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 ...

learn more… | top users | synonyms (2)

4
votes
1answer
62 views

A question regarding monoidal closed categories

If a category $\mathcal{C}$ is (symmetric) monoidal closed, is the opposite category $\mathcal{C}^{\text{op}}$ also monoidal closed? It is not clear to me whether by dualising the natural bijection $...
14
votes
5answers
427 views

Elementary topoi have initial objects, why?

An elementary topos is a category that: has finite limits is cartesian closed has a subobject classifier and one can show (with quite a bit of effort) that it has finite colimits. Is there, ...
3
votes
1answer
60 views

Completeness of 2-category of Monoidal Categories

Is the 2-category of monoidal categories complete? If not, can any conditions be imposed to satisfy completeness?
0
votes
0answers
23 views

About the Definition of Strict $2$-Natural Transformations between Strict $2$-Functors?

A strict $2$-category $\mathcal{C}$ consists of: (i) A horizontal category $\mathcal{C}^h:=(\mathcal{C}_0, \mathcal{C}_2, s_h, t_h, u_h, \circ_h)$; (ii) A vertical category $\mathcal{C}^v:=(\mathcal{...
0
votes
0answers
26 views

Categorical semantics for dynamic epistemic logic

Dynamic epistemic logic tries to reason about knowledge that certain actors (people, machines, etc.) have and how it can change in response to outside events. It is usually possible to discuss such a ...
3
votes
1answer
167 views

Which book would you recommended as help (assistance) for reading the so-called “Tohoku Paper”?

Recently I thought that maybe is a good time to try, read Grothendieck's "Tohoku paper" as a sort of inspiration for the future and to read some of the ideas of this great mathematician, which (among ...
1
vote
1answer
39 views

Different definitions of cartesian closed category?

The following is the definition of a cartesian closed category in Goldblatt's Topoi: Definition 1: A category $C$ is cartesian closed if (1) it is finitely complete, i.e. every finite diagram ...
3
votes
1answer
61 views

What is the pushout of $\mathbf{1} \longleftarrow \mathbf{2} \longrightarrow \mathbf{1}$?

I wonder what the pushout of the following diagram would be $$\mathbf{1} \stackrel{f}{\longleftarrow} \mathbf{2} \stackrel{g}{\longrightarrow} \mathbf{1}$$(here $\mathbf{1}$ and $\mathbf{2}$ denote ...
6
votes
2answers
80 views

Noteworthy examples of finite categories

So far all the finite categories I have encountered fall into one of these c̶a̶t̶e̶g̶o̶r̶i̶e̶s̶ sets: finite monoids finite preorders just formal devices to explain, what a "diagram" in another (...
3
votes
1answer
38 views

Can tensor abelian categories always be embedded into the category of modules?

Let $(\mathcal A, +,\otimes,I)$ a small symmetric monoidal abelian category. I know that $\mathcal A$ can be embedded into the category of $R$-module for a certain ring $R$. But can we make such ...
2
votes
2answers
43 views

Density of Yoneda Embedding - strange conclusion?

I'm trying to prove $y:\mathsf C\longrightarrow \widehat{\mathsf C}$ is dense. For this it suffices to prove the composite $$\widehat{\mathsf C}\overset{\hat y}{\longrightarrow}\widehat{\widehat{\...
4
votes
0answers
108 views

On comparing two different notions of compactly generated space

I have encountered, in different circumstances, the following two slightly different categories: The full category of $\mathsf{Top}$ consisting of all objects that are: a) topological spaces ...
3
votes
1answer
104 views

Moore-Penrose pseudoinverse and Linear relations

I recently came across this website called Graphical Linear Algebra. I feel like there's a lot of insight there, but it's too monolithic for me to be able to extract it by skimming. Episode 27 is ...
2
votes
0answers
47 views

Clarification about ends as weighted limits

The end of a bifunctor $F:\mathsf C^\text{op}\times \mathsf C\longrightarrow \mathsf D$ is the weighted limit $\varprojlim\nolimits ^{\mathsf C(-,-)}\!F$. This makes perfect sense. However, it's also ...
0
votes
1answer
36 views

What are pullbacks of finite-coproduct injections along arbitrary morphisms?

I am studying a definition of an extensive category: An extensive category is a category $E$ with finite coproducts such that pullbacks of finite-coproduct injections along arbitrary morphisms exist ...
0
votes
0answers
14 views

Double Nerve Preserving Filtered Colimits

The double nerve (https://ncatlab.org/nlab/show/double+nerve) is a functor $\mathcal{N}_2:2Cat\rightarrow sSSet$ from the 3-category of 2-categories to the category of bisimplicial sets. Does the this ...
1
vote
0answers
30 views

Interaction of a functor with internal hom

An additive functor between abelian categories $F: \mathscr{C} \to \mathscr{D}$ induces a functor on categories of chain complexes $F: \mathscr{C}^\bullet \to \mathscr{D}^\bullet$. The internal hom ...
1
vote
1answer
45 views

Is there a generally accepted name for the described property of arrow $f$?

