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Dimitri Chikhladze
Dimitri Chikhladze
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Suppose that $A$ is a monad on the object $C$ in $\Span(\cE)$. So, we have an internal category in $\cE$, also denoted by $A$.

An EM-object of $A$ is just the universal $A$-algebra. Algebras between monads in $\Span(\cE)$ are internal profunctors, see https://ncatlab.org/nlab/show/internal+profunctor. While a (one-sided) $A$-algebra consists of an object $D$ of $\cE$ and an internal profunctor from $D$ considered as a discrete internal category (trivial monad in $\Span(\cE)$) to the category $A$. So, an $EM$-object of $A$ consists of an object $S$$P$ of $\cE$ together with an internal profunctor from $P$ (considered as a discrete internal category) to $A$, which induces

$$\mathbf{Prof}(D, A) \cong \cE(D, P)$$

With the existence of "internal presheaf" object $\mathcal{P}A$

$$\mathbf{Prof}(D, A) \cong \cE(D, \mathcal{P}A)$$

and than we can take $P = \mathcal{P}A$. The idea of "internal presheaf" goes to the notions of cosmos and Yoneda structures, for which you can see https://link.springer.com/chapter/10.1007/BFb0063103 and http://www.academia.edu/30587479/Yoneda_structures_on_2-categories. In particular, for "internal presheaf" $\mathcal{P}A$ for an internal category $A$ see Examples (2) in the latter paper. I believe following this direction might lead to precise formulations.

Suppose that $A$ is a monad on the object $C$ in $\Span(\cE)$. So, we have an internal category in $\cE$, also denoted by $A$.

An EM-object of $A$ is just the universal $A$-algebra. Algebras between monads in $\Span(\cE)$ are internal profunctors, see https://ncatlab.org/nlab/show/internal+profunctor. While a (one-sided) $A$-algebra consists of an object $D$ of $\cE$ and an internal profunctor from $D$ considered as a discrete internal category (trivial monad in $\Span(\cE)$) to the category $A$. So, an $EM$-object of $A$ consists of an object $S$ of $\cE$ together with an internal profunctor from $P$ (considered as a discrete internal category) to $A$, which induces

$$\mathbf{Prof}(D, A) \cong \cE(D, P)$$

With the existence of "internal presheaf" object $\mathcal{P}A$

$$\mathbf{Prof}(D, A) \cong \cE(D, \mathcal{P}A)$$

and than we can take $P = \mathcal{P}A$. The idea of "internal presheaf" goes to the notions of cosmos and Yoneda structures, for which you can see https://link.springer.com/chapter/10.1007/BFb0063103 and http://www.academia.edu/30587479/Yoneda_structures_on_2-categories. In particular, for "internal presheaf" $\mathcal{P}A$ for an internal category $A$ see Examples (2) in the latter paper. I believe following this direction might lead to precise formulations.

Suppose that $A$ is a monad on the object $C$ in $\Span(\cE)$. So, we have an internal category in $\cE$, also denoted by $A$.

An EM-object of $A$ is just the universal $A$-algebra. Algebras between monads in $\Span(\cE)$ are internal profunctors, see https://ncatlab.org/nlab/show/internal+profunctor. While a (one-sided) $A$-algebra consists of an object $D$ of $\cE$ and an internal profunctor from $D$ considered as a discrete internal category (trivial monad in $\Span(\cE)$) to the category $A$. So, an $EM$-object of $A$ consists of an object $P$ of $\cE$ together with an internal profunctor from $P$ (considered as a discrete internal category) to $A$, which induces

$$\mathbf{Prof}(D, A) \cong \cE(D, P)$$

With the existence of "internal presheaf" object $\mathcal{P}A$

$$\mathbf{Prof}(D, A) \cong \cE(D, \mathcal{P}A)$$

and than we can take $P = \mathcal{P}A$. The idea of "internal presheaf" goes to the notions of cosmos and Yoneda structures, for which you can see https://link.springer.com/chapter/10.1007/BFb0063103 and http://www.academia.edu/30587479/Yoneda_structures_on_2-categories. In particular, for "internal presheaf" $\mathcal{P}A$ for an internal category $A$ see Examples (2) in the latter paper. I believe following this direction might lead to precise formulations.

Source Link
Dimitri Chikhladze
Dimitri Chikhladze
  • 2.3k
  • 1
  • 13
  • 16

Suppose that $A$ is a monad on the object $C$ in $\Span(\cE)$. So, we have an internal category in $\cE$, also denoted by $A$.

An EM-object of $A$ is just the universal $A$-algebra. Algebras between monads in $\Span(\cE)$ are internal profunctors, see https://ncatlab.org/nlab/show/internal+profunctor. While a (one-sided) $A$-algebra consists of an object $D$ of $\cE$ and an internal profunctor from $D$ considered as a discrete internal category (trivial monad in $\Span(\cE)$) to the category $A$. So, an $EM$-object of $A$ consists of an object $S$ of $\cE$ together with an internal profunctor from $P$ (considered as a discrete internal category) to $A$, which induces

$$\mathbf{Prof}(D, A) \cong \cE(D, P)$$

With the existence of "internal presheaf" object $\mathcal{P}A$

$$\mathbf{Prof}(D, A) \cong \cE(D, \mathcal{P}A)$$

and than we can take $P = \mathcal{P}A$. The idea of "internal presheaf" goes to the notions of cosmos and Yoneda structures, for which you can see https://link.springer.com/chapter/10.1007/BFb0063103 and http://www.academia.edu/30587479/Yoneda_structures_on_2-categories. In particular, for "internal presheaf" $\mathcal{P}A$ for an internal category $A$ see Examples (2) in the latter paper. I believe following this direction might lead to precise formulations.