ct.category theory – Are there categories whose internal hom is somewhat ‘exotic’?

$newcommand{Hom}{mathrm{Hom}}newcommand{Set}{mathrm{Set}}newcommand{hom}{mathrm{hom}}$In this example one cannot really say that the internal Hom is “exotic”, but at least it is morally different from the external Hom.

Let $G$ be a group and let $Set(G)$ be the category of sets with a $G$-action. Morphisms of this category are mappings of sets which commute with the action. This category has finite products, given by the products of the underlying sets. In fact, it is cartesian closed: The internal Hom $hom(X,Y)$ is the set of all mappings $X to Y$, with the $G$-action given by conjugation: $$sigma cdot f := X xrightarrow{cdot sigma^{-1}} X xrightarrow{f} Y xrightarrow{cdot sigma} Y. $$

Note however that in a monoidally closed category, the external Hom can always be recoverd from the internal Hom as the set of its global elements:
$$ Hom(X,Y) cong Gamma(hom(X,Y)) := Hom( I, hom(X,Y)).$$
Here, $I$ is the unit object of the monoidal category.

In the above example, global elements of some $G$-set $Z$ are morphisms from the one-point $G$-set to $Z$. These correspond precisely to the fixed points of $Z$ under the $G$-action: $Gamma(Z) = Z^G$. In particular, the global elements of the internal Hom are mappings invariant under conjugation – i.e. morphisms of $G$-sets.

Elasticsearch displays categories from different storeview

Elasticsearch displays unactivated categories in the search results in the current store view. Click on the result to display a 404 page.

An idea of ​​what's wrong? I use Elasticsuite from Smile.

information architecture – Should I repeat the menu items in the main navigation if they belong to two categories?

I work on navigation for a retail site. I have a typical account menu (My Account, Login, Create Account, etc.) and a menu of hamburgers with department pages and so on.

The experience of the site is greatly improved when the client registers, so I wish to encourage this behavior.

My question is:
I have a menu item Account (My Shopping) in the main hamburger navigation under the title "Popular", I also have a My Account section in the hamburger menu. Should I repeat My purchases under Popular and My account in the burger menu so that the menu structure counts separately or should I remove My purchases from the hamburger section My account because it already appears in another part of the menu structure hamburger?

The structure and the two options are below:

Separate account menu:

Child 1: My purchases
Child 2: x
Child 3: x

Option 1:

Hamburger Menu:

Title of navigation: popular
Child 1: My purchases
Child 2: x
Child 3: x

Title of the navigation: My account
Child 1: My purchases
Child 2: Connection
Child 3: Create an account

Option 2:

Hamburger Menu:

Title of navigation: popular
Child 1: My purchases
Child 2: x
Child 3: x

Title of the navigation: My account
Child 1: Connection
Child 2: Create an account

[GET][NULLED] – Real WordPress Media Library – Media Categories / Folders v4.5.2


(GET) (NULLED) – Real WordPress Media Library – Media Categories / Folders v4.5.2

[GET][NULLED] – Real WordPress Media Library – Media Categories / Folders v4.5.2


(GET) (NULLED) – Real WordPress Media Library – Media Categories / Folders v4.5.2

ct.category theory – Relationship between group representations and evaluation and coevaluation maps for module categories $ vect_ {G} $

Let $ G $ to be a finite group and $ vec_ {G} $ to be the finite dimensional monoidal category $ G $gradient vector spaces.

Given $ vec_ {G} $ module category $ mathcal {M} $ we can define a double module category $ mathcal {M} ^ {*}: = Hom_ {vec} ( mathcal {M}, vec) $, with action given by precomposition (generalizing the double representation of a finite group), as $ mathcal {M} boxtimes mathcal {M} ^ * $ is a $ vec_ {G} $-module under the diagonal action of $ vec_ {G} $ and define an adjunct pair of $ vec_G- $module functors:

$ coev _ { mathcal {M}}: vec rightarrow mathcal {M} boxtimes mathcal {M} ^ * $

$ , , , , , , , , , , , , , , , , , , , , , , , , , , mathbb { C} mapsto oplus_ {i} M_ {i} boxtimes M ^ {i} $

$ ev _ { mathcal {M}}: mathcal {M} boxtimes mathcal {M} ^ * rightarrow with $

$ , , , , , , , , , , , , , , , , , , M, boxtimes M ^ {*} mapsto M ^ * (M $

or $ with $ above is considered the $ vec_ {G} $-module category defined by the forgetful functor $ F: vec_ {G} rightarrow with $ and $ {X ^ {i} } $ is a set of simple objects of $ mathcal {M} $ and $ {M ^ {i} } $ is a set of simple objects of $ mathcal {M} ^ * $ such as $ M ^ {j} (M_ {i}) = mathbb {C} $ for $ i = j $ and the zero vector space otherwise.

