# Hook arrow $\hookrightarrow$ versus Tail arrow $\rightarrowtail$

Sorry, this is a very basic question but I can't seem to find a definite answer:

Whats the difference between $$\rightarrowtail$$ and $$\hookrightarrow$$?

I know that $$\rightarrowtail$$ is used to denote monic arrows. I think that $$\hookrightarrow$$ is used to denote an inclusion, but I'm not entirely sure.

Assuming the above is correct, since inclusions are (always?) monic, could any arrow decorated $$\hookrightarrow$$ be decorated by $$\rightarrowtail$$?

EDIT: here are instances of uses of $$\hookrightarrow$$ and $$\rightarrowtail$$.

$$\hookrightarrow$$ is used at the bottom of https://kerodon.net/tag/0001. Lurie writes $$N_\cdot:\{Categories\}\hookrightarrow \{\infty - Categories \}$$. Another way I have seen such notation used is in maps $$\Lambda^n_i \hookrightarrow \Delta^n$$.

$$\rightarrowtail$$ is defined in Rhiel's "Category Theory in Context" on page 11.

• It's my impression that in category theory people may hesitate to talk about "inclusions", that is, "subobjects" due to the underlying sets being in the relation of inclusion/injection. That is, "monic" is the correct categorical concept. Yes, in some "concrete categories" (=have underlying sets...) these can be interpreted as inclusions-of-sets ... with additional structure preserved. But that's entangling things needlessly. Jun 29 at 19:42
• @paulgarrett I'm not sure I understand the answer to the original question. Does $\hookrightarrow$ mean inclusion? Jun 29 at 19:44
• Yes, that symbol often means inclusion, with the left end of that arrow suggesting the "is a subset of" symbol. Jun 29 at 19:46
• awesome, thank you! Jun 29 at 19:46

There's no universally agreed upon distinction, but yes, you have the idea that $$\hookrightarrow$$ generally indicates a map that's "more like a subset inclusion" in some sense, and $$\rightarrowtail$$ indicates an arbitrary monomorphism. Some authors will use $$\hookrightarrow$$ for any monomorphism–type thing (note that Lurie is using it for a fully faithful functor, which isn't even the same thing as a monomorphism of categories!), but I think nobody will use $$\rightarrowtail$$ only for inclusion-type things.
• @ZhenLin Sure, right, and it could probably be used for any kind of $n$-monomorphism more generally. Jun 30 at 13:06
In category theory, usually $$\rightarrowtail$$ denotes a monomorphism; instead $$\hookrightarrow$$ denotes an injection, that is a concept making sense only inside a "concrete" category (informally, a category whose objects are sets and whose morphisms are selected maps).
So a concrete category always admits a "forgetful" functor, i.e. a faithful functor to $$\sf Set$$ (this is the actual definition of concrete). Since faithful functors reflect monos, any morphism whose underlying map (that is, whose image under the forgetful functor) is injective is also monic, but the converse is not true in general. However in $$\sf Set$$ any mono is injective, so the two concepts are indeed equivalent.