Category management - Category captains

In the UK, the Groceries Code Adjudicator found in her 2015-16 investigation into Tesco plc that some suppliers paid "large sums of money in exchange for category captaincy or participation in a price review". She found some evidence of benefits which suppliers derive from these arrangements but also recorded a concern, to be investigated further, as to whether the purpose of the Groceries Code was being circumvented by these payments.

Category management - Category captains

It is commonplace for a particular supplier in a category to be nominated by the retailer as a category captain. The category captain will be expected to have the closest and most regular contact with the retailer and will also be expected to invest time, effort, and often financial assets into the strategic development of the category within the retailer. In return, the supplier will gain a more influential voice with the retailer. The category captain is often the supplier with the largest turnover in the category. Traditionally the job of category captain is given to a brand supplier, but in recent times the role has also gone to particularly switched-on private label suppliers. In order to do the job effectively, the supplier may be granted access to a greater wealth of data-sharing, e.g. more access to an internal sales database such as Walmart's Retail Link.

Comma category - Slice category

The first special case occurs when, the functor S is the identity functor, and (the category with one object * and one morphism). Then T(*) = A_* for some object A_* in \mathcal{A}. In this case, the comma category is written, and is often called the slice category over A_* or the category of objects over A_*. The objects (A, *, h) can be simplified to pairs (A, h), where. Sometimes, h is denoted by \pi_A. A morphism from (A, \pi_A) to in the slice category can then by simplified to an arrow making the following diagram commute:

Comma category - Arrow category

S and T are identity functors on \mathcal{C} (so ). In this case, the comma category is the arrow category. Its objects are the morphisms of \mathcal{C}, and its morphisms are commuting squares in \mathcal{C}.

Comma category - Coslice category

The dual concept to a slice category is a coslice category. Here, S has domain \textbf{1} and T is an identity functor. In this case, the comma category is often written , where B_*=S(*) is the object of \mathcal{B} selected by S. It is called the coslice category with respect to B_*, or the category of objects under B_*. The objects are pairs with. Given and, a morphism in the coslice category is a map making the following diagram commute:

Category (mathematics) - Dual category

Any category C can itself be considered as a new category in a different way: the objects are the same as those in the original category but the arrows are those of the original category reversed. This is called the dual or opposite category and is denoted C op.

Category 6 cable - Category 6A

The standard for Category 6A is ANSI/TIA-568-C.1, defined by the Telecommunications Industry Association (TIA) for enhanced performance standards for twisted pair cable systems. It was defined in 2009. Cat 6A performance is defined for frequencies up to 500 MHz—twice that of Cat 6. Cat 6A also has an improved alien crosstalk specification as compared to Cat 6, which picks up high levels of alien noise at high frequencies.

Compact closed category - Simplex category

The augmented simplex category can be used to construct an example of non-symmetric compact closed category. The **augmented simplex category** is the category of finite ordinals (viewed as totally ordered sets); its morphisms are order-preserving (monotone) maps. We make it into a monoidal category by moving to the arrow category, so the objects are morphisms of the original category, and the morphisms are commuting squares. Then the tensor product of the arrow category is the original composition operator. The left and right adjoints are the min and max operators; specifically, for a monotonic function f one has the right adjoint

Compact closed category - Rigid category

A monoidal category that is not symmetric, but otherwise obeys the duality axioms above, is known as a rigid category. A monoidal category where every object has a left (resp. right) dual is also sometimes called a left (resp. right) autonomous category. A monoidal category where every object has both a left and a right dual is sometimes called an autonomous category. An autonomous category that is also symmetric is then a compact closed category.

Simplex category - Augmented simplex category

The augmented simplex category, denoted by \Delta_+ is the category of all finite ordinals and order-preserving maps, thus, where. Accordingly, this category might also be denoted FinOrd. The augmented simplex category is occasionally referred to as algebraists' simplex category and the above version is called topologists' simplex category.

