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Let $j: \mathsf A \to \mathsf B$ be a fully faithful and dense functor where $\mathsf A$ is a small category and $\mathsf B$ is cocomplete. Let $(T, \eta, \mu)$ be a monad over $\mathsf A$. $\require{AMScd}$ \begin{CD} \mathsf A @>T>> \mathsf A\\ @V j V V @VV j V\\ \mathsf B @>>\text{lan}_j(jT)> \mathsf B \end{CD}

Is it possible to equip $\text{lan}_j(jT)$ with a monad structure?

Prop. 1 When $\text{lan}_j(jT)$ preserve itself, the answer is yes. Moreover, if the monad $T$ is idempotent, so is $\text{lan}_j(jT)$.

Sketch of Proof. The map $\text{lan}_j(j \circ \_): \text{End}(\mathsf A) \to \text{End}(\mathsf B)$ is strong monoidal on the full subcategory spanned by the monad $T$ and $1_{\mathsf A}$. Indeed $\text{lan}_j(jT^2) \cong \text{lan}_j(jT)^2,$ because $$\text{lan}_j(jT^2) \cong \text{lan}_j(\text{lan}_j(jT)jT) \cong \text{lan}_j(jT) \circ \text{lan}_j(jT) = \text{lan}_j(jT)^2. $$

(Observe that part of the functoriality is due to the fact that $j$ is dense). In particular, it maps monoid object to monoid objects.

Cor. 2 When $j$ is the Yoneda embedding, $\text{lan}_j(jT)$ admits a monad structure.

Proof. By the universal property of the presheaf construction, $\text{lan}_j(jT)$ is cocontinous, thus preserve every (pointwise) Kan extension, in particular, itself.

Cor. 3 When $\text{lan}_j(jT)$ is absolute, $\text{lan}_j(jT)$ admits a monad structure.

1. Is the argument in Prop. 1 correct?
2. How natural is the hypothesis that $\text{lan}_j(jT)$ preserve itself?
3. Are there natural or known assumption under which it is possible to equip $\text{lan}_j(jT)$ with a monad structure?

Let $j: \mathsf A \to \mathsf B$ be a fully faithful and dense functor where $\mathsf A$ is a small category and $\mathsf B$ is cocomplete. Let $(T, \eta, \mu)$ be a monad over $\mathsf A$. $\require{AMScd}$ \begin{CD} \mathsf A @>T>> \mathsf A\\ @V j V V @VV j V\\ \mathsf B @>>\text{lan}_j(jT)> \mathsf B \end{CD}

Is it possible to equip $\text{lan}_j(jT)$ with a monad structure?

Prop. 1 When $\text{lan}_j(jT)$ preserve itself, the answer is yes. Moreover, if the monad $T$ is idempotent, so is $\text{lan}_j(jT)$.

Sketch of Proof. The map $\text{lan}_j(j \circ \_): \text{End}(\mathsf A) \to \text{End}(\mathsf B)$ is strong monoidal on the full subcategory spanned by the monad $T$ and $1_{\mathsf A}$. Indeed $\text{lan}_j(jT^2) \cong \text{lan}_j(jT)^2,$ because $$\text{lan}_j(jT^2) \cong \text{lan}_j(\text{lan}_j(jT)jT) \cong \text{lan}_j(jT) \circ \text{lan}_j(jT) = \text{lan}_j(jT)^2. $$

(Observe that part of the functoriality is due to the fact that $j$ is dense). In particular, it maps monoid object to monoid objects.

Cor. 2 When $j$ is the Yoneda embedding, $\text{lan}_j(jT)$ admits a monad structure.

Proof. By the universal property of the presheaf construction, $\text{lan}_j(jT)$ is cocontinous, thus preserve every (pointwise) Kan extension, in particular, itself.

Cor. 3 Given a locally $\lambda$-presentable category $\mathsf{A}$ and a $\lambda$-colimit preserving monad $T: \mathsf{A}_{\lambda} \to \mathsf{A}_{\lambda}$ (the full subcategory of $\lambda$-presentable objects), then there exists an extension $\mathsf{T}: \mathsf{A} \to \mathsf{A}$ admitting a monad structure. Moreover, $\mathsf{T}$ is cocontinuous.

