Questions tagged [category-theory]

Categories are structures containing objects and arrows between them. Many mathematical structures can be viewed as objects of a category, with structure preserving morphisms as arrows. Reformulating properties of mathematical objects in the general language of category can help one see connections between seemingly different areas of mathematics.

Filter by
Sorted by
Tagged with
0 votes
0 answers
49 views

Categories where $Hom(A,B) = Hom(B,A)$

What do you call categories such that $Hom(A,B) = Hom(B,A)$ for all objects $A,B$?
user avatar
0 votes
1 answer
25 views

Aluffi Algebra Chapter 0 - Small Categories Doubt

In the bottom image (from Aluffi's Algebra: Chapter 0, page 22 in Preliminaries), he writes that the category corresponding to endowing $\mathbb{Z}$ with the $\leq$ relation is small. Is it small ...
user avatar
3 votes
0 answers
48 views

Is it possible to define this object using Kripke-Joyal Semantics

I've been looking into Kripke-Joyal semantics and, for each formula $\phi(x)$ defining an object $\{ x \, | \, \phi(x) \}$. Is it possible to, somehow, "fix a variable"? Let's say I've got a ...
user avatar
  • 93
2 votes
0 answers
18 views

Symmetric monoidal categories and modules over the unit

Consider a symmetric monoidal category $(C, \otimes, I)$, where $I$ is the unit. Then there is a restricted Yoneda functor $$ C \rightarrow Hom(I,I)-mod $$ taking an object $X$ of $C$ to $Hom(I, X)$, ...
user avatar
  • 1,398
3 votes
0 answers
43 views

Clarify on slice categories

Let $\mathbf C$ be a category and, for an object $c$, consider the slice category $\mathbf C/c$. While finding the subobject classifier of $\mathbf C/c$ (assuming that $\mathbf C$ has one), I proved ...
user avatar
0 votes
1 answer
69 views

Meaning of identity in category theory

Reading definitions of a Category, like on wikipedia, I start to wonder what the identity morphism actually means in the following sense: Purely formally, I understand that for object $A$ the identity ...
user avatar
  • 551
2 votes
0 answers
38 views

Unitors in star-autonomous categories

1.Context Let $(C, \otimes, I, a, l,r)$ be a monoidal category. Suppose $S: C^{op} \xrightarrow{\sim} C$ is an equivalence of categories with inverse $S’$. Assume that there are bijections $\phi_{X,Y,...
user avatar
  • 1,819
0 votes
1 answer
41 views

Pullbacks, Terminal objects and Products: proof of a proposition

I am trying to prove proposition 11.13 in Adamek’s Joy of Cats. The proposition says that if, in a pullback square, the sink object is a terminal object, then the pullback is a product. I tried to ...
user avatar
0 votes
0 answers
51 views

What is the identity for "<" relation (in Category)

This is from exercises 3.4 from Algebra Chapter 0 by Paolo Aluffi. The question asks can we define a category in style of reflexive and transitive relation (for example, "$\le$") using the ...
user avatar
  • 123
1 vote
1 answer
44 views

Difficulty with converting Yoneda's natural isomorphy into a group isomorphism in the proof of Cayley's theorem

$\newcommand{\A}{\mathscr{A}}\newcommand{\Gc}{\mathscr{G}}\newcommand{\G}{\mathcal{G}}\newcommand{\s}{\mathsf{Set}}\newcommand{\op}{^{\mathsf{op}}}\newcommand{\sym}{\mathsf{Sym}}$I am having ...
user avatar
  • 8,561
1 vote
0 answers
20 views

Elementary theory of the double category of spans

Is there an existing axiomization of an "Elementary theory of the double category of spans?" Among other nice things a category could be then defined as a monad inside the theory of spans. I ...
user avatar
3 votes
0 answers
53 views

Topos-theoretic proof of the consistency of CH

I hope this is not a duplicate, if so please let me know. So in MacLane Moerdijk‘s book Sheaves in Geometry and Logic it is shown that there is a boolean, two-valued topos with nno and choice, in ...
user avatar
1 vote
0 answers
32 views

How does one construct an affine variety morphism from an algebra homomorphism between their coordinate rings?