My question is, given the composition:

$ coev circ ev: vec rightarrow mathcal {M} boxtimes mathcal {M} ^ * rightarrow vec in End_ {Vect_ {G}} (vec) simeq Rep (G) $.

Is it possible to find a $ vec_ {G} $ module category $ mathcal {M} $ as the product $ coev circ ev $ defines a given representation $ ( rho, V) $ of $ G $? In addition, all representations of $ G $ to be found using the above composition of module functors?

I am also interested in the case of $ vec ^ { alpha} _ {G} $ or $ alpha in H ^ {3} (G, U (1)) $ defines a non-trivial monoidal $ vec_ {G} $ if anyone understands the generalization.

wordpress – price prefix of some product categories with text in Woocommerce

I needed to prefix the price of certain categories of products only on the store page (because the price may be higher depending on the extras that the customer might want).

Here are my requirements for this code:

  • Add the prefix only on the store page, not on the product page;
  • Add only the prefix to certain product categories (hard-coded).

I would like advice on possible improvements and certainly on potential security risks, if any.

Along with that, I've heard that globals are considered bad practices, but I often see them in WordPress. What are the alternatives, if there are any.

That's the code:

add_filter('woocommerce_get_price_html', 'change_product_price_html');
function change_product_price_html($price){

  // Check if we're on the shop page
  // Don't want the prefix on the product pages.
  if( is_shop() ) {
    global $product;

    // ID of categories to add the prefix to
    $categories = ("17");

    // Loop through all the category ID's if they equal one of the values
    // in the array, add the prefix.
    foreach( $product->get_category_ids() as $id ) {
      if( in_array($id, $categories) ) {

        // Add prefix
        $prefixedPrice   = "Starting at ";
        $prefixedPrice  .= $price;

        return $prefixedPrice;
    // Nothing matched in the loop,
    // return the price without the prefix.
    return $price;
  // We're not on the shop page so
  // return the price without the prefix.
  return $price;

web application – How to design a simple and effective layout for a parameter page with at least 200 options covering more than 18 categories

With so many options, the most important addition would probably be a Search establishment.

However, in a logical and consistent way, you manage to group your 200+ options into 20 categories, you will get users who know (pretty much) what they want to define / change, but can not immediately see in what category she could insert. Have a search mechanism where they can enter Color and see the settings for Background color, Text color or Color of the head bar etc. would, I think, be very useful.

Ideally, the "search domain" would include not only the labels used for each of the options, but also any wider "help" text that may be available (for example, text overwritten by a mouse, sections of the documentation). ).

A Z List Help. How to display the pages of certain categories?

Hello, I'm trying to create a list for my website that displays different pages of a certain category. I am able to create an AZ list of all my pages using the AZ list, but I do not know how to display only the pages of a certain category. Does anyone know how to do it or how to do it with another free list plugin? Thank you for all the help you can give us.

monoidal categories – $ A otimes (-) monoidality with $ A $ central monoid

Let $ ( mathcal {C}, otimes) $ a monoidal category, and $ (A, m, e) $ a monoid (where $ m: A otimes A to A $, $ e: I to A $ ecc. ), with $ (A) $ belonging to the center of $ ( mathcal {C}, otimes) $: $ u: A (ot) (-) cong (-) otimes A $, ecc. see (JS) p.38.

Consider the functor (usual) $ F_A (-): = A otimes (-): mathscr {C} to mathscr {C} $there are natural morphisms:

$ alpha_ {X, Y}: F_A (X) otimes F_A (Y) = A otimes X otimes A otimes Y xrightarrow {1u1} A otimes A otimes X otimes Y
xrightarrow {m11} X otimes Y = F_A (X otimes Y) $

$ phi: I cong F_A (I) $ (canonical isomphism).

Question: Does the above data define a monoidal functor? What axioms of consistency (between the monoidal and central object structure) we need?

(JS) Braided tensor categories, A.Joyal, R.Street