22
votes
Accepted
Why are operads sometimes better than algebraic theories?
First - yes, for symmetric set-operads this functor is "injective", though it is not fully faithful. It is faithful on general maps and fully faithful on isomorphisms. Its image can easily ...
- 42.4k
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 ...
- 10.6k
14
votes
Accepted
Is it possible that the number of $\mathcal{T}$-algebras is an arithmetic progression?
Is it possible that the number of $\mathcal{T}$-algebras is an arithmetic progression?
Here is a near miss: Let $\mathcal{V}$ be the variety of $\mathbb Z_2$-sets. These may be thought of as algebras $...
- 14.6k
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 ...
- 10.6k
9
votes
Reference request for Linton's theorems on equational theories
1 and 3 are proved in Appendix A of the book “Algebraic theories”
by Jiří Adámek, Jiří Rosický, Enrico M. Vitale.
1 is Theorem A.37 (and A.41 for the multisorted version).
3 is Theorem A.21 (and A.40 ...
- 37.8k
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 ...
- 10.6k
5
votes
Merging single-sorted and multi-sorted theories
I was about to write a long answer, but varkor stole the scene. I think his answer is pretty good and I'd like to complement it with a bunch of references that I find relevant.
The general mood of the ...
- 9,111
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. (...
- 10.6k
4
votes
Accepted
A syntactic characterisation of morphisms of algebraic theories whose induced algebraic functors admit right adjoints
I thought about this a bit when trying to understand $\lambda$-rings – the same example that Wraith offers – and I think the answer is basically what is remarked in the cited lecture notes: it is ...
- 15.9k
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 ...
- 10.6k
4
votes
Internal logic of locally strongly finitely presentable categories
The comments are getting a bit long (sorry, that is largely my fault), so I think it's worth expanding on Jiří Rosický 's point, which comes very close to completely answering the question.
Adamek, ...
3
votes
Characterisation of presentations for varietal large equational theories
The following are equivalent:
$\mathcal T$ is varietal
Each free algebra $T \in \mathcal T$ is small
For each arity, there are a small number of $E$-equivalence classes of words in the language $\...
- 63.9k
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 ...
- 10.6k
3
votes
Reference request for Linton's theorems on equational theories
A detailed, self-contained and beginner-friendly proof of both theorems can now be found in my new paper on limit sketches in the appendix.
3
votes
Merging single-sorted and multi-sorted theories
A more syntactic kind of presentation is available if the generality of the categories $\mathcal{M}$ is more restricted. Specifically, Leena Subramaniam's thesis From dependent type theory to higher ...
- 66.8k
3
votes
Accepted
What should be required from a model category so that the category of algebraic objects in it has the natural model structure?
The first question is already answered in the comments, so let me focus on the second question. Note first that every (colored) Lawvere theory is in particular a (colored) PROP. This is easy to see ...
- 30.3k
2
votes
Commuting filtered colimits & finite limits in infinitary theories
The category $sSet$ of simplicial sets fits the bill.
As $sSet$ is a topos, it is Barr-exact. Moreover, the coproduct of representables $D = \amalg_{n \in \mathbb N} \Delta^n$ is a projective ...
- 63.9k
2
votes
Why are operads sometimes better than algebraic theories?
Yes, there are many contexts where operads are better than Lawvere theories. For one thing, algebras over a Lawvere theory are defined in the category Set. If you want to study algebras in a different ...
- 30.3k
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