Various structures are studied in category theory using properties of objects and morphisms between them. Many constructions are special cases of categorical limits and colimits (e.g. products in various categories). The notions of functor and natural transformations are very important in category ...

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How to construct a G-extension of a category C?

Note: I'm a physicist so this will be phrased somewhat in physics language. Suppose we have a unitary modular tensor category $\mathcal{C}$. In physics language, we can think of $\mathcal{C}$ as ...
7
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1answer
109 views

Can these definitions of the words “problem” and “solution” be formalized, and if so, has this been done? If so, where can I learn more about it?

I had a thought. Define that: Vague Definition 0. A problem consists of: a set $X$ a set $Y$ a function $f : X \rightarrow Y$ a way $\overline{X}$ of representing the elements of $X$ ...
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0answers
40 views

What would be an arrow in category of Hilbert space?

Let $H,K$ be Hilbert spaces. Let $S$ be a Hilbert basis for $H$. (Which means that $S$ is orthogonal and the span of $S$ is dense in $H$.) For each arrow $S\rightarrow K$, there exists a unique ...
5
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1answer
73 views

Showing epimorphism without using the Freyd-Mitchell Embedding Theorem

In an Abelian category $\mathscr{C}$ consider a commutative diagram as follows: $$\require{AMScd}\begin{CD} 0@>>>\ker f@>\theta>>W @>{f}>> Y\\ @. @. @V{\phi}VV @|{id} \\ @. ...
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44 views

Is there a name for those concrete categories in which every subset / quotient set inherits the structure of an object in at most one way?

The following situation seems to occur a lot in abstract algebra: We have a category $\mathbf{C}$, concrete over $\mathbf{Set}$, that satisfies: For every object $Y$ of $\mathbf{C}$ and every set $...
2
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1answer
63 views

reference request: Category theory

I am sure that a similar question has been asked before, but I make my ideal textbook and situation more specific. I would like a textbook on category theory designed for someone who knows basically ...
1
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2answers
157 views

A categorical first isomorphism theorem

It is known, that for a morphism of universal algebras $f : A \to B$, if $R$ is the congruence relation given by $xRy \Leftrightarrow fx=fy$, then $\operatorname{im} f \cong A/R $. Here is an idea ...
5
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1answer
90 views

The bigger picture the Five Lemma fits into

The Five Lemma is a statement in category theory about certain conditions under which certain maps in exact sequences are isomorphisms. It has a few relatives like the 4 lemmas and maybe the Nine ...
0
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1answer
48 views

Notation for kernel object

When $f: A \mapsto B$ is a morphism in some category with a zero object and limits, we can use $\ker(f)$ to refer to an equivalence class of morphisms to $A$ which satisfy a particular universal ...
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2answers
40 views

How to take a limit of a diagram with more than one category?

The German Wikipedia describes how one can define the quotient field of a ring over a universal property: A quotient field $(\mathrm{Quot}(R), i)$ of a ring $R$ is a field $\mathrm{Quot}(R)$ ...
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1answer
36 views

Is an “$\aleph_0$-limit” a finite limit or a small limit?

I am sure this is a very trivial question. But I do not know anything about cardinals, and the nLab is full of them. I just want to know how to interpret a statement of the form "$\mathcal{C}$ has $\...
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1answer
90 views

Category of Sets with only monomorphisms

I came to work with a category which is the category of sets, "Sets", but for which I consider only arrows that are monomorphisms (i.e. injective maps). This makes sense in particular when expressing ...
1
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1answer
61 views

Reference/Definition of Homotopy in an Abstract Category

Let $\mathscr{C}$ be a complete and cocomplete category, and let $W$ be the collection of weak equivalences relative to some model on $\mathscr{C}$. We can form the homotopy category by localizing at ...
5
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0answers
38 views

Replacing faces of a cube in a quasicategory

I have a question on quasicategories which seems to be unavoidably cubical, and which I haven't been able to locate any information about. Suppose I have a cube $U:\mathbf{2}^3:\to C$ in a ...
0
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1answer
60 views

Geometric Morphism

I am trying to understand the whole concept of toposes and how the geometric morphisms are involved to it. But always i stumble upon a new concept, something that i am not familiar with and a new ...
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0answers
44 views

How to call a “non-strict” monoidal category?

A monoidal category is a category $\mathsf{C}$ equipped with a bifunctor $\otimes : \mathsf{C} \times \mathsf{C} \to \mathsf{C}$, a unit object, an associator, and right and left unitors satisfying a ...
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1answer
41 views

Are arbitrary coproducts filtered colimits?

