1
vote
0answers
47 views

When does a cogenerator determine a variety?

Two varieties of universal algebras are categorically equivalent iff their respective full subcategories of finitely generated free algebras are equivalent. Roughly speaking, this follows because they ...
2
votes
2answers
79 views

Locally finite varieties which are not finitely generated

Let $\Sigma$ be a signature consisting of operations with finite arity. Let $\mathcal{V}$ be a variety of algebras for this signature. Further suppose that $\mathcal{V}$ is locally finite i.e. every ...
3
votes
0answers
127 views

Equational theories determined by “identities without variables”

How to characterize equational theories $T$ which have the following property: for any two terms $t(x_1,...,x_n)$ and $t'(x_1,...,x_n)$ in the signature of $T$, if for any closed terms (i. e. terms ...
5
votes
1answer
143 views

What do algebraic theories with strictly terminal trivial models look like?

By algebraic theory I mean one in the sense of Lawvere, i.e. a collection of finitary operations, including projections, together with a multi-composition satisfying the obvious axioms. (I believe ...
11
votes
3answers
407 views

Varieties where every algebra is free

I'd like to know more about varieties (in the sense of universal algebra) where every algebra is free. Another way to state the condition is that the comparison functor from the Kleisli category to ...
10
votes
0answers
129 views

Does every commutative variety of algebras have a cogenerator?

By a commutative variety $\mathcal{V}$ I mean a classical variety of algebras for some $(\Sigma,E)$, such that each pair of operations in $\Sigma$ commutes. Equivalently (i) every interpretation of ...
3
votes
2answers
278 views

Are algebraic structures uniquely identifed by their free objects?

It might be a naive question, as I am not a specialist in this field. This is a follow-up to this question. I want to study varieties of objects generalizing ordered monoids, in particular using an ...
1
vote
1answer
147 views

Varieties generated by a two element algebra

I have two questions regarding universal algebra, and also its ordered version. If a variety $\mathcal{V}$ is generated by a specific two element algebra $2 = \{0,1\}$, then is that the only ...
3
votes
2answers
159 views

A variety of algebras satisfying some dual conditions

I would like prove that, under the conditions described below, no non-trivial variety exists. Let $\mathcal{V}$ be a variety of algebras e.g. rings, semigroups, semilattices. Further suppose ...
3
votes
2answers
149 views

Coequalizers in an Eilenberg-Moore category

Last month I proved that some category $\mathbf C$ that I happen to care about is isomorphic to the Eilenberg-Moore category for a monad on the category of bounded posets $\mathbf{BPos}$. I know from ...
3
votes
2answers
456 views

Shape of axioms in abstract algebra

When defining abstract algebraic structures (like monoids, groups, etc...), are there some constraints on the shape of the axioms, for the structure to have good properties that we implicitly use in ...
3
votes
1answer
192 views

Iterating Monad-Comonads structures

Let $(T, \mu , \eta )$ a monad on the category $\mathscr{C}$ , with the usual EM (Eilenberg-Moore) adjunction $\langle F_T, U_T, \eta_Y, \epsilon_T \rangle: \mathscr{C}^T \to \mathscr{C}$ where ...
5
votes
1answer
278 views

Is there a general theory of “representation theorems”?

Let $V$ and $W$ be classes of algebraic structures, and suppose we have some canonical way of constructing objects of $W$ from objects of $V$. Let's call this construction $C$, so that for all $A\in ...
10
votes
1answer
275 views

Which categories are the categories of models of a Lawvere theory?

Background: a Lawvere theory $T$ is a category with finite products such that each object is a power of a fixed object $x$. Given a Lawvere theory $T$, the category $\text{Mod}_T$ of models of $T$ is ...
3
votes
1answer
243 views

Classification of plethories over $\mathbb{Q}$

Let $k$ be a commutative ring. For every cocommutative bialgebra $A$ over $k$ the symmetric algebra of the underlying $k$-module $S(A)$ carries the structure of a $k$-plethory (Borger, Wieland, 2.5). ...
6
votes
1answer
367 views

Two questions about commutative theories

Let $\mathcal{T}$ be a commutative algebraic theory (for example sets, abelian groups, commutative monoids, but not groups etc.). References include the nlab and Borceux' Handbook of Categorical ...
3
votes
2answers
177 views

Are the categories of bialgebras and weak biaglebras cocomplete/algebraic?

