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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3
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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
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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
90 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
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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
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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\...
0
votes
2answers
53 views

Constructing a coreflection functor from its components

Let $\mathbf{A}$ be a coreflective subcategory of $\mathbf{B}$ and for all $B$, $A_B\xrightarrow{c_B}B$ an $\mathbf{A}$-coreflection. This is $\forall$ $\mathbf{B}$-objects $B$. I claim that there ...
2
votes
1answer
68 views

Covering by open subfunctors and epimorphisms of sheaves.

I am trying to learn about the functor of points approach to algebraic geometry. Given the category of locally ringed spaces $GSp$ (geometric spaces) we have a functor $$\mathcal{G}: GSp \to Set^{...
4
votes
1answer
36 views

Let $\mathbf{A}$ be a subcategory of $\mathbf{B}$, s.t. $\forall$ $\mathbf{A}$-objects $A$, $A\xrightarrow{id}A$ is an $\mathbf{A}$-reflection

I wish to prove that this implies that $\forall$ $\mathbf{A}$-objects $A$, any $\mathbf{A}$-reflection $A\xrightarrow{r_A}A^{*}$ is an $\mathbf{A}$-isomorphism. What I have managed to show, without ...
4
votes
1answer
46 views

Hamiltonian Mechanics and the Symplectic Category

Are canonical transformations (in the sense of Hamiltonian mechanics) morphisms for a certain category? They seem to fit the archetypal description of morphisms being "structure-preserving maps". ...
7
votes
1answer
330 views

Corollaries of the Yoneda Lemma in Analysis?

I am looking for some simple examples of how the Yoneda Lemma can be applied in analysis and probability theory and related fields. A simple candidate example that I can think of and somewhat ...
4
votes
3answers
189 views

Category of pointed sets and category of sets are not equivalent

Let $\mathbf{Set}$ denote the category of sets, with $\mathrm{hom}(X,Y)=Y^X$, and let $\mathbf{pSet}$ denote the category of pointed sets, with objects of the form $(X,x),\, x\in X$, and $\mathrm{hom}(...
0
votes
1answer
81 views

What is the dual category of topological spaces? [duplicate]

What is the dual category of topological spaces $Top$? I know that the order theoretic dual of a topological space is a closed set system rather than an open set system. However, this doesn't answer ...
-1
votes
2answers
61 views

Is every finite category identifiable with a directed multigraph? (and vice versa?) [duplicate]

What seems implicit in this talk on youtube, is the claim that every directed multigraph (with loops) can be identified with a finite category and vice versa, if we consider the paths of the directed ...
2
votes
1answer
64 views

Show that for each of the following graphs G there exists up to isomorphism precisely one category A with G(A) = G.

I was working through the exercises in Abstract and Concrete Categories: The Joy of Cats (http://katmat.math.uni-bremen.de/acc/acc.pdf) and I was stuck on exercise 3A.(d). It seems to me that the ...
0
votes
1answer
31 views

Let $\mathbf{A}$ be a category, then $\mathcal{P}_\mathbf{A} (\mathrm{dom}(f))\implies\mathcal{Q}_\mathbf{A}(f)$

This is a very trivial question in category theory, and the textbook I'm working from has this as a supposedly trivial example of the dual property for categories, and unfortunately, I can't seem to ...
4
votes
1answer
115 views

Homotopy Colimit of Truncations

Let $\mathcal{A}$ be an additive category with countable coproducts. I am just starting to learn about homotopy colimits and I am struggling with the following example that I am very interested in ...
1
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3answers
114 views

In category theory: Do we need products to define exponentials?

In the HoTT book the type of functions $A\to C$ construction is described first and the product type $A\times B$ construction later, using function types in its definition. So my obvious naive ...
0
votes
0answers
37 views

Does a binormal category always admit an additive structure?

Let $\mathsf{C}$ be a category. We call $\mathsf{C}$ binormal if it has a null object, has all equilizers and coequilizers, all monomorphisms are kernels and all epimorphisms are cokernels (whereby a ...
3
votes
1answer
85 views

Existential axioms for category theory

There are some existential axioms in set theory, for example, axiom schema of specification. It's my understanding that category theory isn't based essentially on set theoretic foundation. If so, I ...
1
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0answers
43 views

“External” Lawvere-Tierney Topologies?

Suppose I have a map $j : \text{Sub}(1) \to \text{Sub}(1)$ from subterminal objects of a topos to themselves which satisfies analogous axioms to those of a Lawvere-Tierney topology, namely $j(1) = 1$, ...
1
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0answers
29 views

Any straightforward proof of “in an abelian category, a pullback yields a monomorphism at cokernel level”?

Here is the question I encountered: $$\require{AMScd} \begin{CD} s @>{f^\prime}>> a @>{\varphi^\prime}>> \bar a\\ @V{g^\prime}VV @V{g}VV @V{\bar g}VV\\ b @>{f}>> c @>{...
6
votes
3answers
156 views

Is there a monoid structure on the set of paths of a graph?

Given a graph G, and the set of paths in G called PathG. Is there a monoid structure on PathG? Will concatenation be the multiplication formula? even if it's not defined for some paths? What about ...
1
vote
2answers
57 views

Can spaces where all singletons are closed and all singletons are open be homeomorphic?

Suppose $(X, \mathfrak{T})$ is a space where all singletons are closed, and $(Y, \mathfrak{J})$ is a space where all singletons are open. Can these two spaces be homeomorphic? My thought is that ...
4
votes
1answer
57 views

$A^A$ in category of graphs

(reference is Lawvere/Schanuel, Session 31, Ex. 1) I'm trying to calculate the exponential object $A^A$ and its evalution map $e \colon A \times A^A \to A$ in the category of graphs, where $A$ is the ...
0
votes
0answers
33 views

Determining whether these categories are toposes

Let $\textbf{Rel}_{\text{lt}}$ be the category whose objects are finite sets and whose morphisms are left-total relations between them. Let $M$ be a finite monoid. I would like to know whether: $\...
2
votes
1answer
35 views

Are pullbacks of regular epimorphisms in an additive category always epic?

