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Colimits is an English word. Below you'll find 10+ example sentences showing how it's used in practice.

Rare word

Colimits in a sentence

Related words

Colimits | Colimit

Colimits meaning

plural of colimit

Using Colimits

  • The main meaning on this page is: plural of colimit
  • In the example corpus, colimits often appears in combinations such as: and colimits, of colimits, colimits of.

Context around Colimits

  • Average sentence length in these examples: 18.4 words
  • Position in the sentence: 4 start, 7 middle, 6 end
  • Sentence types: 17 statements, 0 questions, 0 exclamations

Often before:

small 1× takes 1× weak 1× necessarily 1× preserves 1× take 1×

Often after:

seealso 1× except 1× categorical 1×

Sentence patterns with Colimits

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

all small colimits 1× and weak colimits are defined 1× category has colimits of type 1× cones are colimits and co 1× definitions for colimits 1× lifting of colimits 1× limits and colimits 1× limits and colimits can also 1×

Sentence openings and endings

Common sentence openings:

a category has 1× any colimit functor 1× examples every representable 1× for contravariant functors 1× for this reason 1× it follows that 1×

Common sentence endings:

all small colimits 1× and co cones 1× are equally valid 1× colimits to limits 1× definitions for colimits 1× in category theory 1×

Words around Colimits

Just before it in the word index:

colimit 8 colima 7 colijn 2 colihan 2 coligny 9 coliform 12

Just after it in the word index:

colin 1.121 colina 11 colinas 8 colindale 3 colindres 2 colins 3

Similar in corpus frequency:

aav 21 abdicating 21 abductor 21 abet 21 abh 21 abhorred 21

Corpus analysis for Colimits

  • In this selection, "colimits" usually appears in the middle of the sentence. The average example has 18.4 words, and this corpus slice is mostly made up of statements.
  • Around the word, small, takes, weak, seealso, except and categorical stand out and add context to how "colimits" is used.
  • Recognizable usage signals include all small colimits and and weak colimits are defined. That gives this page its own corpus information beyond isolated example sentences.
  • By corpus frequency, "colimits" sits close to words such as aav, abdicating and abductor, which helps place it inside the broader word index.

Example types with colimits

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

One can make analogous definitions for colimits. (7 words)

The dual statements for colimits are equally valid. (8 words)

There are dual definitions for the lifting of colimits. (9 words)

Longer example sentences

Any colimit functor is left adjoint to a corresponding diagonal functor (provided the category has the type of colimits in question), and the unit of the adjunction provides the defining maps into the colimit object. (35 words)

This follows, in part, from the fact the covariant Hom functor Hom(N, ) : C → Set preserves all limits in C. By duality, the contravariant Hom functor must take colimits to limits. (31 words)

A category has colimits of type J if every diagram of type J has a colimit in C. A cocomplete category is one that has all small colimits. (28 words)

Example sentences (17)

A category has colimits of type J if every diagram of type J has a colimit in C. A cocomplete category is one that has all small colimits.

For contravariant functors the corresponding notions would be a functor that takes colimits to limits, or one that takes limits to colimits.

For this reason one often speaks of the limit of F. Colimits seeAlso The dual notions of limits and cones are colimits and co-cones.

Weak limit and weak colimits are defined like limits and colimits, except that the uniqueness property of the mediating morphism is dropped.

Any colimit functor is left adjoint to a corresponding diagonal functor (provided the category has the type of colimits in question), and the unit of the adjunction provides the defining maps into the colimit object.

Examples * Every representable functor C → Set preserves limits (but not necessarily colimits).

It follows that any functor which preserves limits will take terminal objects to terminal objects, and any functor which preserves colimits will take initial objects to initial objects.

Limits and colimits Categorical products are a particular kind of limit in category theory.

Advertentie

More general concepts are the limits and colimits of category theory.

One can generalize the above example to arbitrary limits and colimits.

One can make analogous definitions for colimits.

Since many common constructions in mathematics are limits or colimits, this provides a wealth of information.

The dual statements for colimits are equally valid.

There are dual definitions for the lifting of colimits.

The terminology is somewhat confusing: inverse limits are limits, while direct limits are colimits.

This follows, in part, from the fact the covariant Hom functor Hom(N, ) : C → Set preserves all limits in C. By duality, the contravariant Hom functor must take colimits to limits.

Variations Limits and colimits can also be defined for collections of objects and morphisms without the use of diagrams.

Advertentie

Common combinations with colimits

These word pairs occur most frequently in English texts:

Frequently asked questions

How do you use "colimits" in a sentence?
An example: "A category has colimits of type J if every diagram of type J has a colimit in C. A cocomplete category is one that has all small colimits." This page contains 10+ example sentences with the word "colimits" from authentic English texts.
What does "colimits" mean?
Colimits means: plural of colimit
How many example sentences with "colimits" are there?
Voorbeeldzinnen.info contains at least 10+ example sentences with "colimits", drawn from a database of millions of English sentences.