$K$ denotes an algebraically closed field. My course states that the category of affine varieties is equivalent to the opposite if the category of finitely generated reduced $K$-algebras. If $V \...
user avatar
  • 1,588
3 votes
2 answers
63 views

Coherence in closed monoidal categories

Let $(M, \otimes, I)$ be a left-closed (non-symmetric) monoidal category with left-internal hom $\underline{\operatorname{hom}}(-,-)$. Denote by $\sigma_{A,B,C}: M(A\otimes B, C) \xrightarrow{\sim} M(...
user avatar
  • 1,819
0 votes
1 answer
30 views

Additive category without finite limits

I was wondering if they are easy examples of additive categories wherein the equalizer or the pullback do not always exist ?
user avatar
  • 898
1 vote
1 answer
32 views

When the tensor product by a $K$-algebra is a faithful functor on the category of $K$-modules?

This is a quite general question, I know. Given a commutative ring $K$ and an algebra $K\to A$, there's an endofunctor $A\otimes_K-\colon K\text{-}\mathrm{Mod}\to K\text{-}\mathrm{Mod}$. I was ...
user avatar
  • 844
3 votes
0 answers
52 views

Showing that the category of epimorphisms with same codomain in an abelian category is cofiltrant

This is part (i) of exercise 1.7 in Kashiwara and Schapira's "Sheaves on Manifolds". Let $\mathcal C$ be an abelian category. For any object $Z$ of $\mathcal C$, let $\mathcal P(Z)$ be the ...
user avatar
  • 31
3 votes
1 answer
76 views

Exercise about exact sequence and pushout

The following is a commutative diagram in an abelian category. Assume that the rows are exact and that $h,k$ are epic. $\require{AMScd}$ $$\begin{CD} 0@>>>a@>{f}>> b @>{g}>> ...
user avatar
2 votes
1 answer
60 views

Cocomplete abelian category with enough injectives has exact coproducts

In this post it is claimed that for any (cocomplete) abelian category with enough injectives, the coproduct functor is exact, that is for a family of short exact sequences $0 \to A_i \to B_i \to C_i \...
user avatar
  • 57
3 votes
2 answers
146 views

Difference between products and coproducts.

I am struggling with understanding the difference between products and coproducts in category theory. For the category of abelian groups what part of the universal property of coproduct implies the ...
user avatar
  • 437
1 vote
0 answers
84 views

"Operations" in category theory that are not defined for arrows

Functors in category theory are defined for both objects and arrows. Depending on how they treat arrows, functors are characterized as either covariant or contravariant. Some "operations" ...
user avatar
  • 11
9 votes
1 answer
85 views

In a regular category a pullback is a pushout if its sides are regular epi

$\require{AMScd}$If in a regular category a square, whose sides $f,g,h,k$ are regular epis, is cartesian, is it cocartesian too? I would say yes, but I'm not sure that the proof below holds only under ...
user avatar
3 votes
0 answers
20 views

If Rel is enriched over suplattices what is Span enriched over?

In Rel the category of relations between sets you can "and", "or" and do several other operations over relations in nice ways. I don't fully get the details here but we can say the ...
user avatar
2 votes
1 answer
178 views

Is the term "category" in Category theory entirely different from the category in topological spaces?

$(X, \tau) $ be a topological space. $A\subset X$ is said to be first category (meager) if it can be expressed as a countable union of nowhere dense sets. Otherwise we call the set $A$ second ...
user avatar
  • 4,216
-1 votes
0 answers
39 views

Section in Abelian Category [closed]

In $\mathcal{A}$ a morphism $f: A \longrightarrow B$ is a section if and only if it is an injective function and $f(A)$ is a direct summand of $B$. Can anyone help me with an idea of ​​how I could ...
user avatar
  • 168
1 vote
0 answers
20 views

When two semilinear morphisms are said to be equal? [closed]

Suppose $(S,A)$ and $(T,B)$ are two left semigroup acts. A pair of mappings $(\mu,f):(S,A) \to (T,B)$ is called a semilinear morphism if $\mu$ is a semigroup homomorphism and $f$ is a function ...
user avatar
1 vote
1 answer
78 views

What are benefits of fixing a Grothendieck universe?

