Not signed in

Want to take part in these discussions? Sign in if you have an account, or apply for one below

• Username
• Password
• Remember me

• Sign in using OpenID
• Remember me

• Forgot your password?
• Apply for membership

2-category 2-category-theory abelian-categories adjoint algebra algebraic algebraic-geometry algebraic-topology analysis analytic-geometry arithmetic arithmetic-geometry book bundles calculus categorical categories category category-theory chern-weil-theory cohesion cohesive-homotopy-type-theory cohomology colimits combinatorics complex complex-geometry computable-mathematics computer-science constructive cosmology definitions deformation-theory descent diagrams differential differential-cohomology differential-equations differential-geometry digraphs duality elliptic-cohomology enriched fibration foundation foundations functional-analysis functor gauge-theory gebra geometric-quantization geometry graph graphs gravity grothendieck group group-theory harmonic-analysis higher higher-algebra higher-category-theory higher-differential-geometry higher-geometry higher-lie-theory higher-topos-theory homological homological-algebra homotopy homotopy-theory homotopy-type-theory index-theory integration integration-theory k-theory lie-theory limits linear linear-algebra locale localization logic mathematics measure-theory modal modal-logic model model-category-theory monad monads monoidal monoidal-category-theory morphism motives motivic-cohomology nforum nlab noncommutative noncommutative-geometry number-theory of operads operator operator-algebra order-theory pages pasting philosophy physics pro-object probability probability-theory quantization quantum quantum-field quantum-field-theory quantum-mechanics quantum-physics quantum-theory question representation representation-theory riemannian-geometry scheme schemes set set-theory sheaf simplicial space spin-geometry stable-homotopy-theory stack string string-theory superalgebra supergeometry svg symplectic-geometry synthetic-differential-geometry terminology theory topology topos topos-theory tqft type type-theory universal variational-calculus

1. • CommentRowNumber1.
   • CommentAuthornLab edit announcer
   • CommentTimeJan 24th 2023
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexstarting page on lists by a suggestion on the discussion page for [[free monoid]] Anonymous <a href="https://ncatlab.org/nlab/revision/list/1">v1</a>, <a href="https://ncatlab.org/nlab/show/list">current</a>

   starting page on lists by a suggestion on the discussion page for free monoid

   Anonymous

   v1, current

2. • CommentRowNumber2.
   • CommentAuthorUrs
   • CommentTimeJan 24th 2023
   • PermaLink
   Author: Urs
   Format: MarkdownItexThanks. If you can, please try to start the Idea section with at least one sentence on the general idea of "lists" (and maybe their relation to free monoids), before mentioning HoTT. We must remember that the readership here is not just HoTT specialists. Also, I think it's good for you HoTT nerds to try to step back once in a while and try to articulate subject matters in broader terms.

   Thanks.

   If you can, please try to start the Idea section with at least one sentence on the general idea of “lists” (and maybe their relation to free monoids), before mentioning HoTT. We must remember that the readership here is not just HoTT specialists. Also, I think it’s good for you HoTT nerds to try to step back once in a while and try to articulate subject matters in broader terms.

3. • CommentRowNumber3.
   • CommentAuthornLab edit announcer
   • CommentTimeJan 24th 2023
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexstarting page on lists by a suggestion on the discussion page for [[free monoid]] Anonymous <a href="https://ncatlab.org/nlab/revision/list/1">v1</a>, <a href="https://ncatlab.org/nlab/show/list">current</a>