Category (Kant) - Meaning of "category"

A category is that which can be said of everything in general, that is, of anything that is an object. John Stuart Mill wrote: "The Categories, or Predicaments—the former a Greek word, the latter its literal translation in the Latin language—were believed to be an enumeration of all things capable of being named, an enumeration by the summa genera (highest kind), i.e., the most extensive classes into which things could be distributed, which, therefore, were so many highest Predicates, one or other of which was supposed capable of being affirmed with truth of every nameable thing whatsoever."

Category management - Modified category management

For MRP-based manufacturing industries, the predominant cost-saving methodology in category management (CM) involves the integration of market intelligence with leveraged spending (for a given category of product or service). In industries where asset operation and preservation bear more significance to the procurement process than do product manufacturing – such as in an MRO environment – demonstrable benefit can still be achieved with category management but is best approached with some manner of adjustment to CM’s usual processes for analysis and strategy development. The first challenge becomes incorporating analytical processes and value drivers that are largely indigenous to the MRP world in a manner that makes sense to an MRO environment. The second (and no less important) challenge becomes avoiding a trap where the CM processes are perceived to be more important than their outcome – a scenario that can result in significant analytical delay, and even complete process paralysis. An excellent example of an MRO environment warranting adjustment to classical category management is nuclear power generation in the United States, where the adjusted approach to category management has been coined "MCM" – standing for MRO-based Category Management or Modified Category Management. Not only does electricity generation epitomize an MRO-driven environment, the nuclear energy source adds numerous dimensions of supply and procurement complexity – including federal and state regulatory compliance, nuclear industry standards compliance, nuclear-unique system and component design, and a tightly-audited (and very small) supply base, amongst others. Due to the nature and quantity of discrete characteristics native to nuclear power generation, it can easily be argued that nuclear power generation, in and of itself, should be a distinct category of procurement within a category management project. The fundamental adjustment made between the classical category management approach and the nuclear MCM approach is a shift from procurement strategies focused on leveraged spending to procurement strategies embracing nuclear value drivers, technology innovation, risk management, and strategic sourcing.

Simplex category - Augmented simplex category

The augmented simplex category provides a simple example of a compact closed category.

Simplex category - Augmented simplex category

A contravariant functor defined on \Delta_+ is called an augmented simplicial object and a covariant functor out of \Delta_+ is called an augmented cosimplicial object; when the codomain category is the category of sets, for example, these are called augmented simplicial sets and augmented cosimplicial sets respectively.

Simplex category - Augmented simplex category

The augmented simplex category, unlike the simplex category, admits a natural monoidal structure. The monoidal product is given by concatenation of linear orders, and the unit is the empty ordinal [-1] (the lack of a unit prevents this from qualifying as a monoidal structure on \Delta). In fact, \Delta_+ is the monoidal category freely generated by a single monoid object, given by [0] with the unique possible unit and multiplication. This description is useful for understanding how any comonoid object in a monoidal category gives rise to a simplicial object since it can then be viewed as the image of a functor from to the monoidal category containing the comonoid; by forgetting the augmentation we obtain a simplicial object. Similarly, this also illuminates the construction of simplicial sets from monads (and hence adjoint functors) since monads can be viewed as monoid objects in endofunctor categories.

Quasi-category - The homotopy category

Given a quasi-category C, one can associate to it an ordinary category hC, called the homotopy category of C. The homotopy category has as objects the vertices of C. The morphisms are given by homotopy classes of edges between vertices. Composition is given using the horn filler condition for n = 2.

Quasi-category - The homotopy category

For a general simplicial set there is a functor \tau_1 from sSet to Cat, known as the fundamental category functor, and for a quasi-category C the fundamental category is the same as the homotopy category, i.e. .

Category (Kant) - Meaning of "category"

The word comes from the Greek κατηγορία, katēgoria, meaning "that which can be said, predicated, or publicly declared and asserted, about something." A category is an attribute, property, quality, or characteristic that can be predicated of a thing. "…I remark concerning the categories…that their logical employment consists in their use as predicates of objects." Kant called them "ontological predicates."

Category management - Rationale for category management

A third reason was that the collaboration with the supplier meant that supplier's expertise about the market could be drawn upon, and also that a considerable amount of workload in developing the category could be delegated to the supplier.

Category O - Definition of category O

Morphisms of this category are the -homomorphisms of these modules.