Tagged Questions

45 views

Commutative diagrams with antiparallel arrows

A diagram in category theory is said to commute when for all objects $A$ and $B$ in it, every the composite morphism resulting from a possible path from $A$ to $B$ are the identical. Does that mean ...
45 views

How would you describe category $\mathsf{Rel}$?

I encountered two definitions for a category denoted by $\mathsf{Rel}$: Objects are pairs $\left(A,R\right)$ where $A$ is a set and $R$ a relation on $A$. Arrows in ...
61 views

What does it mean to categorify something? [closed]

What does it mean to categorify something? Is there any simple illustrative example of this process? How it is done and what is it useful for?
35 views

Definition of quotient category

Is there any reason why only gluing of morphisms sharing domain and codomain is usually allowed in the definition of quotient category?
66 views

What is $1 / \mathbf{Set}$ if $1$ is a one-element set and $\mathbf{Set}$ a category?

What does $1 / \mathbf{Set}$ denote? A pointed set is a set $X$ equipped with an element (a basepoint) $x \in X$. Let $\mathbf{Set_*}$ be the category of pointed sets and basepoint-preserving ...
46 views

What is the desirable function identification when setting up arrows in the category of types?

My question is which functions can not be allowed in a statically typed programming language, so that the "canonical" category is less coarse than what you get if you define it's arrows to be ...
145 views

Has the opposite category exactly the same morphisms as the original?

This is actually a question about categories; not only about the category that I mention here specifically. I only use category $\mathsf{Rel}$ as an example. How to describe a morphism that ...
53 views

Collection vs set in this textbook about category theory, and some related questions.

What is the meaning of collection in this context ? Is it here a synonym of set ? Can someone please explain what the author means by "A moment's though shows that, as sets of functions, these two ...
186 views

Definition of forgetful functor [duplicate]

Is there an actual definition of "forgetful functor?" Wolfram MathWorld defines it in terms of functors from algebraic categories to the category of sets, but then says, "Other forgetful functors..." ...
125 views

Categorial definition of free products?

If $X$ and $Y$ are objects of a concrete category $\mathcal{C}$, is there an accepted definition of "free product of $X$ and $Y$," generalizing the in the special case where $\mathcal{C}$ is the ...
390 views

Difference between being faithful and being injective on arrows

I'm studying the concept of faithful functors, but I cannot grasp the difference between being faithtful and being injective on arrows. Could someone explain the difference and provide some examples? ...
76 views

Is there way to formalize the idea that a category can be “cocomplete from the inside”?

Let $\mathrm{KSet}$ denote the category of all countable sets, including the finite ones. Then $\mathrm{KSet}$ is finitely complete. Furthermore, $\mathrm{KSet}$ admits all countable colimits, or, ...
21 views

Is there a term for an endomorphism defined up to conjugation by an automorphism?

Is there a standard term to designate the equivalence class of endomorphisms where two endomorphisms $\phi$ and $\psi$ are considered equivalent if there exists an automorphism $\alpha$ such that ...
501 views

What is category theory?

I have read the page about category theory in wikipedia carefully, but i don't really get what this theory is. Is category theory a content in ZFC-set theory? (Just like measure theory, group theory ...
39 views

Indecomposable groups vs. indecomposable objects

An object $X$ in a category $\cal C$ with an initial object is called indecomposable if $X$ is not the initial object and $X$ is not isomorphic to a coproduct of two noninitial objects. A group $G$ ...
52 views

Double categories

So, I wanted to ask a question about double groupoids until I find myself having the answer, meanwhile I wrote a lot of stuff about double categories, so I decided to create a question and answer my ...
116 views

Definition of truth values in a topos.

I'm trying to understand what is meant exactly by a "truth value" in a topos. Take for example the topos of irreflexive graphs. It is known that the classifying morphism can take nodes to 2 different ...
97 views

How does this definition capture the intuitive notion of an algebra?

On page 15 of this document, the author writes: Definition 1.1.1. Let $\mathcal{E}$ be any category. Given an endofunctor $\Gamma : \mathcal{E} \rightarrow \mathcal{E}$, a $\Gamma$-algebra ...
124 views

Should axioms be viewed as part of the signature?

I included category theory in the tags in order to get feedback from the categorial logic community. It goes without saying that this isn't really category theory. A semigroup can be defined as a ...
47 views

Definition for the action of a category on a set.

I'm trying to understand the definition of the action of a category on a set which is given in nLab, more particularly the first one. If one has a functor $\rho: C \to Set$, one takes the set S as the ...
74 views

Definitions of adjoints (functional analysis vs category thy)

If I have a linear operator $f$ on a Hilbert space, then I define the adjoint of $f$ to be $f^*$ where, $(fx,y)=(x,f^*y)$ for all $x,y$. I am confused because this definitions is very different to ...
48 views

How to see a 2-group as a 2-category with only one object?

We'll take the following definition of a 2-group: A 2-group $\mathsf{G}$ is a category internal to $\mathsf{Grp}$ Namely, it is a group $\mathsf{G_0}$ of objects, a group $\mathsf{G_1}$ of ...
63 views

79 views

In category theory, is there any such thing as “compatibility” for arrow composition?

Is this a properly defined category? Objects $\{P, R, S\}$ Arrows $f_{1} : P \rightarrow R$ $f_{2} : P \rightarrow R$ $g : R \rightarrow S$ $h_{1} : P \rightarrow S$ $h_{2} : P \rightarrow S$ ...
241 views

Definition of a point and object

Is there any theory in which a point has a definition? What is the definition of "object" as seen in category theory?
45 views

Reducts of categories

There are several ways to reduce a category. The skeleton of a category is the category with isomorphic objects collapsed into one i.e. the only isomorphisms that remain are the identities. What's ...
376 views

Exact sequence in a nonabelian category [previously: “Exact sequence for topological groups?”]

If $A$, $B$, and $C$ are topological groups, and $f: A \to B$ and $g: B \to C$ are two continuous group homomorphisms, what does it usually mean for 1 \to A \stackrel{f}{\to} B \stackrel{g}{\to} C ...
91 views

Definition of a groupoid of fractions

The title sums it up : I am looking for a definition of "a groupoid of fractions for a category". I would be interested in any example someone might have as well...