A coproduct in a category $\mathcal{C}$ is a colimit over a diagram $F:S\to\mathcal{C}$, where $S$ is a set. So my question is equivalent to asking whether a set is a filtered category. The nLab ...
0
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1answer
38 views

Contruction of Weighted Colimit in a 2-category

On page 306 of Kelly's Elementary Observations on 2-categorical Limits, it is explained that a weighted limit $\{F, G\}$ in a 2-category can be constructed as the equalizer of $v$ and $w$ in $$ (3.2) \...
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1answer
29 views

Naturality as functoriality on arrow category

Here it is said that a natural transformation $\varphi:F\Rightarrow G$ is the same as a function $\varphi_0:\mathrm{Ob}\mathcal A\rightarrow \mathrm{Mor}\mathcal B$ satisfying ${\rm dom}\circ\varphi=F,...
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1answer
31 views

Adjunction with reversed elements

There is an adjunction between $L$ and $R$ when: $$ \text{Hom}(LA,B) \approx \text{Hom}(A,RB) $$ Is there something related we can say when instead we have: $$ \text{Hom}(B,LA) \approx \text{Hom}(A,...
4
votes
2answers
109 views

What are the group objects in the category of finite sets and bijections, and its functor category?

An object $G$ in a category $\mathcal{C}$ is called a group object if, given any object $X$ in $\mathcal{C}$, there is a group structure on the morphisms $\operatorname{hom}\left(X,G\right)$ such that ...
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1answer
35 views

A linear category is a Vect-module

I would like to know how to show that any linear category is a $\mathrm{Vec}$-module. Here $\mathrm{Vect}$ denotes a category of finite dimensional vector spaces. More general statement can be found ...
3
votes
1answer
37 views

Existence of adjoints with commutativity condition

Are there (non-trivial) examples of adjunctions $F \operatorname{\dashv} G$ with unit and counit $\eta$, $\epsilon$ such that $$\eta GF = GF \eta$$ and $$FG \epsilon = \epsilon FG,$$ and if so, what ...
4
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0answers
63 views

Deducing the existence of particular functions $\mathbb{N}\longrightarrow\mathbb{Q}$ in the context of Tom Leinster's “Rethinking Set Theory”

This question concerns the set theory given by Tom Leinster in his paper "Rethinking Set Theory," available here: http://arxiv.org/abs/1212.6543 In this paper, axioms for set theory are given in the ...
5
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1answer
88 views

Orthogonal Factorization Systems and functoriality

In definition 1.1 of these notes on factorization systems, (III) calls the factorization functorial if given the solid diagram described, there's a unique horizontal arrow making both squares commute. ...
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1answer
68 views

What limits/colimits are preserved by the Yoneda embedding?

I know that the contravariant Yoneda embedding $X\mapsto \mathcal{C}(-,X)$ preserves all small limits that exist in $\mathcal{C}$. I guess it follows that the covariant Yoneda embedding preserves all ...
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1answer
56 views

Associativity of the tensor product of bimodules

Let $A_0,\dots,A_n$ be algebras over some fixed commutative ring $k$ (you may assume $k=\mathbb{Z}$ for simplicity). Let $M_i$ be an $(A_{i-1},A_i)$-bimodule for $i=1,\dots,n$. A multilinear map from $...
3
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0answers
39 views

An asymmetry in the Galois connection between topologies and sequential convergences?

On a set $X$ consider a relation $c \subseteq X^{\mathbb{N}} \times X$ and for $((x_n), x) \in c$ write $x_n \to_c$ x. Such a relation $c$ is a sequential convergence if (i) $x \to_c x$ for all $x \in ...
2
votes
1answer
39 views

Simplicial homotopy's exponential law

The following was cited from Simplicial Homotopy Theory by John F. Jardine and Paul Gregory Goerss. We have an adjunction $$\hom_{\mathbf{S}}(X\times Y,Z)\simeq \hom_{\mathbf{S}}(Y,\mathbf{Hom}(X,Z))$$...
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44 views

Functorial construction of the convolution algebra of measures on a group

Let $G$ be a lcoally compact group and $C_c(G) = \lim_{K \subset G} C(K)$ its space of continuous functions with compact support endowed with the topology of the limit of banach spaces $C(K)$ with $K$ ...
0
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1answer
35 views

Action groupoid as $G\rightrightarrows \textrm{Bij}(X)$?

Let $G$ be a group and $X$ a set. A left action of $G$ on $X$ can be thought either as a map $G\times X\longrightarrow X$, $(g, x)\longmapsto g\cdot x$, satisfying: $(i)$ $g\cdot (h\cdot x)=(gh)\cdot ...
3
votes
2answers
53 views

Epimorphisms and faithful functors in a rigid abelian tensor category

Let $\mathsf{C}$ be a rigid abelian tensor category in the sense of Deligne and Milne's notes (p.9). Let $\mathbf{1}$ denote the identity object in $\mathsf{C}$ with respect to $\otimes$. The abelian ...
0
votes
2answers
42 views

Subcategory of category of Module satisfies SSA?

