# 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.

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

What do you call categories such that $Hom(A,B) = Hom(B,A)$ for all objects $A,B$?
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### 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 ...
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### 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 ...
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### 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 $Hom(I, X)$, ...
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### 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 ...
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### 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 ...
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### 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 ...
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### "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" ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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....
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $M$, ...
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### 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 ...
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### 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) ...
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### 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 ...
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### 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.
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### 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? ...
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### 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 ...
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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 $\... • 2,022 2 votes 0 answers 59 views ### Why do we care about non-equal but isomorphic objects in a category? Note: this is more of a philosophical question. Said differently, what if we were to ask Categories to be extensional in the sense that if two objects are isomorphic then they are equal? Is there any ... • 21 2 votes 0 answers 71 views ### 2-categorical universal property of the classifying category of a type theory For example let us say we are in the setting of cartesian closed categories and the simply typed$\lambda$-calculus. Let$\mathtt{strCCCat}$denote the$2$-category of strict cartesian closed ... • 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 ... • 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 : \...
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 ...