Proof. Similar to Cor. 2.

Cor. 4 When $\text{lan}_j(jT)$ is absolute, $\text{lan}_j(jT)$ admits a monad structure.

1. Is the argument in Prop. 1 correct?
2. How natural is the hypothesis that $\text{lan}_j(jT)$ preserve itself?
3. Are there natural or known assumption under which it is possible to equip $\text{lan}_j(jT)$ with a monad structure?

Let $j: \mathsf A \to \mathsf B$ be a fully faithful and dense functor where $\mathsf A$ is a small category and $\mathsf B$ is cocomplete. Let $(T, \eta, \mu)$ be a monad over $\mathsf A$. $\require{AMScd}$ \begin{CD} \mathsf A @>T>> \mathsf A\\ @V j V V @VV j V\\ \mathsf B @>>\text{lan}_j(jT)> \mathsf B \end{CD}

Is it possible to equip $\text{lan}_j(jT)$ with a monad structure?

Prop. 1 When $\text{lan}_j(jT)$ preserve itself, the answer is yes. Moreover, if the monad $T$ is idempotent, so is $\text{lan}_j(jT)$.

Sketch of Proof. The map $\text{lan}_j(j \circ \_): \text{End}(\mathsf A) \to \text{End}(\mathsf B)$ is strong monoidal on the full subcategory spanned by the monad $T$ and $1_{\mathsf A}$. Indeed $\text{lan}_j(jT^2) \cong \text{lan}_j(jT)^2,$ because $$\text{lan}_j(jT^2) \cong \text{lan}_j(\text{lan}_j(jT)jT) \cong \text{lan}_j(jT) \circ \text{lan}_j(jT) = \text{lan}_j(jT)^2. $$

(Observe that part of the functoriality is due to the fact that $j$ is dense). In particular, it maps monoid object to monoid objects.

Cor. 2 When $j$ is the Yoneda embedding, $\text{lan}_j(jT)$ admits a monad structure.

Proof. By the universal property of the presheaf construction, $\text{lan}_j(jT)$ is cocontinous, thus preserve every Kan extension, in particular, itself.

1. Is the argument in Prop. 1 correct?
2. How natural is the hypothesis that $\text{lan}_j(jT)$ preserve itself?
3. Are there natural or known assumption under which it is possible to equip $\text{lan}_j(jT)$ with a monad structure?

Let $j: \mathsf A \to \mathsf B$ be a fully faithful and dense functor where $\mathsf A$ is a small category and $\mathsf B$ is cocomplete. Let $(T, \eta, \mu)$ be a monad over $\mathsf A$. $\require{AMScd}$ \begin{CD} \mathsf A @>T>> \mathsf A\\ @V j V V @VV j V\\ \mathsf B @>>\text{lan}_j(jT)> \mathsf B \end{CD}

Is it possible to equip $\text{lan}_j(jT)$ with a monad structure?

Prop. 1 When $\text{lan}_j(jT)$ preserve itself, the answer is yes. Moreover, if the monad $T$ is idempotent, so is $\text{lan}_j(jT)$.

Sketch of Proof. The map $\text{lan}_j(j \circ \_): \text{End}(\mathsf A) \to \text{End}(\mathsf B)$ is strong monoidal on the full subcategory spanned by the monad $T$ and $1_{\mathsf A}$. Indeed $\text{lan}_j(jT^2) \cong \text{lan}_j(jT)^2,$ because $$\text{lan}_j(jT^2) \cong \text{lan}_j(\text{lan}_j(jT)jT) \cong \text{lan}_j(jT) \circ \text{lan}_j(jT) = \text{lan}_j(jT)^2. $$

(Observe that part of the functoriality is due to the fact that $j$ is dense). In particular, it maps monoid object to monoid objects.

Cor. 2 When $j$ is the Yoneda embedding, $\text{lan}_j(jT)$ admits a monad structure.

Proof. By the universal property of the presheaf construction, $\text{lan}_j(jT)$ is cocontinous, thus preserve every (pointwise) Kan extension, in particular, itself.