Let $\mathcal{D}=Mod-A$ be a category of module over an Algebra $A$. and $\mathcal{C}$ be a subcategory of $\mathcal{D}$, I have the following questions:- Is $C$ complete category? Is $\mathcal{C}$ ...
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4answers
126 views

Composing functors with natural transformations

So I'm doing a project in Category Theory. I fully understand natural transformations and functors, but what does it mean to compose them, for example in the monad axioms where you have something like ...
2
votes
0answers
25 views

Question Janelidze and Tholen's 'Beyond Barr Exactness: Effective Descent Morphisms'

I have some questions about Janelidze and Tholen's paper Beyond Barr Exactness: Effective Descent Morphisms. First of all, they remark at the end of the introduction: Throughout this chapter $\...
2
votes
1answer
41 views

Polynomial functors preserving ω-colimits

I'm working my way through Steve Awodey's Category Theory book and on p.271, Proposition 10.12 says: If the category $S$ has an initial object $0$ and colimits of diagrams of type $\omega$ (call them ...
0
votes
2answers
71 views

Is there a category in which, between any two objects, there is a unique morphism?

I am interested in knowing information about such a category (if it is well-defined). Does there exist a category $\mathcal{C}$, in which there is a unique morphism between any two objects in it?
2
votes
1answer
46 views

adjoint functor and subcategories

First at all i have to say I am very new to categories (just basic definition). My Question:- We have two categories ( $\mathcal{C},\mathcal{R}$) object in both of these two categories are module ( ...
2
votes
0answers
33 views

Extension Operator.

I am working on my thesis about completion and extensions from an algebraic point of view. We have the closure operator which takes subsets to subsets with 3 criterias to meet $X\subseteq C(X)$ $X\...
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vote
1answer
46 views

Presheaf and copresheaf categories on finite sets

Briefly: do they agree? In more detail: denote by $\mathbf{Finset}$ the category of finite sets, and by $\mathbf{Set}$ the category of sets. I want to know what the functor categories $[\mathbf{...
4
votes
2answers
63 views

Non-monadic adjunction

Could someone give some examples of a non-monadic adjunctions please? Possibly explaining why they are not monadic and how they contradict the monadicity theorem? Thanks!
0
votes
1answer
42 views

Establish canonical isomorphism in Set category

Using notation $A^B := \mathsf{Mor}_{\mathcal{SET}}(B, A)$ establish canonical isomorphisms for any sets $X, Y$ and $Z$: $$ (Z^Y)^X \cong Z^{Y \times X} \; , \;(Z \times Y)^X \cong Z^X \times Y^X.$$ ...
0
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1answer
71 views

Formal Proof that $f^{-1} \circ f = id_x \ , \forall f$

Given $f$ as an invertible function with domain $X$ and codomain $Y$, then we can say $$f^{-1}(f(x)) = x $$ Or since they are both logically equivalent $$ f(f^{-1}(x)) = x $$ This can also be ...
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1answer
25 views

Adjointness of internal contravariant Hom in symmetric monoidal categories.

Let consider a closed symmetric monoidal cateogry, $\mathscr C,\otimes$, with adjunction $(X\otimes-)\dashv\mathrm{Hom}(X,-)$ for all objects $X$. The following isomorphis, valid in categories such ...
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0answers
31 views

Functor between Category and its Free Strict Monoidal Category

Let $C$ be a category and let $\sum(C)$ denote the free strict monoidal category over $C$. According to Wikipedia, the operation $\sum: C\rightarrow\sum(C)$ extends to a 2-monad on $Cat$. Can someone ...
0
votes
1answer
61 views

Terminal object in the category of sheaves?

Let $\mathsf{C}$ denote one of the categories $\mathsf{Set}$, $\mathsf{Group}$, $\mathsf{Ring}$, or $\mathsf{Mod}_R$, or some other sensible category in which we would like sheaves to take their ...
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0answers
37 views

split object as coproduct

Let $X$ be an object and $f : Y \to X$ a monomorphism in some category with coproducts. Is there a categorical characterization for objects $Z$ such that $Y + Z \cong X$ and left injection composed ...
1
vote
1answer
38 views

If monoid satisifes universal mapping property over $X$, then $X$ generates the monoid

A monoid $M$ satisfies the universal mapping property (UMP) over $X$, if $X \subseteq M$ and for every map $\varphi : X \to N$, where $N$ is another monoid, there exists a unique homomorphism $\varphi ...
4
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1answer
42 views

Question about limit of cosimplicial diagram associated with a sheaf

Let $F$ be a presheaf of sets on the usual topological site. Let $\left\{U_i\rightarrow U\right\}$ be covering of $U$. On the second page of Hypercovers and Simplicial Presheaves the authors write the ...
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0answers
33 views

How to adjoin a terminal object to a category such that each constant morphism factors through it

Let $\mathbf{C}$ be any category. Is there a way to adjoin a terminal object $\ast$ to $\mathbf{C}$ such that each constant morphism factors through any terminal object, i.e., for each constant $f:A\...