# Questions tagged [forgetful-functors]

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10 questions
54 views

### Do subobjects in concrete categories correspond to subsets?

A concrete category is a category $C$ endowed with a faithful functor $U:C\rightarrow Set$. And if $a$ is an object in $C$, then a subobject of $a$ is an isomorphism class of monomorphisms with ...
116 views

### Is there a notion of a transversal of subobjects?

If $a$ is an object in a category $C$, then a subobject of $a$ is an isomorphism class of monomorphisms with codomain $a$. The subobjects of all the objects in $C$ partitions the class of ...
28 views

### Is the category of small graphs a concrete category?

Let $\mathbf{Grf}$ be the category whose objects are small graphs and whose arrows are graph homomorphisms. A small graph is tuple $G = \langle V, E, \operatorname{src}, \operatorname{trg}\rangle$ ...
76 views

### Composition of monadic functors isn't monadic

Disclaimer: this question already has a solution here: Composition of monadic functors may not be monadic . However, I would like to understand how to solve this using another characterisation of ...
78 views

### $\lambda$-accessible categories, unclear proof

In the context of $\lambda$-accessible categories consider the proof of the proposition $1.22$ here. How/where did we use the fact that $\cal D$ in the last but one line has less than $\lambda$ ...
51 views

### Underlying set of the free monoid, does it contain the empty string?

In the free monoid over a set the unique sequence of zero elements, often called the empty string is the identity element. Is the empty string an element of the underlying set of the free monoid? In ...
80 views

### How to see that endotransformations of fiber functor have a coalgebra structure?

This question is based on section 5.2 in Tensor Categories, by Etingof et al. Note also that the question is pretty much in the title and what follows is just some background along with my fruitless ...
31 views

### Subfunctor from a category to Set

I'm looking for the definition of a subfunctor $R$ of a functor $U:\cal K \to \mathbb {Set}$. Are there any other conditions beyond that $R(A)\subseteq U(A)$ for all objects $A\in \cal K$ ?