Let $F:\mathcal A\to\mathcal E$ denote a functor. Let $f:a\to b$ be an arrow in $\mathcal A$ that has the following property: For every arrow $g:a\to c$ in $\mathcal A$ and every arrow $h:Fb\to Fc$ ...
0
votes
0answers
38 views

categorical product of two objects

I know that the product of two objects $A,B$ in a category $\mathscr{C}$ is defined up to isomorphism. This means that when $${<}{ P,f,g}{>}\text{ and }{<}{ P',f',g'}{>}$$ are two ...
3
votes
1answer
63 views

Does the associated bundle functor have left or right adjoints?

Let $\mathsf{Prin}_G$ be the category of (right) $G$-principal bundles, with a morphism from the bundle $p: P \to M$ to the bundle $p': P' \to M'$ being a pair of arrows $\chi: P \to P'$ and $\bar{\...
1
vote
1answer
18 views

Is the kernel of the product of cokernels a pullback?

I am currently reading Paolo Aluffi's textbook "Algebra: Chapter 0". I was working on exercise IX.1.18, which says: Formulate a notion of 'intersection' of two monomorphisms with a common target ...
4
votes
1answer
66 views

Codense if and only if truncated Yoneda embedding is fully faithful

I want to prove a functor $F:\mathsf C\rightarrow \mathsf D$ is codense if and only if the truncated (dual of the) Yoneda embedding $$\mathsf D^\text{op} \rightarrow [\mathsf D,\mathsf{Set}] \to [\...
3
votes
4answers
78 views

Does this thing I'm calling 'the operationalization of $x$' have an accepted name?

Given a set $X$ and an element $x \in X$, we can turn $x$ into a function denoted $\tilde{x}$ as follows: for any set $Y$ and any function $f : X \rightarrow Y$, define $$\tilde{x}(f) = f(x).$$ For ...
3
votes
1answer
45 views

Cartesian closed subcategories of compact Hausdorff topological spaces?

The category of compact Hausdorff topological spaces is famously not cartesian closed. I was wondering how much more one has to assume to actually arrive to a cartesian closed category. For example, ...
3
votes
2answers
134 views

Question on inverse limits

1.7. Remark. The inverse limit of an inverse system of non-empty sets might be empty as the following example shows: Let $I:=\mathbb{N}$ and $X_n:=\mathbb{N}$ for every $n\in\mathbb{N}$. Let $\...
3
votes
2answers
62 views

Regarding the injectivity of units of monads on $\mathbf{Set}.$

Given a monad $T$ on $\mathbf{Set}$, it usually seems to be the case that the unit of $T$ is componentwise injective (meaning that for all objects $X$ of $\mathbf{Set}$, the map $\eta^T_X : X \...
2
votes
1answer
60 views

Delooping of a ring?

I'm no expert on category theory, so the definition of delooping in the nlab article is a bit over my head. However, I do understand the practical idea that we can think of a group $G$ as a one-object ...
1
vote
4answers
126 views

What exactly are the meaning of the followings in the definition of a category?

In Awodey's Category Theory a category is defined as follows. A category consists of the following data, Objects: $A, B, C,\ldots$ Arrows: $f,g,h,\ldots$ For each arrow $f$ there are ...
3
votes
3answers
108 views

category-theory, right group action

Let $G$ be a group. We observe the category $(Set)_G$ of right group actions. a) Let $X\times G\to X$ and $Y\times G\to Y$ be two transitive right group actions with $x\in X$ and $y\in Y$. Find a ...
3
votes
1answer
42 views

Left adjoint for the “strings category” functor

Let $\mathbf{Cat}$ be the category of small categories and let $\mathbf{sCat}$ denote the category of simplicial objects in $\mathbf{Cat}$. We have a functor $$ \text{str}\colon \mathbf{Cat}\...
1
vote
1answer
70 views

Universal property, localization of rings and modules, and initial element in a category

Sorry for the confusing title. I just started learning category theory and am very confused about the concept "universal property". I am not even sure whether my "proof" is a proof or is just a ...
0
votes
3answers
25 views

Lift a morphism onto the category theoretical product

For three sets $X, Y, Z$, we can lift a morphism $f:Y\to Z$ onto the cartesian product with $X$, such that we get a morphism $\bar f:X\times Y\to X\times Z$ (obviously $\bar f(x,y)=(x,f(y))$). How ...
1
vote
1answer
25 views

Finite product exists implies finite coproduce exist.

