nLab
colimits in categories of algebras

Context

Limits and colimits

Higher algebra

Introduction
Reflexive coequalizers and cocompleteness
Relatively free functors
References

Introduction

Let $T$ be a monad on a category $C$ , and let $C^{T}$ denote the Eilenberg-Moore category of $T$ . Let

U : C^{T} \to C

be the usual underlying or forgetful functor, with left adjoint $F : C \to C^{T}$ , unit $η : 1_{C} \to U F$ , and counit $ε : F U \to 1_{C^{T}}$ . It is well-known that $U$ reflects limits, so that if $C$ is complete, then $C^{T}$ is also complete and $U$ is continuous.

The situation with regard to colimits is more complicated. It is not generally true that if $C$ is cocomplete, then $C^{T}$ is also. In this article we collect some partial results which address these issues.

Reflexive coequalizers and cocompleteness

A simple but basic fact is the following. Suppose $J$ is a small category, and suppose that the monad $T$ preserves colimits over $J$ , that is, suppose that for every $F : J \to C$ the canonical map

{colim}_{J} T \circ F \to T ({colim}_{J} F)

is an isomorphism.

Proposition

Under these hypotheses, $U : C^{T} \to C$ reflects colimits over $J$ .

Here are some sample applications of this proposition which arise frequently in practice. Let $J$ be the generic reflexive fork, having exactly two objects $0, 1$ , generated by three non-identity arrows

0 \to 1 \vec{\to} 0,

and subject to the condition that the two composites from $0$ to $0$ are the identity. A colimit over $J$ is called a reflexive coequalizer. It frequently happens that a monad $T : C \to C$ preserves reflexive coequalizers; in this case, if $C$ has reflexive coequalizers, then so does $C^{T}$ .

Theorem

If $C$ is cocomplete and $C^{T}$ has reflexive coequalizers, then $C^{T}$ is cocomplete.

Proof

First observe that if $(c, ξ : T c \to c)$ is a $T$ -algebra, then $ξ$ is the coequalizer of the reflexive fork

F U c \overset{F η U c}{\to} F U F U c \overset{\overset{ε F U c}{\to}}{\underset{F U ε c}{\to}} F U c

To show $C^{T}$ has coproducts, let $(c_{i}, ξ_{i})$ be a collection of algebras. Then $F (\sum_{i} U c_{i})$ is the coproduct $\sum_{i} F U c_{i}$ in $C^{T}$ (since $F$ preserves coproducts and $C$ has them). We have a reflexive fork

\sum_{i} F U c_{i} \overset{\sum_{i} F η U c_{i}}{\to} \sum_{i} F U F U c_{i} \overset{\overset{\sum_{i} ε F U c_{i}}{\to}}{\underset{\sum_{i} F U ε c}{\to}} \sum_{i} F U c_{i}

and it is not difficult to show that the coequalizer in $C^{T}$ of this diagram is the coproduct $\sum_{i} c_{i}$ .

Finally, general coequalizers in $C^{T}$ are constructed from coproducts and reflexive coequalizers: given a parallel pair $f, g : c \vec{\to} d$ in $C^{T}$ , the coequalizer of $f$ and $g$ is the colimit of the reflexive fork

d \to c + d \overset{\overset{(f, 1_{d})}{\to}}{\underset{(g, 1_{d})}{\to}} d

where the first arrow is the coproduct coprojection.

Corollary

If $T$ is a monad on $Set$ , then ${Set}^{T}$ is cocomplete.

Proof

It is enough to show that ${Set}^{T}$ has coequalizers. Suppose given a pair of algebra maps $f, g : A \vec{\to} B$ whose coequalizer we wish to construct. Let $R$ be the $T$ -algebra relation

R = ⟨ f, g ⟩ : A \to B \times B

and then let $E$ be the smallest $T$ -congruence (equivalence relation that is a $T$ -subalgebra map $E ↪ B \times B$ ) through which $R$ factors. (This is the intersection of all $T$ -congruences through which $R$ factors, and may be calculated in $Set$ , where it is reflected in $T$ - $Alg$ .) The coequalizer as calculated in $Set$ ,

E \overset{\overset{π_{1}}{\to}}{\underset{π_{2}}{\to}} B \overset{p}{\to} Q

is a split coequalizer, because every quotient of an equivalence relation in $Set$ is a split coequalizer. (This requires the axiom of choice. A splitting is given by any splitting $i : Q \to B$ of $p$ , which picks a representative in each equivalence class, together with $⟨ i p, 1 ⟩ : B \to E$ .) It is therefore an absolute colimit, which the monad $T$ preserves. Hence the top row in

\begin{matrix} T E & \overset{\overset{T π_{1}}{\to}}{\underset{T π_{2}}{\to}} & T B & \overset{T p}{\to} & T Q \\ ↓ & ↓ & ↓ \\ E & \overset{\overset{π_{1}}{\to}}{\underset{π_{2}}{\to}} & B & \to & Q \end{matrix}

(the first two vertical arrows being algebra structure maps) is a coequalizer in ${Set}^{T}$ . The last vertical arrow making the diagram commute gives $Q$ a $T$ -algebra structure, and the split coequalizer in the bottom row is reflected in ${Set}^{T}$ .

Corollary

If $T$ is a monad on a complete and cocomplete category $C$ that preserves reflexive coequalizers, then $C^{T}$ is complete and cocomplete.

The hypotheses of the preceding corollary hold when $T$ is a monad on a complete, cocomplete, cartesian closed category that is induced from a finitary algebraic theory. (The key observation being that the finitary power functors $x \mapsto x^{n}$ preserve reflexive coequalizers.)

