nLab precategory

Contents

This entry is about a category whose objects form a type rather than a set. For the specific type of paracategory also known as a precategory, in which all the composites are generated from binary and nullary ones, see paracategory.

Context

Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

type theory (dependent, intensional, observational type theory, homotopy type theory)

syntax object language

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logicset theory (internal logic of)category theorytype theory
propositionsetobjecttype
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction setinternal homfunction type
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian productdependent productdependent product type
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijectionisomorphism/adjoint equivalenceequivalence of types
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels

semantics

Category theory

Contents

Idea

In homotopy type theory (or more generally intensional type theory), when defining a category (meaning a 1-category), one has to decide whether to impose an h-level restriction on the type of objects. A precategory is a category in which no such restriction is imposed; thus the type of objects may be a set (i.e. a 0-type), or a 1-type, or an even higher type.

Surprisingly much category theory can be done purely with precategories, and some authors call them merely “categories”. However, there are notable differences between precategories and the familiar notion of “category” from set-theoretic foundations. Moreover, the two do not coincide under the standard interpretation of homotopy type theory into set theory (e.g. the model in simplicial sets). Indeed, the semantics of a precategory is the 1-categorical analogue of a Segal space.

Alternative notions of “1-category” in homotopy type theory include strict categories, whose type of objects is a set, and univalent categories, whose type of objects is a 1-type whose identifications coincide with isomorphisms for the category structure. The latter are generally better-behaved and include naturally-occurring examples, but the former are sometimes technically useful.

There are also “even more pre-” notions of category in intensional type theory, such as E-categories.

Definition

With a family of morphisms

A precategory 𝒞\mathcal{C} consists of

  • a type of objects Ob(𝒞)Ob(\mathcal{C}),
  • for each object A:Ob(𝒞)A:Ob(\mathcal{C}) and B:Ob(𝒞)B:Ob(\mathcal{C}), a set Hom(A,B)Hom(A, B) of morphisms
  • for each object A:Ob(𝒞)A:Ob(\mathcal{C}), B:Ob(𝒞)B:Ob(\mathcal{C}), and C:Ob(𝒞)C:Ob(\mathcal{C}), a binary function
    () A,B,C():Hom(B,C)×Hom(A,B)Hom(A,C)(-)\circ_{A, B, C}(-) :Hom(B, C) \times Hom(A, B) \to Hom(A, C)
  • for each object A:Ob(𝒞)A:Ob(\mathcal{C}), a morphism id A:Hom(A,A)\mathrm{id}_A:Hom(A, A)
  • such that
    • the composition of morphisms is associative: for each object A:Ob(𝒞)A:Ob(\mathcal{C}), B:Ob(𝒞)B:Ob(\mathcal{C}), C:Ob(𝒞)C:Ob(\mathcal{C}), and D:Ob(𝒞)D:Ob(\mathcal{C}), and for each morphism f:Hom(A,B)f:Hom(A, B), g:Hom(B,C)g:Hom(B, C), and h:Hom(C,D)h:Hom(C, D),
      h A,C,D(g A,B,Cf)=h B,C,D(g A,B,Df)h \circ_{A, C, D} (g \circ_{A, B, C} f) = h \circ_{B, C, D} (g \circ_{A, B, D} f)
    • the composition of morphisms satisfies the left and right unit laws: for each object A:Ob(𝒞)A:Ob(\mathcal{C}) and B:Ob(𝒞)B :Ob(\mathcal{C}) and morphism f:Hom(A,B)f:Hom(A, B), id B A,B,Bf=f\mathrm{id}_B \circ_{A, B, B} f = f and f A,A,Bid A=ff \circ_{A, A, B} \mathrm{id}_A = f.

Of course, the latter two axioms are actually inhabitants of the identity type, hence are data included in the definition just like the first four. However, since each Hom(A,B)Hom(A,B) is a set, such equalities are (typally) unique whenever they exist, so in most cases their presence as data can be ignored.