Let $C$ be a category such that the law composition of morphisms is bilinear, and there exists a zero object $0$, and the products exists for arbitrary finite sets of objects of $C$. Then the ...
1
vote
0answers
82 views

Chain Homotopy in abelian category

When dealing with complexes of modules or groups, the following lemma is pretty easy: If $f,g :E\rightarrow E'$ are homotopic, i.e. $f-g=d'h+hd$ for some h, then $f,g$ induce the same homomorphism ...
2
votes
1answer
44 views

A question about the definition of adjoint functors

Aluffi's "Chapter 0", on pg. 492, says the following: Let $C$ and $D$ be categories, and let $\mathcal{F}:C\to D, \mathcal{G}:D\to C$ be functors. We say that $\mathcal{F}$ and $\mathcal{G}$ are ...
1
vote
1answer
19 views

homology theory for $C^*$-algebras, map is natural wrt morphisms of short ecact sequences

I want to assure me if I understand the part of the exactness axiom that "$\delta$ is natural" corretly and if not, then my question is: what does it mean? The setting is the following definition: ...
2
votes
1answer
58 views

A question about monoidal categories

I am learning about monoidal categories and I am a bit confused about the following: Suppose $(A,\otimes,I)$ is a monoidal category. What can be said about the opposite $A^{\text{op}}$? Is it ...
2
votes
1answer
55 views

Meaning of functorial

It's known that for a short exact sequence of complexes, $0\rightarrow E'\rightarrow E\rightarrow E''\rightarrow 0$, it associates a homology sequences $...\rightarrow H(E')\rightarrow H(E)\rightarrow ...
2
votes
0answers
32 views

Show that the law of the excluded middle does not hold in a BCCC

I want to show that the law of the excluded middle do not hold in a bicartesian closed category (BCCC), interpreted as follows: In general, there need not be a morphism $1 \to A + 0^A$ for $A \in \...
2
votes
1answer
34 views

Recursive definition in kernel of a morphism.

I find the following definition in Serge Lang's Algebra on p 133: Given a morphism $f:E\rightarrow F$ in an additive category, we define a kernel of $f$ to be a morphism $E'\rightarrow E$ such ...
1
vote
1answer
38 views

Completing a Category definition with Nodes as sets and Arrows as a triples of a Set and two functions

I would like to study a category that: Objects: are (finite) Sets. Arrows: are triples of the form $(A, src:A\rightarrow B,trg:A\rightarrow C)$, such that A, B, C are sets and src and trg are ...
3
votes
2answers
42 views

About the minimal equivalence relation identifying some points.

I am solving a problem where I have a set $X$ together with a subset of elements that I want to identify. To do this I consider the minimal equivalence relation identifying these points. I have a ...
2
votes
1answer
70 views

Graphic intuition for generalizing to weighted limits

One of the ways to define a limit of a functor $F:\mathsf C\longrightarrow\mathsf D$ is a representation of $\mathsf{Nat}(\Delta-,F)$. Along the journey of generalization to the enriched setting, one ...
1
vote
0answers
36 views

Existence of an illegitimate conglomerate of isomorphisms of $\mathbf{Set}$ onto itself

Definitions: A conglomerate $\mathbf{K}$ is legitimate if there exists a class $\mathbf{C}$, and a surjection $f:\mathbf{C}\to\mathbf{K}$, or alternatively, if there exists an injection $f:\mathbf{K}\...
7
votes
1answer
78 views

Is there a category theory notion of the image of an axiom or predicate under a functor?

Let me first state that I am a category theory novice so your patience is appreciated. I might be making some very basic conceptional mistakes and I might just need a simpler language to do what I ...
5
votes
1answer
89 views

Categorical semantics explained – what is an interpretation?

I’ve been really having a hard time trying to understand categorical semantics. In fact, I am confused to the point I am afraid I don't know how to ask this question! I’ve been reading textbooks like ...
2
votes
2answers
83 views

$\text{Ext}_R^n(\bigoplus_{i\in I} M_i, N) = \prod_{i\in I}\text{Ext}_R^n(M_i,N)$

Let $(M_i)_{i\in I}$ be a collection of $R$-moduls. Show that for all $N\in \text{Ob}(_R\text{Mod})$ is $$ \text{Ext}_R^n(\bigoplus_{i\in I} M_i, N) = \prod_{i\in I}\text{Ext}_R^n(M_i,N). $$ My ...
1
vote
2answers
35 views

Composition of a unique arrow with the inverse of another

Suppose we have the arrows $u:T \rightarrow Q$, $v:T \rightarrow P$ and $f:P \rightarrow Q$. Furthermore, suppose $u$ is unique and $f$ is iso. I understand that we can say that $v = u;f^{-1}$, but ...
2
votes
0answers
25 views

How to calculate braiding eigenvalues in a fusion category?

Statements like this are found in published articles: The context: Assume $\mathcal{C}$ is a complex fusion category (i.e. complex linear, finitely semisimple, monoidal, with duals, with simple ...
2
votes
2answers
80 views

nonequivalence of category of Groups and category of Pointed Groups

Am I correct in thinking that the category pGrp, whose objects are pairs $(G,g)$ where $G$ is a group, $g \in G$ and $$\hom\left((G,g),(H,h) \right) = \{ \varphi: G \rightarrow H \hspace{1mm} \big\...