Here is a more difficult result given in Toposes, Theories, and Triples (theorem 3.9, p. 267):

Proposition

If $C$ has coequalizers and equalizers of arbitrary sets of parallel morphisms, and if a monad $T : C \to C$ preserves colimits of countable chains $ω \to C$ , then $C^{T}$ has coequalizers.

Corollary

If $C$ is complete and cocomplete and $T : C \to C$ preserves filtered colimits, then $C^{T}$ is complete and cocomplete.

Relatively free functors

Theorem

Suppose that $θ : S \to T$ is a morphism of monads on $C$ , and suppose that $C^{T}$ has coequalizers. Then the forgetful functor

C^{θ} : C^{T} \to C^{S}

has a left adjoint.

Proof

Since the following diagram is commutative:

\begin{array}{cccc} C^{T} & \overset{C^{θ}}{\to} & C^{S} \\ ^{U^{T}} ↓ & ↓^{U^{S}} \\ C & = & C \end{array}

(using an obvious notation), it follows immediately from a corollary to the adjoint lifting theorem that if $C^{T}$ has coequalizers of reflexive pairs, then $C^{θ}$ has a left adjoint and is, in fact, monadic.

This completes the proof, but it is interesting to observe that there is a concrete description of the left adjoint in this case: The left adjoint sends an $S$ -algebra $(c, ξ : S c \to c)$ to the (reflexive) coequalizer of the pair

T S c \overset{T ξ}{\to} T c T S c \overset{T θ c}{\to} T T c \overset{μ c}{\to} T c

where $μ : T T \to T$ is the monad multiplication. (If $u : 1_{C} \to S$ is the unit of $S$ , then $T u c : T c \to T S c$ is a common right inverse of both arrows of the pair.) This coequalizer is analogous to the construction of the left adjoint $B \otimes_{A} -$ to the functor ${Ab}^{f} : {Ab}^{B} \to {Ab}^{A}$ between module categories given by restricting scalar multiplication; the coequalizer will be denoted $T \circ_{S} c$ to underline the analogy.

To see that $T \circ_{S} -$ is the left adjoint, let $(d, α : T d \to d)$ be a $T$ -algebra. Any map $f : c \to d$ in $C$ induces a unique $T$ -algebra map $ϕ : T c \to d$ :

ϕ = (T c \overset{T f}{\to} T d \overset{α}{\to} d)

and the claim is that $f : c \to d$ is an $S$ -algebra map $c \to C^{θ} (d)$ if and only if $ϕ$ coequalizes the pair of above, i.e., if $ϕ$ factors (uniquely) through a $T$ -algebra map $T \circ_{S} c \to d$ .

Indeed, the condition that $f$ is an $S$ -algebra map is satisfaction of the equation

(S c \overset{ξ}{\to} c \overset{f}{\to} d) = (S c \overset{S f}{\to} S d \overset{θ d}{\to} T d \overset{α}{\to} d)

and now we have a long train of equations

\begin{matrix} α \circ T f \circ μ c \circ T θ c & = & α \circ μ d \circ T T f \circ T θ c \\ = & α \circ μ d \circ T θ d \circ T S f \\ = & α \circ T α \circ T θ d \circ T S f \\ = & α \circ T (α \circ θ d \circ S f) \\ = & α \circ T (f \circ ξ) \\ = & α \circ T f \circ T ξ \end{matrix}

which gives $ϕ \circ μ c \circ T θ c = ϕ \circ T ξ$ . This completes one direction. In the other direction, suppose $ϕ \circ μ c \circ T θ c = ϕ \circ T ξ$ . Then

\begin{matrix} α \circ θ d \circ S f & = & α \circ T f \circ θ c \\ = & ϕ \circ θ c \\ = & ϕ \circ μ c \circ η T c \circ θ c \\ = & ϕ \circ μ c \circ T θ c \circ η S c \\ = & ϕ \circ T ξ \circ η S c \\ = & ϕ \circ η c \circ ξ \\ = & α \circ T f \circ η c \circ ξ \\ = & α \circ η d \circ f \circ ξ \\ = & f \circ ξ \end{matrix}

as desired. This completes the proof.

References

Michael Barr and Charles Wells, Toposes, Theories, and Triples, Reprints in Theory and Applications of Categories (2005), 1-289. (online pdf)

Revised on April 8, 2013 19:57:49 by Todd Trimble (67.81.93.26)

nLab
colimits in categories of algebras

Context

Limits and colimits

1-Categorical

2-Categorical

(∞,1)-Categorical

Model-categorical

Higher algebra

Algebraic theories

Algebras and modules

Higher algebras

Model category presentations

Geometry on formal duals of algebras

Theorems

Contents

Introduction

Reflexive coequalizers and cocompleteness

Proposition

Theorem

Proof

Corollary

Proof

Corollary

Proposition

Corollary

Relatively free functors

Theorem

Proof

References

nLab colimits in categories of algebras

Context

Limits and colimits

1-Categorical

2-Categorical

(∞,1)-Categorical

Model-categorical

Higher algebra

Algebraic theories

Algebras and modules

Higher algebras

Model category presentations

Geometry on formal duals of algebras

Theorems

Contents

Introduction

Reflexive coequalizers and cocompleteness

Proposition

Theorem

Proof

Corollary

Proof

Corollary

Proposition

Corollary

Relatively free functors

Theorem

Proof

References

nLab
colimits in categories of algebras