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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If two monoids have equivalent action categories, are they isomorphic?

If $G,H$ are groups such that $G\mathsf{-Set} \simeq H\mathsf{-Set}$, then $G \cong H$; see math.SE/1375309 for a proof by Zhen Lin. Question. Does this also hold when $G,H$ are monoids? Since there ...
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23 views

Free lattice in three generators

By general results for every set $X$ there is a free bounded lattice $L(X)$ on $X$. I would like to understand the element structure of this lattice. The cases $X=\emptyset$, $X=\{x\}$ and $X=\{x,y\}$ ...
2
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1answer
76 views

Are there any free or fascist boolean algebras?

A boolean algebra is an algebra with the binary operations $\wedge$, $\vee$, an unary operation $\neg$, and constants $0$, and $1$, satisfying axioms. A heyting algebra is an algebra with the binary ...
2
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0answers
41 views

How to compute (co)limits of enriched categories?

Let $\mathscr{V}$ be a monoidal category. Let $\mathbf{Cat}_{\mathscr{V}}$ be the category of (small) categories. I would like to know how to compute (co)limits in $\mathbf{Cat}_{\mathscr{V}}$. This ...
5
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47 views

What are the universally effective epimorphisms of topological spaces?

An effective epimorphism in a category is a morphism that is the coequaliser of its kernel pair, and a universally effective epimorphism is a morphism $f : X \to Y$ such that, for every pullback ...
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1answer
30 views

sequence of colimit diagrams

Suppose we have a finite diagram with a colimit. Is it possible to then take the colimit diagram as a new base diagram and then have a new colimit of this new diagram? We could build up diagrams ...
4
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1answer
64 views

Functorial properties of the compact open topology.

Let $X,Y,Y'$ be topological spaces and $A\subseteq Y$ a subspace. Every set of continuous maps is equipped with the compact-open topology. Is the canonical map ...
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1answer
46 views

Prove that a morphism $\alpha$ of $Fun(\mathcal{A},\mathcal{B})$ is an isomorphism iff each component $\alpha_A$, is an isomorphism in $\mathcal{B}$

I'm a computational engineer starting with a course of Introduction to Category Theory, and perhaps is extremely basic what I'm asking but I'm trying to learn how to make proofs in category theory ...
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1answer
43 views

A question about combinatorial commutative diagrams.

In the comments of this question the directed graphs that can appear as commutative diagrams are axiomatized: A graph is a combinatorial commutative diagram iff it's nonempty and such that any ...
3
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1answer
79 views

Which directed graphs correspond to “algebraic” diagrams?

Any diagram for which the question of commutativity make sence is a directed graph, but not any directed graph make the question meaningful. $\require{AMScd}$ \begin{CD} A @>>> B @. A ...
6
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1answer
109 views

Is “Categories and Sheaves” a good followup to Aluffi's “Algebra: Chapter 0”?

I'm about to finish Aluffi's "algebra: chapter 0" and am a bit confused as to what should be my next move. I've been planning to read Tom Dieck's Algebraic Topology for some time now. I glimpsed at it ...
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1answer
49 views

Topology on compactly supported smooth functions

I'm confused by a set of lecture notes I'm reading and would like help in understanding what's going on. First, there is the following nice theorem. Theorem. The topology of a locally convex space is ...
2
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1answer
87 views

How similar are pullbacks to products?

Please excuse me if this is a trivial question. Let $f:A\to B$ and $g:C\to B$ be morphisms in a category and consider their pullback. I have seen books that say $projections$ for the morphisms ...
5
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0answers
40 views

Why don't we use the constant sheaf as the dualizing sheaf in Verdier Duality?

The Verdier dual of a complex of sheaves $F$ on a complex manifold $X$ is defined in the derived category $D_b(X)$ as $\mathcal{RHom}(F,\mathbb{D}_X)$, where $\mathbb{D}_X$ is the dualizing "sheaf" ...
4
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1answer
65 views

The algebra of natural transformations of the n-th power tensor functor

Let $k$ be a $0$ characteristic field, $n$ an positive integer and $S_n$ the $n$-th symmetric group. Let's work in the symmetric monoidal category of $k$-vector spaces and linear maps that we denote ...
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1answer
37 views

The inverse limit of C$^*$-algebras and whether it commutes with taking the minimal tensor product

Suppose we are given a C$^*$-algebra $A$ and a family of C$^*$-ideals $\mathfrak{I}$ that is upwards directed when ordered by reverse inclusion (i.e. for any $I_1,I_2\in\mathfrak{I}$ there exists a ...
1
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1answer
21 views

Does the inverse image sheaf have a left adjoint for $\mathsf{Set}$-valued sheaves?

It's known that for sheaves with values in modules, the inverse image sheaf functor $j^\ast$ for $j:U\subset X$ an inclusion of an open set has a left adjoint which is extension by zero. Is there any ...
6
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1answer
75 views

Does an equivalence of $G$-sets and $H$-sets imply an isomorphism of $G$ and $H$?

Here $G$-sets denote the category of sets which have a left $G$-action. So the question is whether a functor $F \colon \text{$G$-sets} \to \text{$H$-sets}$ implies that we have an isomorphism of ...
2
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0answers
45 views

Pullbacks in filtered categories?

