Questions tagged [algebraic-theory]
The algebraic-theory tag has no usage guidance.
18
questions
16
votes
2
answers
2k
views
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 ...
- 6,484
14
votes
3
answers
700
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 ...
- 61.5k
12
votes
3
answers
656
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 ...
- 61.5k
9
votes
1
answer
289
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 ...
- 8,755
9
votes
0
answers
176
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 ...
- 8,755
7
votes
1
answer
457
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 ...
- 8,755
7
votes
0
answers
185
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 ...
- 8,755
7
votes
0
answers
331
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 ...
- 11.7k
6
votes
0
answers
115
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 ...
- 8,755
6
votes
0
answers
136
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 / ...
- 8,755
5
votes
1
answer
501
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 ...
- 26.4k
5
votes
0
answers
101
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 $\DeclareMathOperator\dom{dom}\dom$, $\...
- 1,031
4
votes
1
answer
203
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 ...
- 6,484
4
votes
1
answer
544
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, \...
- 1,964
3
votes
1
answer
171
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 ...
- 295
3
votes
0
answers
56
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 $M$...
- 6,484
2
votes
1
answer
185
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 ...
- 6,484
2
votes
1
answer
107
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 ...
- 8,755