   starting page on lists by a suggestion on the discussion page for free monoid

   Anonymous

   v1, current

4. • CommentRowNumber4.
   • CommentAuthorUrs
   • CommentTimeJan 24th 2023
   • (edited Jan 24th 2023)
   • PermaLink
   Author: Urs
   Format: MarkdownItexNow it looks like you are not just anonymous but also not reading here.(?) In any case, I went ahead and re-wrote the Idea-section from scratch. Currently it reads as follows: *** In [[mathematics]] and specifically in [[formal logic]], by a "list" one means the fairly evident formalization of the colloquial notion of a "list of elements": If a [[set]] $E$ of admissible elements is given --- also called an "[[alphabet]]", in this context --- then a list of $E$-elements is a [[tuple]] $(e_1, e_2, \cdots, e_\ell)$ of elements $e_i \,\in\, E$, of any length $\ell \,\in\, \mathbb{N}$ --- also called a *[[word]]* in the given alphabet. Typically one will write "$List(E)$" or similar for the [[set]] of all lists with entries in $E$. For instance, for $E$ the ([[underlying]] set of) a [[field]], the component-expressions of [[vectors]] in the [[vector space]] $k^n$ may be thought of as lists of length $n$ with elements in $k$. However, doing so means to disregard the [[vector space]]-[[structure]] on the collection of all such lists. Instead, the notion of *list* is a [[concept with an attitude]]: While lists are just [[tuples]] of any length, calling them lists indicates that one intends to consider the operation of *[[concatenation]]* of lists to larger lists, in particular the operations of appending an element to a list. It is evident that the operation of concatenation makes $List(E)$ a *[[monoid]]* (the [[neutral element]] under concatenation is the [[empty list]], i.e. the unique list of length [[zero]]). In fact, a moment of reflection shows that, as such, $\big(List(E), conc\big)$ is the *[[free monoid]]* on $E$. Therefore, in much of the mathematical literature, lists are understood as [[free monoids]]. Specifically in [[computer science]] one commonly deals with the corresponding notion of the [[data type]] of lists (with entries in a prescribed data type), which serve as a basic and common kind of [[data structure]]. In [[programming languages]] supporting something like a [[calculus of constructions]], this [[data type]], essentially with the [[free monoid]]-[[structure]] as above, may be defined as an *[[inductive type]]* in an evident way (made explicit [below](#Definition)). On the other hand, in [[dependent type theories]] which have a [[homotopy type theory]]-interpretation --- in that they do not verify [[uniqueness of identity proofs]] (or "[[axiom K (type theory)|axiom K]]") --- one may want to distinguish between lists and [[free monoids]]: Because, in such theories it may happen that the given [[alphabet]] [[type]] $E$ is not actually a [[set]] but a higher [[homotopy type]], in that it is not [[0-truncated]]. In this case it still makes good sense to speak of *lists* of elements ([[terms]]) of $E$, only that now the corresponding [[type]] $List(E)$ is itself not [[0-truncated]]. But since the term "[[monoid]]" carries with it a connotation of being [[0-truncated]], one may no longer want to call this the *[[free monoid]]* on $E$. It is, of course, still a free monoid in the proper [[higher algebra|higher algebraic]] sense (cf. *[[monoid in a monoidal (infinity,1)-category|monoid in a monoidal $\infty$-category]]*). *** <a href="https://ncatlab.org/nlab/revision/diff/list/3">diff</a>, <a href="https://ncatlab.org/nlab/revision/list/3">v3</a>, <a href="https://ncatlab.org/nlab/show/list">current</a>

   Now it looks like you are not just anonymous but also not reading here.(?)

   In any case, I went ahead and re-wrote the Idea-section from scratch. Currently it reads as follows:

   ---

   In mathematics and specifically in formal logic, by a “list” one means the fairly evident formalization of the colloquial notion of a “list of elements”:

   If a set $E$ of admissible elements is given — also called an “alphabet”, in this context — then a list of $E$-elements is a tuple $(e_1, e_2, \cdots, e_\ell)$ of elements $e_i \,\in\, E$, of any length $\ell \,\in\, \mathbb{N}$ — also called a word in the given alphabet.

   Typically one will write “$List(E)$” or similar for the set of all lists with entries in $E$.

   For instance, for $E$ the (underlying set of) a field, the component-expressions of vectors in the vector space $k^n$ may be thought of as lists of length $n$ with elements in $k$. However, doing so means to disregard the vector space-structure on the collection of all such lists.

   Instead, the notion of list is a concept with an attitude: While lists are just tuples of any length, calling them lists indicates that one intends to consider the operation of concatenation of lists to larger lists, in particular the operations of appending an element to a list.

   It is evident that the operation of concatenation makes $List(E)$ a monoid (the neutral element under concatenation is the empty list, i.e. the unique list of length zero). In fact, a moment of reflection shows that, as such, $\big(List(E), conc\big)$ is the free monoid on $E$.

   Therefore, in much of the mathematical literature, lists are understood as free monoids.

   Specifically in computer science one commonly deals with the corresponding notion of the data type of lists (with entries in a prescribed data type), which serve as a basic and common kind of data structure. In programming languages supporting something like a calculus of constructions, this data type, essentially with the free monoid-structure as above, may be defined as an inductive type in an evident way (made explicit below).

   On the other hand, in dependent type theories which have a homotopy type theory-interpretation — in that they do not verify uniqueness of identity proofs (or “axiom K”) — one may want to distinguish between lists and free monoids:

   Because, in such theories it may happen that the given alphabet type $E$ is not actually a set but a higher homotopy type, in that it is not 0-truncated. In this case it still makes good sense to speak of lists of elements (terms) of $E$, only that now the corresponding type $List(E)$ is itself not 0-truncated. But since the term “monoid” carries with it a connotation of being 0-truncated, one may no longer want to call this the free monoid on $E$. It is, of course, still a free monoid in the proper higher algebraic sense (cf. monoid in a monoidal $\infty$-category).

   ---

   diff, v3, current

5. • CommentRowNumber5.
   • CommentAuthornLab edit announcer
   • CommentTimeJan 25th 2023
   • PermaLink
   Author: nLab edit announcer
   Format: MarkdownItexadded links to [[James construction type]] and [[free monoid]] Anonymous <a href="https://ncatlab.org/nlab/revision/diff/list/7">diff</a>, <a href="https://ncatlab.org/nlab/revision/list/7">v7</a>, <a href="https://ncatlab.org/nlab/show/list">current</a>

   added links to James construction type and free monoid

   Anonymous

   diff, v7, current