Random word

On this page you'll find 9 example sentences with Counit. Discover the meaning, how to use the word correctly in a sentence.

Rare word

Counit in a sentence

Counit meaning

In an adjunction, a natural transformation from the composition of the left adjoint functor with the right adjoint functor to the identity functor of the domain of the right adjoint functor.

Using Counit

  • The main meaning on this page is: In an adjunction, a natural transformation from the composition of the left adjoint functor with the right adjoint functor to the identity functor of the domain of the right adjoint functor.
  • In the example corpus, counit often appears in combinations such as: the counit, and counit.

Context around Counit

  • Average sentence length in these examples: 28.4 words
  • Position in the sentence: 2 start, 3 middle, 4 end
  • Sentence types: 9 statements, 0 questions, 0 exclamations

Often before:

first 1×

Often after:

unit 4× map 1×

Sentence patterns with Counit

These short patterns come directly from the example sentences and show how the word is often embedded:

called the counit 1× form a counit unit adjunction 1× normalizing the counit by dividing 1× of the counit can be 1× than the counit unit definition 1× the first counit unit equation 1× the second counit unit equation 1× trace the counit map with 1×

Sentence openings and endings

Common sentence openings:

additionally every pair 1× equivalences of categories 1× for example naturality 1× for example one 1× formally one can 1× in practice in 1×

Common sentence endings:

be the identity 2× called the counit 1× corresponds to dimension 1× counit are isomorphisms 1× counit unit definition 1× functor preserves limits 1×

Words around Counit

Just before it in the word index:

counihan 2 coundon 2 cound 2 councilwomen 5 councilwoman 161 councilthe 4

Just after it in the word index:

counos 3 counrty 3 counsel 2.643 counseled 37 counseling 663 counsell 36

Similar in corpus frequency:

aakash 12 aanholt 12 aardwolf 12 abadan 12 abati 12 abcc 12

Corpus analysis for Counit

  • In this selection, "counit" usually appears near the end of the sentence. The average example has 28.4 words, and this corpus slice is mostly made up of statements.
  • Around the word, first, unit and map stand out and add context to how "counit" is used.
  • Recognizable usage signals include called the counit and form a counit unit adjunction. That gives this page its own corpus information beyond isolated example sentences.
  • By corpus frequency, "counit" sits close to words such as aakash, aanholt and aardwolf, which helps place it inside the broader word index.

Example types with counit

The same corpus examples are grouped by length and sentence type, making it easier to see the contexts in which the word appears:

Short example sentences

The second counit-unit equation says that for each group X the composition : should be the identity. (17 words)

Additionally, every pair of adjoint functors comes equipped with two natural transformations (generally not isomorphisms) called the unit and counit. (20 words)

For example, naturality and terminality of the counit can be used to prove that any right adjoint functor preserves limits. (20 words)

Longer example sentences

Equivalences of categories If a functor F: C←D is one half of an equivalence of categories then it is the left adjoint in an adjoint equivalence of categories, i.e. an adjunction whose unit and counit are isomorphisms. (39 words)

Formally, one can compose the trace (the counit map) with the unit map of "inclusion of scalars " to obtain a map mapping onto scalars, and multiplying by n. Dividing by n makes this a projection, yielding the formula above. (39 words)

In practice, in bialgebras one requires that this map be the identity, which can be obtained by normalizing the counit by dividing by dimension ( ), so in these cases the normalizing constant corresponds to dimension. (34 words)

Example sentences (9)

Then, a direct verification that they form a counit-unit adjunction is as follows: The first counit-unit equation says that for each set Y the composition : should be the identity.

Additionally, every pair of adjoint functors comes equipped with two natural transformations (generally not isomorphisms) called the unit and counit.

Equivalences of categories If a functor F: C←D is one half of an equivalence of categories then it is the left adjoint in an adjoint equivalence of categories, i.e. an adjunction whose unit and counit are isomorphisms.

For example, naturality and terminality of the counit can be used to prove that any right adjoint functor preserves limits.

For example, one may have an algebra A with maps (the inclusion of scalars, called the unit) and a map (corresponding to trace, called the counit ).

Formally, one can compose the trace (the counit map) with the unit map of "inclusion of scalars " to obtain a map mapping onto scalars, and multiplying by n. Dividing by n makes this a projection, yielding the formula above.

In practice, in bialgebras one requires that this map be the identity, which can be obtained by normalizing the counit by dividing by dimension ( ), so in these cases the normalizing constant corresponds to dimension.

The second counit-unit equation says that for each group X the composition : should be the identity.

This definition is a logical compromise in that it is somewhat more difficult to satisfy than the universal morphism definitions, and has fewer immediate implications than the counit-unit definition.

Common combinations with counit

These word pairs occur most frequently in English texts:

Frequently asked questions

How do you use "counit" in a sentence?
An example: "Then, a direct verification that they form a counit-unit adjunction is as follows: The first counit-unit equation says that for each set Y the composition : should be the identity." This page contains 9 example sentences with the word "counit" from authentic English texts.
What does "counit" mean?
Counit means: In an adjunction, a natural transformation from the composition of the left adjoint functor with the right adjoint functor to the identity functor of the domain of the right adjoint functor.
How many example sentences with "counit" are there?
Voorbeeldzinnen.info contains 9 example sentences with "counit", drawn from a database of millions of English sentences.