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 ...

10
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 ...

8
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 ...

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

### 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, ...

Community wiki

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 $\...

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 ...

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.

Community wiki

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 ...

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 ...

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 ...

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