Presenting Functors by Operations and Equations

(draft)

The main theorem is that a functor on a variety (in the sense of universal algebra) has a presentation by operations and equations if and only if the functor preserves sifted colimits.

The theory of sifted colimits is well explained in Adamek-Rosicky-Vitale. We only cover some highlights.

In all of the following

$\mathcal{A}$ is a variety in the sense of univeral algebra, that is, a category of algebras specified (finitary) operations equations. (We will assume all operations to be finitary in the following, even if we drop the qualifier.)

Sifted Colimits

Filtered categories are precisely categories

$\mathcal{D}$ such that colimits over

$\mathcal{D}$ commute with finite limits in Set. There is also a characterization of filtered categories independent of sets – a category

$\mathcal{D}$ is filtered if and only if the diagonal functor

$\mathcal{D}\to {\mathcal{D}}^{\mathcal{I}}$ is final for each finite category

$\mathcal{I}$.

Sifted categories are the categories

$\mathcal{D}$ such that colimits over

$\mathcal{D}$ commute with finite products. These categories are characterized by the property that the diagonal functor

$\mathcal{D}\to \mathcal{D}\times \mathcal{D}$ is final.

Facts:

• Every filtered colimit is sifted.
• Reflexive coequalisers are sifted but not filtered.
• Sifted colimit preserving functors preserve filtered colimits.

An object

$A$ of a category

$\mathcal{C}$ is finitely presentable if its hom-functor

$hom(A,-):\mathcal{C}\to Set$ preserves filtered colimits. A category

$\mathcal{C}$ is locally finitely presentable (lfp) if it is cocomplete and has a set

$\mathcal{X}$ of finitely presentable objects such that each object of

$\mathcal{C}$ is a filtered colimit of objects from

$\mathcal{X}$.

An object

$A$ is strongly finitely presentable if its hom-functor

$hom(A,-):\mathcal{C}\to Set$ preserves sifted colimits. A category

$\mathcal{C}$ is strongly locally finitely presentable (slfp) if it is cocomplete and has a set

$\mathcal{X}$ of strongly finitely presentable objects such that each object of

$\mathcal{C}$ is a sifted colimit of objects from

$\mathcal{X}$.

Let

$\mathcal{A}$ be a variety in the sense of universal algebra.

Facts:

• Finitely presentable objects in
  $\mathcal{A}$ are algebras finitely presentable in a usual sense.
• Strongly finitely presentable algebra = retract of finitely generated free algebra.
• Finitely presentable algebra = reflexive coequalizer of strongly finitely presentable ones.
• Sifted colimit preserving functors on
  $\mathcal{A}$ are determined by their action on finitely generated free algebras.

Let

$Alg(L)$ be the categories of algebras for the functor

$L:\mathcal{A}\to \mathcal{A}$.

Theorem: If

$\mathcal{A}$ is a variety and

$L:\mathcal{A}\to \mathcal{A}$ preserves sifted colimits then

$Alg(L)$ is a variety.

Proposition:

• Let
  $\mathcal{A}$ be a variety such that every finitely presentable algebra is projective. Then any functor
  $L:\mathcal{A}\to \mathcal{A}$ preserving filtered colimits preserves sifted colimits.
• If
  $T:Set\to Set$ preserves filtered colimits then
  $T$ preserves sifted colimits.
• For any filtered colimit preserving functor
  $L:BA\to BA$ there is a sifted colimit preserving functor
  $L\prime :BA\to BA$ such that
  $L$ and
  $L\prime$ are isomorphic when restricted to the full subcategory of BA without
  $1$. Moreover,
  $Alg(L)=Alg(L\prime )$.

Presenting Functors on Varieties

Introduction

Given an adjunction

$L⊣R:\mathcal{C}\to \mathcal{X}$ such that the right-adjoint

$R$ is monadic (or of descent type) all objects

$A\in \mathcal{C}$ have a presentation "by generators and relations"

$$LRLRA\rightrightarrows LRA\to A.$$

In words,

$A$ is the quotient of the free algebra

$LRA$ over generators

$RA$ by equations in

$LRLRA$.

We apply this "monadic presentation" now to the situation where

$A$ is an endofunctor. In fact, we apply it twice and combine two representations:

Step 1: We represent a sifted colimit preserving endofunctor on a variety

$\mathcal{A}$ such that the "generators" and "relations" are given by sifted colimit preserving endofunctors on Set.

Step 2: We represent a sifted colimit on Set as the quotient of polynomial functor.