Let's assume the Tarski's axiom: For all set $u$, there is a Grothendieck universe $\mathbb U$ such that $u\in\mathbb U$. From now on I will drop "Grothendieck" and just write "universe....
user avatar
  • 1,277
-2 votes
1 answer
26 views

G be a finite group, C^× the non-zero complex numbers with trivial G-module structure. then H^2(G, C×) is finite.

Let G be a finite group, C^× the non-zero complex numbers with trivial G-module structure. Show that H^2(G, C^×) is finite. H2(G/N, C*); C* is the G/N-trivial module of the group of nonzero complex ...
user avatar
2 votes
1 answer
41 views

When representables are adjoints

Let $\mathbf C$ be a complete category and let $U:\mathbf C\to\mathbf{Set}$ be a representable functor. Show that $U$ preserves limits. In general representables preserve limits, but the hypothesis ...
user avatar
0 votes
1 answer
42 views

For two objects X and Y in a category, with Y dominating X, prove that X dominates a Zero Object in our category.

If we have a category, $C$, and two objects $X$ and $Y$ , and $Y$ dominates $X$. I think with definition there is morphisms $f : X → Y$ and $g : Y → X$ in our category such that $g ◦ f = Id_X$. I want ...
user avatar
0 votes
2 answers
43 views

Canonical morphism $\text{im}(f)\to\text{ker}(g)$ for exact sequence in an abelian category

Let $\mathcal{A}$ be an abelian category. Suppose we have objects $A$, $B$ and $C$, and morphisms $f:A\to B$, $g:B\to C$ with $g\circ f=0$ i.e. is equal to the zero morphism $A\to C$. In order to make ...
user avatar
  • 2,530
1 vote
0 answers
21 views

Internal language of an enriched category

I think I need a gadget along the lines of an operad enriched over the category of bounded lattices. But I'm having trouble thinking through what features enrichment would correspond to in an internal ...
user avatar
2 votes
4 answers
85 views

What is the right adjoint to the functor $\sf{Psh}\to\sf{Set}$ which evaluates the presheaf on the whole space?

$\newcommand{\O}{\mathcal{O}}\newcommand{\T}{\mathcal{T}}\newcommand{\op}{^{\sf{op}}}\newcommand{\set}{\sf{Set}}\newcommand{\ps}{\sf{Psh}_{\T}}$Let $\T$ be a topological space and $\O(\T)$ the poset ...
user avatar
  • 8,561
4 votes
2 answers
85 views

Build a "rich" first-order logic within a given category

I would like to know a mathematical framework with an internal logic where isomorphic objects can be considered equal. For example, consider the rationals $\mathbb{Q}$. With this set we can construct ...
user avatar
  • 679
3 votes
1 answer
96 views

How do natural transformations capture a "lack of arbitrarity"/uniqueness?

I've read natural transformations are the main motivation for category theory so I'm trying to get a handle on them, but having difficulty. In this question and this quora question it is asked what ...
user avatar
  • 372
1 vote
0 answers
28 views

Cocartesian edge gives contractible choice of filling in commutative diagram

This appears on p. 192 of Land's Introduction to Infinity-Categories which is the first page of his section on Straightening-Unstraightening. Let $p:\mathscr{E} \to \mathscr{C}$ be a cocartesian ...
user avatar
  • 6,822
1 vote
1 answer
67 views

Why is the preservation of identities a funtoriality axiom?

I'm a newbie at category theory and just started reading Emily Riehl's Category Theory in Context. I got to the definition of functors, which contains the following two axioms: if $F:C\to D$ is a ...
user avatar
  • 640
3 votes
1 answer
56 views

Why is a closed monidal category enriched over itself?

