Let $\mathbf{Top}$ denote the $2$-category of topological spaces, continuous mappings, and homotopies between them. Let $\mathbf{C}$ denote a wide subcategory of $\mathbf{Top}$. Then we get a wide sub-$2$-category $\overline{\mathbf{C}}$ of $\mathbf{Top}$ as follows. Firstly, the underlying category of $\overline{\mathbf{C}}$ is just $\mathbf{C}$. Secondly, the $2$-cells are precisely those homotopies between $\mathbf{C}$-arrows that "remain" in $\mathbf{C}$ through the entire deformation process. For example, if $\mathbf{C}$ is the wide subcategory of $\mathbf{Top}$ where the morphisms are injective continuous functions, then a $\mathbf{C}$-morphism $X \rightarrow Y$ is an "$X$-shaped" knot in the space $Y$, and the $2$-cells of $\overline{\mathbf{C}}$ are "witnesses" that two knots are equivalent. Take $X$ to be the circle and $Y$ to be $\mathbb{R}^3$ to recover the usual definitions.

Question. What, if anything, does higher category theory have to say about $2$-categories $\mathbf{T}$ equipped with a special way of extending each subcategory $\mathbf{C}$ into a sub-$2$-category $\overline{\mathbf{C}}$?

E.g. What are the appropriate axioms, definitions, terms and phrases, generalizations, etc.

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    $\begingroup$ Do you have any example where a wide subcategory of a 2-category is not the underlying category of a sub-2-category? (I mean, can't you just restrict to identity 2-cells?) $\endgroup$ – Najib Idrissi Jan 7 '16 at 12:54
  • $\begingroup$ @NajibIdrissi, don't quite follow. Can you say it in different words? $\endgroup$ – goblin Jan 7 '16 at 13:48
  • $\begingroup$ Does there exist a 2-category $\mathbf{D}$ and a wide sub-1-category $\mathsf{C} \subset \mathbf{D}$ such that there does not exist a sub-2-category $\bar{\mathsf{C}} \subset \mathbf{D}$ with underlying 1-category (when you forget all 2-cells) $\mathsf{C}$? Maybe I misunderstood your question? What do you mean by "a subcategory induces a sub-2-category"? $\endgroup$ – Najib Idrissi Jan 7 '16 at 13:55
  • $\begingroup$ @NajibIdrissi, have you read the whole question? $\endgroup$ – goblin Jan 7 '16 at 13:59
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    $\begingroup$ No, I didn't read either the question nor the title, I just typed random words... What the hell kind of question is that? The word "induces" appears only once in your question and isn't defined. Instead of assuming I didn't read your question, perhaps consider that it may not be as clear as possible. Have a good day. $\endgroup$ – Najib Idrissi Jan 7 '16 at 14:00

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