Cor. 3 When $\text{lan}_j(jT)$ is absolute, $\text{lan}_j(jT)$ admits a monad structure.

1. Is the argument in Prop. 1 correct?
2. How natural is the hypothesis that $\text{lan}_j(jT)$ preserve itself?
3. Are there natural or known assumption under which it is possible to equip $\text{lan}_j(jT)$ with a monad structure?

Let $j: \mathsf A \to \mathsf B$ be a fully faithful and dense functor where $\mathsf A$ is a small category and $\mathsf B$ is cocomplete. Let $(T, \eta, \mu)$ be a monad over $\mathsf A$. $\require{AMScd}$ \begin{CD} \mathsf A @>T>> \mathsf A\\ @V j V V @VV j V\\ \mathsf B @>>\text{lan}_j(jT)> \mathsf B \end{CD}

Is it possible to equip $\text{lan}_j(jT)$ with a monad structure?

Prop. 1 When $\text{lan}_j(jT)$ preserve itself, the answer is yes. Moreover, if the monad $T$ is idempotent, so is $\text{lan}_j(jT)$.

Sketch of Proof. The map $\text{lan}_j(j \circ \_): \text{End}(\mathsf A) \to \text{End}(\mathsf B)$ is strong monoidal on the full subcategory spanned by the monad $T$ and $1_{\mathsf A}$. Indeed $\text{lan}_j(jT^2) \cong \text{lan}_j(jT)^2,$ because $$\text{lan}_j(jT^2) \cong \text{lan}_j(\text{lan}_j(jT)jT) \cong \text{lan}_j(jT) \circ \text{lan}_j(jT) = \text{lan}_j(jT)^2. $$

(Observe that part of the functoriality is due to the fact that $j$ is dense). In particular, it maps monoid object to monoid objects.

Cor. 2 When $j$ is the Yoneda embedding, $\text{lan}_j(jT)$ admits a monad structure.

Proof. $\text{lan}_j(jT)$ is cocontinous, and thus preserve every Kan extension, in particular itself.

1. Is this argument in Prop. 1 correct?
2. How natural is the hypothesis that $\text{lan}_j(jT)$ preserve itself?
3. Are there natural or known assumption under which it is possible to equip $\text{lan}_j(jT)$ with a monad structure?

Let $j: \mathsf A \to \mathsf B$ be a fully faithful and dense functor where $\mathsf A$ is a small category and $\mathsf B$ is cocomplete. Let $(T, \eta, \mu)$ be a monad over $\mathsf A$. $\require{AMScd}$ \begin{CD} \mathsf A @>T>> \mathsf A\\ @V j V V @VV j V\\ \mathsf B @>>\text{lan}_j(jT)> \mathsf B \end{CD}

Is it possible to equip $\text{lan}_j(jT)$ with a monad structure?

Prop. 1 When $\text{lan}_j(jT)$ preserve itself, the answer is yes. Moreover, if the monad $T$ is idempotent, so is $\text{lan}_j(jT)$.

Sketch of Proof. The map $\text{lan}_j(j \circ \_): \text{End}(\mathsf A) \to \text{End}(\mathsf B)$ is strong monoidal on the full subcategory spanned by the monad $T$ and $1_{\mathsf A}$. Indeed $\text{lan}_j(jT^2) \cong \text{lan}_j(jT)^2,$ because $$\text{lan}_j(jT^2) \cong \text{lan}_j(\text{lan}_j(jT)jT) \cong \text{lan}_j(jT) \circ \text{lan}_j(jT) = \text{lan}_j(jT)^2. $$

(Observe that part of the functoriality is due to the fact that $j$ is dense). In particular, it maps monoid object to monoid objects.

Cor. 2 When $j$ is the Yoneda embedding, $\text{lan}_j(jT)$ admits a monad structure.

Proof. By the universal property of the presheaf construction, $\text{lan}_j(jT)$ is cocontinous, thus preserve every Kan extension, in particular, itself.

1. Is the argument in Prop. 1 correct?
2. How natural is the hypothesis that $\text{lan}_j(jT)$ preserve itself?
3. Are there natural or known assumption under which it is possible to equip $\text{lan}_j(jT)$ with a monad structure?