Let $M$ be a left-closed monidal category (Assume more such as symmetry if needed). Let $i$ be the unity object of $M$. Let $\alpha,\lambda,\rho$ be the associator, left unitor, right unitor of $M$, ...
user avatar
  • 1,277
2 votes
1 answer
51 views

Quotient map on S^1 such that that the quotient is an uncountable space with the indiscrete toplogy.

Based on Leinster Basic Category Theory page 135, problem 5.2.22 (b) I am trying to find a quotient map on the circle $S^1$ which results in the quotient having the indiscrete topology. Using the ...
user avatar
0 votes
1 answer
26 views

Basic property of a simplest fibered product

Let $X_1$ and $X_2$ be sets and let $\phi:X_2\to X_1$ be a map. The identity map $\text{id}:X_1\to X_1$ along with the map $\phi:X_2\to X_1$ gives rise to the fibered product (in the category of sets) ...
user avatar
2 votes
1 answer
49 views

Is There a Notion of Diagram in Multicategories and/or Operads?

In ordinary category theory there is a notion of a diagram in a category $\mathsf{C}$ which is usually described as a functor $F: \mathsf{J \to C}$ where $\mathsf{J}$ is some small category. Based on ...
user avatar
  • 12.2k
1 vote
2 answers
77 views

Category without pushouts or pullbacks

Can anybody try to give a example of a category with no pushouts or pullbacks? I want to find my own but I think an example would help.
user avatar
1 vote
0 answers
62 views

If $𝑓∘𝑔∘ℎ=𝑓 ∧ 𝑔∘ℎ∘𝑓=𝑔$ then must $ℎ∘𝑓∘𝑔=ℎ$?

If not, then What can be said of each $𝑓,𝑔,ℎ$ and are there any simpy-definable minimal conditions imposable upon one or more of the indexable functions that would ensure this symmetric closure? ...
user avatar
4 votes
0 answers
71 views

Motivating Grothendieck Toposes generated by Souslin Objects

I am currently reading about the independence of the continuum hypothesis from ZFC following the topos theoretic proof given in Chapter VI.3 of MacLane Moerdijk's Sheaves in Geometry and Logic. The ...
user avatar
0 votes
0 answers
21 views

Recovering composition in an enriched category

Let $(\mathcal{V},\otimes,e)$ be a closed symmetric monoidal category and $\underline{\mathcal{A}}$ a tensored $\mathcal{V}$-category. We will write $\mathcal{A}$ for the underlying category of $\...
user avatar
  • 2,022
2 votes
0 answers
59 views

Why do we care about non-equal but isomorphic objects in a category?

Note: this is more of a philosophical question. Said differently, what if we were to ask Categories to be extensional in the sense that if two objects are isomorphic then they are equal? Is there any ...
user avatar
  • 21
2 votes
0 answers
71 views

2-categorical universal property of the classifying category of a type theory

For example let us say we are in the setting of cartesian closed categories and the simply typed $\lambda$-calculus. Let $\mathtt{strCCCat}$ denote the $2$-category of strict cartesian closed ...
user avatar
  • 1,368
3 votes
1 answer
108 views

Schröder–Bernstein theorem in category.

In set theory, Schröder–Bernstein theorem assert for every set $A$ and $B$ if there exists injections from $A$ into $B$, from $B$ into $A$. Then there exists a bijection from $A$ onto $B$. I want to ...
user avatar
  • 33
2 votes
0 answers
23 views

Are the cylinders in the sketch for the tensor product of two $\mathcal F$-theories colimiting cylinders?

In section 6.5 of Basic Concepts of Enriched Category Theory Kelly writes If $\lambda : F \to \mathcal A(T-,M)$ is a cylinder in $\mathcal A$, where $F : \mathcal L^{op} \to \mathcal V$ and $T : \...
user avatar
0 votes
0 answers
38 views

Finite product and coproduct are the same in abelian categories

I am working with the following definition of abelian category. a) It has a $0$ object. b) Every morphism has a kernel and a cokernel. Every monomorphismis a kernel and every epimorphism is a ...
user avatar

1
2 3 4 5
256