Fix a additive category $\mathsf C$ (that is to say, $\mathsf C$ admits an additive structure and has a zero object all biproducts). Given four morphisms in $\mathsf C$ as follows: $$\require{AMScd} ...
2
votes
0answers
64 views

How to verify commutativity of a diagram?

Let $C$ be a category, and let all objects $X_i, Y_j$ belong to $ob(C)$, and morphisms $f_{ij}, h_i, g_{ij}$ be morphisms between them in $C$. Let us have a diagram then: How do we verify it's ...
1
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1answer
35 views

inverse limit as a functor

I have a question about inverse limits based on https://en.wikipedia.org/wiki/Inverse_limit . In the section "general definition" there is noted that an inverse system is a contravariant functor $I\...
1
vote
1answer
71 views

Yoneda lemma confusion

The (extended) Yoneda lemma is about a natural isomorphism of functors. I'm trying to write down these functors, but am getting stuck. The usual Yoneda lemma gives an isomorphism $\mathsf{Nat}(h_A,G)\...
1
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0answers
51 views

Sheafification by the small object argument

Is the sheafification functor constructed as a double application of the plus construction a special case of the small object argument? I thought it might be since I can't think of any other general ...
2
votes
0answers
45 views

F-algebras in 2-categories

Given a functor $F : C \to C$, one can usually study the $F$-algebras: morphisms $\alpha : F X \to X$. Where can I read about its generalisation to 2-categories? I think that one can consider now "lax"...
0
votes
1answer
45 views

Checking if two diagrams have isomorphic colimits

Consider the infinite diagrams: $$ C_1\to C_2 \to \cdots \to C_n \to \cdots $$ $$ D_1\to D_2 \to \cdots \to D_n \to \cdots $$ in some category, and suppose both colimits exist. How do I check if ...
0
votes
1answer
29 views

Ind- and pro-objects, reference request

Can someone point me to a good exposition of ind- and pro-objects, the intuition behind, and how one "in practice" works with them (i.e. prove things)? The nlab page is nice (especially for the ...
1
vote
1answer
30 views

When are two ind-objects isomorphic?

Two diagrams may be different, but they may still have the same isomorphic limits (or colimits). Ind-objects are, so to speak, formal colimits of diagrams, even if the actual limit may not exist. ...
1
vote
1answer
67 views

Elements in commutative diagram

The same way I define a function, by explicitly including the image of an element: $$ \begin{aligned} \mathbb{R} & & \overset{\exp}{\longrightarrow} & & \mathbb{R} \\ x & & \...
3
votes
1answer
51 views

Characterization of internal groupoids via pullbacks

The most intuitive way (for me) to define an internal groupoid is as an internal category with extra structure, namely an involution on the object of morphisms which "produces inverses". In Borceux ...
3
votes
1answer
72 views

Dualizing the statement “A functor is monadic”.

This is another example of my struggle with the dualizing principle in Category theory. There are two notions, monadicity and comonadicity. I want to see how exactly they are dual to each other. ...
4
votes
0answers
37 views

Some lemmas on overcategory adjunctions

I'm reading chapter 5 of Borceux and Janelidze's Galois Theories and I think the formulation of some lemmas in section 5.1 require unnecessary conditions. Let $F\dashv G$ where $\mathsf C \stackrel{F}...
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0answers
27 views

Retraction of a map

It's an exercise from Lawvere's book, "Sets for Mathematics" on page 53. Suppose S $\xrightarrow{i}$ B $\xrightarrow{j}$ E, $i$, $j$ are maps in category. Prove that $i$ has a retraction if $ji$ ...
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1answer
37 views

Functorial Morphism in Top / Topological Question

Is there any natural function which assignes to any function $f:X\to X$ a function $\epsilon_{\tau}(f):X\to X$ which is continuous in the topological space $(X,\tau)$?
2
votes
0answers
56 views

Forgetfull functor from Set and Top admits adjoints

Is there any book or article where I can find some completely explained examples of adjunctions? I just don't know how to make a complete prove that, for instance in $\mathbf{Set}$ and $\mathbf{Top}$ ...
7
votes
1answer
227 views

Is There a Concept of Fractional Composition?

Does there exist a concept of fractional composition for functions? Continuous or differentiable functions?
3
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0answers
66 views

How exactly do natural transformations relate to independence of choice of basis?

Let $V$ be a finite dimensional vector space over some field $K$. When I did not know anything about category theory, I learned about an isomorphism $(V^*)^* \cong V$ that 'does not depend on choice ...
3
votes
1answer
56 views

Difference between product projections and split epis in $\mathbf{Top}$

I don't understand the excerpt below. The trivial bundles were defined as split epimorphisms, and since the ground category was additive with kernels (in fact abelian), they are the same as ...
6
votes
1answer
115 views

map of hom-sets as morphism

I am an undergraduate math student, and while studying cartesian closed categories i encountered a problem i could not solve: Suppose we have a (locally small) cartesian closed category $C$, and let ...
4
votes
1answer
57 views

If $D$ is a triangulated category, and $E_i$ is a set of generators, then $D$ is equivalent to $D(End(\oplus E_i))$?

I am looking for a result along the lines of the following statement: If $D$ is a triangulated category, and $E_i$ is a set of generators (every object can be obtained up to isomorphism by shifts and ...