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14
votes
Accepted
Strict toposes as a finite limit theory
The original reference for the essential algebraicity of elementary toposes is Freyd's Aspects of topoi (in which the notion of essentially algebraic theory, which is equivalent to that of a finite li …
11
votes
Accepted
Reference request for Linton's theorems on equational theories
(1, 2, 3) Though Linton's An outline of functorial semantics does contain the essence of the results and proofs of the monad–theory correspondence (see in particular Theorems 8.1 and 9.1 – 9.3), it is …
9
votes
0
answers
181
views
Michel Thiébaud's thesis ("Self-Dual Structure-Semantics and Algebraic Categories")
I am looking for a copy of Michel Thiébaud's 1971 thesis Self-Dual Structure-Semantics and Algebraic Categories, which appears to be an early reference for the relationship between the Kleisli constru …
9
votes
1
answer
301
views
Internal logic of locally strongly finitely presentable categories
There is a duality between locally strongly finitely presentable (LSFP) categories and (Cauchy complete) cartesian categories, i.e. multisorted algebraic theories [1]. The internal logic of cartesian …
9
votes
Merging single-sorted and multi-sorted theories
The answer to your question really depends on just how closely you want the theory of "algebraic theories relative to $\mathcal M$" to mirror the theory of classical ($S$-sorted) algebraic theories.
P …
7
votes
0
answers
188
views
Were algebraic theories and abstract clones defined independently?
Algebraic theories (by which I mean the formalism based on bijective-on-objects functors) and abstract clones both capture universal algebraic structure, and are well-known to be equivalent. Algebraic …
7
votes
1
answer
551
views
Characterisation of essentially algebraic theories as monads
The following correspondence between algebraic theories and monads on $\mathbf{Set}$ is well-known (see, for example, Algebraic Theories: A Categorical Introduction to General Algebra).
The catego …
7
votes
1
answer
77
views
A syntactic characterisation of morphisms of algebraic theories whose induced algebraic func...
Let $f : S \to T$ be a morphism of algebraic theories. Such a morphism induces a monadic functor $f^* : \mathrm{Mod}(T) \to \mathrm{Mod}(S)$ (hence $f^*$ has a left adjoint). We may view $f$ syntactic …
6
votes
0
answers
120
views
Original reference for the correspondence between commutative algebraic theories and commuta...
Commutative algebraic theories were introduced by Linton in the 1966 paper Autonomous Equational Categories. Commutative monads were introduced by Kock in the 1970 paper Monads on symmetric monoidal c …
6
votes
0
answers
137
views
Characterisation of essentially algebraic theories with a fixed set of sorts
It is well known (e.g. Palmgren–Vickers's Partial Horn logic and cartesian categories) that many-sorted essentially algebraic theories (equivalently partial Horn theories / quasi-equational theories / …
5
votes
Accepted
Literature about the category of finitary monads
These claims are proven more generally for the category $\mathrm{Mnd}_f(\mathscr A)$ of finitary monads on a locally presentable category $\mathscr A$ in Lack's On the monadicity of finitary monads. ( …
4
votes
Pseudo-morphisms in essentially algebraic theories
This is one of the main themes of Arkor–Bourke–Ko's paper Enhanced 2-categorical structures, two-dimensional limit sketches and the symmetry of internalisation. In particular, in §5, the authors intro …
3
votes
Accepted
Characterisation of essentially algebraic theories as monads
I'm going to give a partial answer to my question, which addresses a misconception I had and illustrates why many of the existing generalisations of theory–monad correspondence are not sufficient to p …
2
votes
1
answer
112
views
Characterisation of presentations for varietal large equational theories
Let $T : \mathbf{Set}^\mathrm{op} \to \mathscr T$ be a large equational theory (i.e. a bijective-on-objects product-preserving functor). Following Linton in Some Aspects of Equational Categories, we c …