With one type of morphism

A precategory 𝒞\mathcal{C} consists of

  • a type Ob(C)Ob(C) of objects;

  • a type Mor(C)Mor(C) of morphisms (or arrows);

  • functions s:Mor(C)Ob(C)s:Mor(C) \to Ob(C), called the source or domain, and t:Mor(C)Ob(C)t:Mor(C) \to Ob(C), called its target or codomain, such that ss and tt are jointly 0-truncated: the type of functions u:Mor(C)(Ob(C)×Ob(C))u:Mor(C) \to (Ob(C) \times Ob(C)) such that the following conditions are met:

    • the fibers of uu at each pair of objects a:Ob(C)×Ob(C)a:Ob(C) \times Ob(C) is a set

    • for the universal product projection functions π 1:(Ob(C)×Ob(C))Ob(C)\pi_1:(Ob(C) \times Ob(C)) \to Ob(C) and π 2:(Ob(C)×Ob(C))Ob(C)\pi_2:(Ob(C) \times Ob(C)) \to Ob(C) out of the product type and for every morphism f:Mor(C)f:Mor(C), there are identifications leftuniversal(f):π 1(u(f))=s(f)\mathrm{leftuniversal}(f):\pi_1(u(f)) = s(f) and rightuniversal(f):π 2(u(f))=t(f)\mathrm{rightuniversal}(f):\pi_2(u(f)) = t(f)

is contractible:

isContr( u:Mor(C)(Ob(C)×Ob(C)) f:Mor(C)isSet( a:Ob(C)×Ob(C)u(f)=a)×(π 1(u(f))=s(f))×π 2(u(f))=t(f))))\mathrm{isContr}\left(\sum_{u:Mor(C) \to (Ob(C) \times Ob(C))} \prod_{f:Mor(C)} \mathrm{isSet}\left(\sum_{a:Ob(C) \times Ob(C)} u(f) = a\right) \times (\pi_1(u(f)) = s(f)) \times \pi_2(u(f)) = t(f)))\right)

This recalls the definition of a preorder, where the source and target functions are required to be jointly monic, i.e. jointly (-1)-truncated;

  • a function

    ()():( f:Mor(C) g:Mor(C)t(f)=s(g))Mor(C)(-) \circ (-): \left(\sum_{f:Mor(C)} \sum_{g:Mor(C)} t(f) = s(g)\right) \to Mor(C)

    called composition;

  • a function id:Ob(C)Mor(C)id:Ob(C) \to Mor(C), called the identity;

  • such that the following properties are satisfied:

    • source and target are respected by composition: s(gf)=s(f)s(g \circ f) = s(f) and t(gf)=t(g)t(g\circ f) = t(g);

    • source and target are respected by identities: s(1 x)=xs(1_x) = x and t(1 x)=xt(1_x) = x;

    • composition is associative: (hg)f=h(gf)(h \circ g)\circ f = h\circ (g \circ f) whenever t(f)=s(g)t(f) = s(g) and t(g)=s(h)t(g) = s(h);

    • composition satisfies the left and right unit laws: if s(f)=xs(f) = x and t(f)=yt(f) = y, then 1 yf=f=f1 x1_y \circ f = f = f \circ 1_x.

Remarks

  • The interpretation of the notion of precategory in the model of homotopy type theory in simplicial sets yields a Segal space in which the morphism A 1A 0×A 0A_1 \to A_0 \times A_0 has homotopically discrete fibers.

  • The definition of precategory is evidently a type-theoretic notion of the definition of category with a family of collections of morphisms. An equivalent definition analogous to the definition of category with one collection of morphisms could be given, but it would be much less convenient to use. In particular, the “type of all morphisms” A,B:Ob(𝒞)Hom(A,B)\sum_{A,B:Ob(\mathcal{C})} Hom(A,B) in a precategory is not itself a set, since it can incorporate nontrivial identifications from Ob(𝒞)Ob(\mathcal{C}); the “hom-types are sets” condition in a one-type-of-morphisms definition of precategory would have to be stated as “the function 𝒞 1𝒞 0\mathcal{C}_1 \to \mathcal{C}_0 is 0-truncated” (i.e. its fibers are sets).

Properties

Every precategory is a locally univalent E-category.

In univalent type theory there is a structure identity principle? for precategories:

Equality of precategories in a univalent universe is the same as having a fully faithful equivalent-on-objects functor between precategories in the univalent universe.

This is because fully faithful functors establish an equivalence of hom-sets between the two precategories, and equivalent-on-objects functors establish an equivalence of object types between the two precategories, which means that fully faithful equivalent-on-objects functors is the right notion of equivalence between precategories.

Only in univalent categories are such functors the same as adjoint equivalence.

See also

References

Last revised on September 26, 2022 at 05:32:39. See the history of this page for a list of all contributions to it.