Questions tagged [algebraic-theory]

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16 votes
2 answers
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Why are operads sometimes better than algebraic theories?

Question 1: Are there any contexts in which replacing the category of (non-symmetric or symmetric) operads (in some monoidal category or symmetric monoidal category, respectively) with the category of ...
Arshak Aivazian's user avatar
14 votes
3 answers
683 views

Reference request for Linton's theorems on equational theories

This is a reference request for the following "well-known" theorems in category theory: There is an equivalence of categories between finitary monads on $\mathbf{Set}$ and finitary Lawvere ...
Martin Brandenburg's user avatar
11 votes
3 answers
652 views

IBN for algebraic theories

Let us say that a finitary algebraic theory $\tau$ has IBN (invariant basis number) if the free functor $F : \mathsf{Set} \to \mathsf{Mod}(\tau)$ reflects the isomorphism relation: If $S,T$ are sets ...
Martin Brandenburg's user avatar
9 votes
0 answers
168 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 ...
varkor's user avatar
  • 7,889
8 votes
1 answer
425 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 ...
varkor's user avatar
  • 7,889
8 votes
1 answer
234 views

Internal logic of locally strongly finitely presentable categories

There is a duality between locally strongly finitely presentable categories and (Cauchy complete) cartesian categories, i.e. multisorted algebraic theories. The internal logic of cartesian categories ...
varkor's user avatar
  • 7,889
7 votes
0 answers
181 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 ...
varkor's user avatar
  • 7,889
7 votes
0 answers
329 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 ...
Jacques Carette's user avatar
6 votes
0 answers
110 views

Original reference for the correspondence between commutative algebraic theories and commutative monads

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 ...
varkor's user avatar
  • 7,889
6 votes
0 answers
133 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 / ...
varkor's user avatar
  • 7,889
5 votes
1 answer
495 views

Request for reference: Banach-type spaces as algebraic theories.

Sparked by Yemon Choi's answer to Is the category of Banach spaces with contractions an algebraic theory? I've just spent a merry time reading and doing a bit of reference chasing. Imagine my delight ...
Andrew Stacey's user avatar
5 votes
0 answers
95 views

Pseudo-morphisms in essentially algebraic theories

Categories with terminal objects can be written as an essentially algebraic theory or a generalized algebraic theory: There is one sort $M$ with unary operations $\DeclareMathOperator\dom{dom}\dom$, $\...
Trebor's user avatar
  • 779
4 votes
1 answer
198 views

What should be required from a model category so that the category of algebraic objects in it has the natural model structure?

I have two reference questions What should be required of a category with finite products so that a (multi-sorted, finitary) Lawvere theory induces a monadic adjunction on it? This should be ...
Arshak Aivazian's user avatar
4 votes
1 answer
515 views

Question about an implication of Thomason's étale descent spectral sequence

On page 5 of this paper by Dwyer and Mitchell, it is said that Thomason's étale descent spectral sequence from his paper Algebraic K-theory and étale cohomology, which reads $$H^p_{\acute{e}t}(X, \...
xir's user avatar
  • 1,942
3 votes
1 answer
161 views

Commuting filtered colimits & finite limits in infinitary theories

Filtered colimits & finite limits commute in categories that are finitary monadic over Set (i.e. algebras of finitary algebraic theories). Results such as Fred Linton's result that if categories ...
Oddly Asymmetric's user avatar
3 votes
1 answer
95 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 ...
varkor's user avatar
  • 7,889
3 votes
0 answers
55 views

Right transferred model structure on the category of algebras in the Grothendieck topos

Let M be a model category that is a Grothendieck topos. Let $T$ be a multisorted finitary algebraic theory. Does there exist the right transferred model structure on the category of $T$-algebras in $M$...
Arshak Aivazian's user avatar
2 votes
1 answer
175 views

Literature about the category of finitary monads

This answer states that the category of finitary monads is locally presentable and monadic over the category $\mathrm{Set}^{\mathbb{N}}$. Where can I find proof of this claim? More generally: I've ...
Arshak Aivazian's user avatar