Fix an algebraic theory; denote its free models by $T^k$. There are two possible definitions of what it means for a coequalizer $T^m\twoheadrightarrow M$ to be a finite presentation of $M$. $f\colon T^m\twoheadrightarrow M$ is the coequalizer of some pair of morphisms $T^k\rightrightarrows T^m\twoheadrightarrow M$. This is the standard definition in category theory textbooks. If […]

Let $\mathbf{A}$ be an algebra (in the sense of universal algebra) of some signature $\Sigma$. By quasi-identity I mean the formula of the form $$(\forall x_1) (\forall x_2) \dots (\forall x_n) \left(\left[\bigwedge_{i=1}^{k}t_i(x_1, \dots, x_n)=s_i(x_1, \dots, x_n)\right]\rightarrow t(x_1, \dots, x_n)=s(x_1, \dots, x_n) \right) \;, $$ where $t_i(x_1, \dots, x_n), s_i(x_1, \dots, x_n), t(x_1, \dots, x_n), s(x_1, […]

There is a consensus, that “isomorphy” (the “is isomorphic to”-relation) is the right kind of sameness between universal algebras (say groups, (single-sorted) vector spaces, lattices, …), because isomorphic objects share all their “algebraic properties”. I wonder though: Is there a (meta)theorem, that tells us exactly which properties, I can possibly come up with in the […]

Let $T$ denote a Lawvere theory and $X$ denote a $T$-algebra. Under my preferred definitions: A subalgebra of $X$ consists of a $T$-algebra $Y$ together with an injective homomorphism $Y \rightarrow X$. A quotient of $X$ consists of a $T$-algebra $Y$ together with a surjective homomorphism $X \rightarrow Y$. Also: A congruence on $X$ is […]

By “(possibly many-sorted) algebraic category”, I mean “category of models of a (possibly many sorted) Lawvere theory. I have arguments for both affirmative and negative answers, and I can’t find a hole in either of them. Let $\mathsf{TF}$ denote the category of torsionfree abelian groups. For background, though, note that $\mathsf{TF}$ is locally finitely presentable: […]

A binomial ring is a ring (for the purposes of this question all rings are commutative and unital) which is torsion-free and has, for each $n$, a binomial function $\binom{x}{n}$ satisfying $n!\binom{x}{n} = x(x-1) \cdots (x-n+1)$. The category $\mathsf{BRing}$ of binomial rings is a full subcategory of the category $\mathsf{Ring}$ of rings. This paper shows […]

In a slice category $C/A$ of a category $C$ over a given object $A$, what is the role of the $C$ identity morphism, $A\to A$ ($1_A$), in $C/A$, particularly with respect to composition? I understand that as an arrow targeting $A$, it is an object of $C/A$. However, in $C$, $1_A$ is not practically composable, […]

I am looking for some mathematical software that can help me with a very common task in the realm of universal algebra (as far as I know programs like prover9/mace4 and uacalc do not help with this issue). The input of this software should be two different finite algebras, each of them given through the […]

I think a useful combination of resources for universal algebra would ideally, when taken together: Provide ample motivation behind the various developments in the field. Either provide powerful intuitions, or avoid them altogether, in an effort to not cripple the learner’s imagination. Provide a clear formal development (or, minimizes the number of “rabbits pulled out […]

From what I understand, the singular homology groups of a topological space are defined like so: Topological Particulars. There’s a covariant functor $F : \mathbb{\Delta} \rightarrow \mathbf{Top}$ that assigns to each natural number $n$ the corresponding $n$-simplex. This yields a functor $$\mathbf{Top}(F-,-) : \Delta^{op} \times \mathbf{Top} \rightarrow \mathbf{Set}.$$ Hence to each topological space $X$, we […]

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Little Bézout theorem for smooth functions
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