  "position": 1, "level": 2, "children": "", "created_at": "2018-11-03 01:32:45", "updated_at": "2019-01-14 06:33:52", "path": "1/2/28", "available_sort_by": [], "include_in_menu": true, "custom_attributes": [   {     "attribute_code": "display_mode",   "value": "PRODUCTS" }, {   "attribute_code": "is_anchor",   "value": "1" }, {   "attribute_code": "path",   "value": "1/2/28" }, {   "attribute_code": "children_count",   "value": "0" }, {   "attribute_code": "custom_use_parent_settings",   "value": "0" }, {   "attribute_code": "custom_apply_to_products",   "value": "0" }, {   "attribute_code": "url_key",   "value": "smart-phones" }, {   "attribute_code": "url_path",   "value": "smart-phones" },  {   "attribute_code": "tamil",   "value": "பேக்குகள் & லக்கேஜ்" }, {   "attribute_code": "telugu",   "value": "బ్యాగ్స్ & లగేజీ" } ] } 

In which “attribute_code”: “telugu”, and “attribute_code”: “tamil”, is custom attribute. and it is coming inside the custom Attribute array. i want this parameter outside of custom attribute along with NAME and ID. please help me to achieve this.