Preliminaries

Let

$S(\mathcal{C})$ be the category of sifted colimit preserving functors

$\mathcal{C}\to \mathcal{C}$.

Fact: The category

$S(\mathcal{C})$ is slfp if

$\mathcal{C}$ is.

A functor

$H:\mathcal{A}\to \mathcal{B}$ between slfp categories is called algebraically exact provided that it preserves limits and sifted colimits.

Fact: If

$H$ is algebraically exact it has a left adjoint.

Step 1

Let

$U:\mathcal{A}\to Set$ be a variety and

$F⊣U$. Define

$$\mathrm{\Psi }:S(\mathcal{A})\to S(Set)$$

via

$\mathrm{\Psi }L=ULF$ and

$$\mathrm{\Phi }:S(Set)\to S(\mathcal{A})$$

via

$\mathrm{\Phi }T=FTU$.

Fact:

$\mathrm{\Psi }$ is algebraically exact and

$\mathrm{\Phi }⊣\mathrm{\Psi }$.

After the first step, we obtain a quotient

$FULFU\to L$. Here,

$FULFU$ results from applying the free construction

$\mathrm{\Phi }=F-U$ to the Set-functor

$ULF$. What we need next is a presentation by operations and equations of

$ULF$.

Step 2

Every sifted colimit preserving on

$Set$ (=filtered colimit preserving functor=finitary functor) can be presented as the quotient of a polynomial functor.

Indeed, if

$T:Set\to Set$ is finitary then we have a natural transformation

$$\begin{array}{rl}\coprod _{n\in \mathbb{N}}Tn\times X^n& \stackrel{q_X}{⟶}TX\\ (\sigma ,v)& \mapsto T(v)(\sigma )\end{array}$$

with each

$q_X$ being surjective (even for infinite sets

$X$).

Note that

$v$ can be seen both as a tuple

$(x_1,\dots x_n)$ and as a function

$n\to X$. Hence we can evaluate the term

$\sigma (x_1,\dots x_n)$ by applying

$Tv:Tn\to TX$ to

$\sigma$.

This gives us a representation of

$T$ as a functor by operations and equations where the set of

$n$-ary operations is

$Tn$ and the set of equations in

$n$ variables is the kernel of

$q_n$.

The Presentation

Theorem: A functor on a variety has a presentation by operations and equations iff the functor preserves sifted colimits.

To prove "if", one shows that every sifted colimit preserving functor

$L$ on a variety is a quotient

$$FGUA\stackrel{q_A}{⟶}LA$$

where

$GX=\coprod _{n\in \mathbb{N}}ULFn\times X^n$.

• The
  $n$-ary modal operators
  $\sigma$ are the elements of
  $ULFn$.
• $q_A$ maps
  $(\sigma ,v:n\to UA)$ to
  $L(v^{†})(\sigma )$ where
  $v^{†}:Fn\to A$ is the adjoint transpose of
  $v$.^[1]
• The equations in
  $n$ variables are the kernel of
  $q_{Fn}$.

In particular the

$(\sigma ,v:n\to UA)$ are modal formulas

$$\mathrm{\Delta }(a_1,\dots a_n)$$

where

$\mathrm{\Delta }=\sigma$ and

$a_i=v(i)$.

The set

$\mathrm{\Sigma }$ of operations and the set

$E$ of equations constitute the presentation

$⟨{\mathrm{\Sigma }}_L,E_L⟩$ of

$L$.

Theorem: Let

$\mathcal{A}\cong Alg({\Sigma }_{\mathcal{A}},E_{\mathcal{A}})$ be a variety and

$⟨{\mathrm{\Sigma }}_L,E_L⟩$ a presentation of

$L:\mathcal{A}\to \mathcal{A}$. Then

$Alg({\Sigma }_{\mathcal{A}}+{\mathrm{\Sigma }}_L,E_{\mathcal{A}}+E_L)$ is isomorphic to Alg(L), where equations in

$E_{\mathcal{A}}$ and

$E_L$ are understood as equations over

${\mathrm{\Sigma }}_{\mathcal{A}}+{\mathrm{\Sigma }}_L$.

Example: The variety of modal algebras is presented by the operations and equations of Boolean algebra plus a unary operation

$◻$ and two equations specifying that

$◻$ preserves finite meets.

References

Kurz-Rosicky: Strongly Complete Logics for Coalgebras, 2012.

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1. Note that

   $(\sigma ,v:n\to UA)$ is a modal formula
   $\mathrm{\Delta }(a_1,\dots a_n)$ where
   $\mathrm{\Delta }=\sigma$ and
   $a_i=v(i)$. ↩︎