I am trying to verify whether the category of bialgebras and then the category of weak bialgebras are cocomplete. We know that algebraic categories are cocomplete (Thm. 4.5 of this book), so I have ...
7
votes
0answers
248 views

When can I assure that the representation theory of a PROP is faithful?

Recall that a PROP is a symmetric monoidal category whose object-set is freely $\otimes$-generated by a single object $x\in P$. A representation or algebra for a prop $P$ in a symmetric monoidal ...
3
votes
1answer
472 views

Presenting Lawvere theories?

(Re)Reading Lawvere theories versus classical universal algebra, I was reminded of a question I have had for quite some time: What is the best way to present Lawvere theories? By present, I mean ...
12
votes
0answers
330 views

When can we tell if PROPs, Algebraic Theories, etc. are faithfully detected in a given category?

I am interested in understanding a certain phenomenon. I am hoping this sort of problem has been studied before, but I don't know the proper terminology and am having trouble finding answers. I am ...
12
votes
3answers
2k views

What is the theory of polynomials?

We all know what polynomials are, along with their elementary properties and many effective algorithms for different representations of polynomials. The question here is more of a universal algebra ...
3
votes
2answers
997 views

Why Lawvere theories have finite products? and more

According to Wikipedia, a Lawvere theory consists of a small category $L$ with (strictly associative) finite products and a strict identity-on-objects functor $I:\aleph_0^\text{op}\rightarrow L$ ...
21
votes
3answers
1k views

Lawvere theories versus classical universal algebra

A Lawvere theory is a small category with finite products such that every object is isomorphic to a finite product of copies of a distinguished object x. A model of the theory in a category with ...
13
votes
3answers
1k views

Relation between monads, operads and algebraic theories

I've begun to interest in algebraic theories and their categorical models: in particular monads, generalized multicategories and operads, lawvere theories and their generalization. Is there any ...
5
votes
3answers
810 views

Monad arising from operad

It's known that from every operad arises a cartesian monad whose algebras are the algebras for the operad. Leinster proved that there are different operads from which arise the same monad, in this way ...
6
votes
5answers
469 views

Computing colimits in a Lawvere theory

Let Prod(C, D) be the set of finite-product preserving functors from C to D. Is it true that for any Lawvere theory L, Prod(L, Set) has small colimits? This seems to be the case, as it is invoked ...
1
vote
4answers
744 views

Is monomorphism going in both directions sufficient for isomorphism?

In category theory, it seems that a monomorphism from $A$ to $B$ and one from $B$ to $A$ should be enough to guarantee isomorphy, but it doesn't seem to be so. (If I'm right then there's something ...
4
votes
0answers
259 views

The graph of algebraic theories

Fix a logic $L$ and consider the category $\mathbf{AlgTh}_L$ with theories of $L$ as objects and theory interpretations as morphisms. For nice enough logics, this category has pushouts (which we will ...
16
votes
6answers
996 views

Do non-associative objects have a natural notion of representation?

A magma is a set $M$ equipped with a binary operation $* : M \times M \to M$. In abstract algebra we typically begin by studying a special type of magma: groups. Groups satisfy certain additional ...
8
votes
2answers
602 views

Right actions of operads and monads

Given an operad $A$, there is an associated monad $M_A$ given by $M_A(X) = \coprod_n A(n) \otimes X^{\otimes n}$ such that being an $A$-algebra and being an algebra over the monad $M_A$ is the same ...
2
votes
1answer
448 views

Higher-order, multi-sorted, non purely equational version of universal algebra ?

I have been looking around, unsuccessfully, for generalizations of universal algebra based on higher-order logic (rather than first order) and where the relations are not purely equational. ...
1
vote
1answer
480 views

Coequalizer in the category of primitive recursive functions

What does a coequalizer in the category of primitive recursive functions look like? I know that in Set, a coequalizer is a minimum congruence but...what is it in particular in the category of ...