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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 their corresponding monads leads to undesirable behavior? (I know that for non-symmetric operads this functor is non-injective and so at least in one of the versions the question makes sense. Is it injective for symmetric operads?)

Question 2: Let $C$ be a monoidal category with projections and diagonal mappings that satisfies reasonably good conditions (a perfect example: an elementary topos with a Cartesian monoidal structure, but I'm targeting a much wider class). Are there contexts in which replacing algebraic theories defined by (symmetric/nonsymmetric) operads in this monoidal category with all algebraic theories in it leads to undesirable behavior? The motive of this question in general is: whenever the richness of the monoidal structure makes it possible to define a wider class of theories - are there reasons not to go to it? Thus, the transition from non-symmetric to symmetric operads is embedded in this motif when it comes to a symmetric monoidal category.

I read several sources for three months (and in parallel dealt with operads a little in one course at the university) and I have not yet developed a coherent vision of why people in all contexts do not write "algebraic monad"(in the sense of the monad of the algebraic theory of expressible in the considered monoidal category) instead of "operad", when they can.

P.S. I understand that there is not always a clear answer to this kind of question, but sometimes it is known and therefore its existence is part of the question.

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 their corresponding monads leads to undesirable behavior? (I know that for non-symmetric operads this functor is non-injective and so at least in one of the versions the question makes sense. Is it injective for symmetric operads?)

Question 2: Let $C$ be a monoidal category with projections and diagonal mappings that satisfies reasonably good conditions (a perfect example: an elementary topos with a Cartesian monoidal structure, but I'm targeting a much wider class). Are there contexts in which replacing algebraic theories defined by (symmetric/nonsymmetric) operads in this monoidal category with all algebraic theories in it leads to undesirable behavior? The motive of this question in general is: whenever the richness of the monoidal structure makes it possible to define a wider class of theories - are there reasons not to go to it? Thus, the transition from non-symmetric to symmetric operads is embedded in this motif when it comes to a symmetric monoidal category.

I read several sources for three months (and in parallel dealt with operads a little in one course at the university) and I have not yet developed a coherent vision of why people in all contexts do not write "algebraic monad"(in the sense of the monad of the algebraic theory that expressible in the considered monoidal category) instead of "operad", when they can.

P.S. I understand that there is not always a clear answer to this kind of question, but sometimes it is known and therefore its existence is part of the question.

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 their corresponding monads leads to undesirable behavior? (I know that for non-symmetric operads this functor is non-injective and so at least in one of the versions the question makes sense. Is it injective for symmetric operads?)

Question 2: Let $C$ be a monoidal category with projections and diagonal mappings that satisfies reasonably good conditions (a perfect example: an elementary topos with a Cartesian monoidal structure, but I'm targeting a much wider class). Are there contexts in which replacing algebraic theories defined by (symmetric/nonsymmetric) operads in this monoidal category with all algebraic theories in it leads to undesirable behavior? The motive of this question in general is: whenever the richness of the monoidal structure makes it possible to define a wider class of theories - are there reasons not to go to it? Thus, the transition from non-symmetric to symmetric operads is embedded in this motif when it comes to a symmetric monoidal category.

I read several sources for three months (and in parallel dealt with operads a little in one course at the university) and I have not yet developed a coherent vision of why people in all contexts do not write "Lawvere theory" instead of "operad", when they can.

P.S. I understand that there is not always a clear answer to this kind of question, but sometimes it is known and therefore its existence is part of the question.

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 their corresponding monads leads to undesirable behavior? (I know that for non-symmetric operads this functor is non-injective and so at least in one of the versions the question makes sense. Is it injective for symmetric operads?)

Question 2: Let $C$ be a monoidal category with projections and diagonal mappings that satisfies reasonably good conditions (a perfect example: an elementary topos with a Cartesian monoidal structure, but I'm targeting a much wider class). Are there contexts in which replacing algebraic theories defined by (symmetric/nonsymmetric) operads in this monoidal category with all algebraic theories in it leads to undesirable behavior? The motive of this question in general is: whenever the richness of the monoidal structure makes it possible to define a wider class of theories - are there reasons not to go to it? Thus, the transition from non-symmetric to symmetric operads is embedded in this motif when it comes to a symmetric monoidal category.

I read several sources for three months (and in parallel dealt with operads a little in one course at the university) and I have not yet developed a coherent vision of why people in all contexts do not write "algebraic monad"(in the sense of the monad of the algebraic theory of expressible in the considered monoidal category) instead of "operad", when they can.

P.S. I understand that there is not always a clear answer to this kind of question, but sometimes it is known and therefore its existence is part of the question.

Question 1: Are there any contexts in which replacing the category of (non-symmetric or symmetric) operads with the category of their corresponding monads leads to undesirable behavior? (I know that for non-symmetric operads this functor is non-injective and so at least in one of the versions the question makes sense. Is it injective for symmetric operads?)

Question 2: Let $C$ be a monoidal category with projections and diagonal mappings that satisfies reasonably good conditions (a perfect example: an elementary topos with a Cartesian monoidal structure, but I'm targeting a much wider class). Are there contexts in which replacing algebraic theories defined by (symmetric/nonsymmetric) operads in this monoidal category with all algebraic theories in it leads to undesirable behavior? The motive of this question in general is: whenever the richness of the monoidal structure makes it possible to define a wider class of theories - are there reasons not to go to it? Thus, the transition from non-symmetric to symmetric operads is embedded in this motif when it comes to a symmetric monoidal category.

I read several sources for three months (and in parallel dealt with operads a little in one course at the university) and I have not yet developed a coherent vision of why people in all contexts do not write "Lawvere theory" instead of "operad", when they can.

P.S. I understand that there is not always a clear answer to this kind of question, but sometimes it is known and therefore its existence is part of the question.

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 their corresponding monads leads to undesirable behavior? (I know that for non-symmetric operads this functor is non-injective and so at least in one of the versions the question makes sense. Is it injective for symmetric operads?)

Question 2: Let $C$ be a monoidal category with projections and diagonal mappings that satisfies reasonably good conditions (a perfect example: an elementary topos with a Cartesian monoidal structure, but I'm targeting a much wider class). Are there contexts in which replacing algebraic theories defined by (symmetric/nonsymmetric) operads in this monoidal category with all algebraic theories in it leads to undesirable behavior? The motive of this question in general is: whenever the richness of the monoidal structure makes it possible to define a wider class of theories - are there reasons not to go to it? Thus, the transition from non-symmetric to symmetric operads is embedded in this motif when it comes to a symmetric monoidal category.

I read several sources for three months (and in parallel dealt with operads a little in one course at the university) and I have not yet developed a coherent vision of why people in all contexts do not write "Lawvere theory" instead of "operad", when they can.

P.S. I understand that there is not always a clear answer to this kind of question, but sometimes it is known and therefore its existence is part of the question.