A sufficient condition for the inclusion of a full subcategory $\mathsf C\hookrightarrow \mathsf D$ to be cofinal is that: Every object of $\mathsf D$ has an arrow into some object of $\mathsf C$. ...
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0answers
40 views

topos have colimits

Define an (elementary) topos to be a cartesian closed category with all finite limits and subobject classifiers. I'm looking for a proof of the fact that a topos also has all finite colimits. I know ...
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1answer
44 views

The Relationship between Separable Functors and Faithful Functors

Consider the adjunction $\mathcal{C} \mathrel{\substack{\mathcal{F}\\\rightleftarrows\\ \mathcal{G}}} \mathcal{D} $ together with unit $\eta: I_{\!_{\mathcal{C}}} \rightarrow \mathcal{G} \mathcal{F} $ ...
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3answers
65 views

Defining a coproduct in $\mathsf{Grp}$ using group presentations

I've encountered this exercise in Aluffi's Algebra: Chapter 0. It might be helpful to say that the book doesn't introduce functors at this stage,, and that the book defines a presentation of a group ...
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1answer
23 views

Copower functor

Computing copowers and "tensoring with sets" often means the same thing. If a locally small category $\mathcal{C}$ has coproducts and if $S$ is a set then for any object $C\in\mathcal{C}$ the copower ...
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37 views

Category theory,split idempotents,Set [closed]

Let $e:A\to A$ be a morphism in the category Set, with $e \cdot e=e$. How do we construct functions $i$ and $p$ with $e=i \cdot p$ and $id=p \cdot i$ ?
8
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1answer
192 views

Objects Too Big To Care About?

I was wondering if in certain fields of math (denoted by some set of axioms describing some class of objects), that there is a cap on size beyond which the existence of larger objects is "irrelevant" ...
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26 views

Conjecture concerning involutions in a unitary braided fusion category/Grothendieck ring

Despite the categorical setup, a solution to this question may require no categorical tools (see Conjecture 2). Let $\mathcal C$ be a unitary braided fusion category, $I$ be its set of isomorphism ...
6
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1answer
42 views

Let $T$ be the set of full binary trees. In what way $T^7 \cong T$?

I was reading the slides of a talk by Tom Leinster. I have trouble understanding the last line of page 17 (pages 1-15 are irrelevant and can be skipped). Could someone please explain it to me? If I ...
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1answer
53 views

What am I working with? [Inferring a theory in Category Theory using associativity of Cartesian Product]

In the category of sets, there is a "natural isomorphism," given three sets $A$, $B$, and $C$, from the set $(A \times B) \times C$ to the set $A \times (B \times C)$, where $\times$ is a Cartesian ...
4
votes
1answer
96 views

When $N \to M \otimes_R N$ is not an embedding.

Can someone please provide an example of the following (or tell me why such an example doesn't exist): Let $R$ be a (not necessarily commutative) ring, $M$ an $R$-$R$-bimodule containing a copy of ...
2
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1answer
53 views

Intuition on the Representable Functor

Given a locally small category C, and an object $C$, the functor: \begin{equation} \mbox{Hom}_\textbf{C}(C,-):\textbf{C} \longrightarrow \textbf{Sets} \end{equation} that sends objects to hom-sets ...
3
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1answer
58 views

Adjoint squares

I'm reading Mac Lane's Categories for the Working Mathematician and I'm having some trouble with exercise 5 in part IV.7. To avoid introducing adjoint squares I will only formulate the question in ...
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33 views

coherence of inverses in 2-groupoids

Given a 2-groupoid $G$, two objects $a,b$, and two 1-morphisms $f,g:a\rightarrow b$ and a 2-morphism $\alpha : f \rightarrow g$, is it the case that there always exists a 2-morphism $\beta : f^{-1} ...
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0answers
39 views

Left Kan extension of a $\mathsf{Set}$-valued finite-product-preserving functor

I've been told that the following is true: Proposition. Consider $\mathcal A,\mathcal B$ small categories with finite products and $j\colon \mathcal A \to \mathcal B$ preserving them. Then for any ...
4
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1answer
109 views

Fiber bundles with category morphisms as fibers

Given a total space $E = M \times F$ of a fiber bundle where $M$ is a smooth manifold and $F$ is the fiber. The fiber $F_x$ corresponding to the point $x \in M$ is the set of morphisms between objects ...
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1answer
46 views

Categorical Products Question

Im currently reading about categorical products in categories. The product of topological spaces etc, the product of graphs, but my stupid question is, if a categorical product is unique? Edit: Also ...
1
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1answer
69 views

Category of Sets w/ 17 Elements: There does not exist a direct product? (Lots of questions here)

I'm having a pretty hard time with this. I'm asked to show that, in the category of sets with exactly 17 elements, no two objects have either a direct product nor direct sum. Part of me doesn't even ...
8
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3answers
419 views

Definition of Category

In Spanier's book of algebraic topology, there is a definition of "categories" which entails "a class of objects". I realize that the vagueness of the concept of "class of objects" is exactly used ...
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1answer
66 views

Terminological conventions regarding group actions

Suppose: $G$ is a group $T$ is a Lawvere theory ("algebraic theory") and $X$ is a $T$-algebra What conventions surround the phrase: "action of $G$ on $X$"? In particular, does this mean: a ...
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1answer
49 views

What is a superfluous epimorphism?

In the definition of projective cover, the term superfluous epimorphism is used. Let $\mathcal{C}$ be a category and $X$ an object in $\mathcal{C}$. A projective cover is a pair $(P,p)$, with $P$ ...
2
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0answers
28 views

Unorthodox definition of semi-abelian category

I recently stumbled upon the book Derived Functors in Functional Analysis by Wengenroth. In it, he defines semi-abelian categories quite differently from the nlab: An $\mathsf{Ab}$-category ...
3
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1answer
63 views

Equivalence of definitions of “group object” using the Yoneda lemma

I have just started learning category theory and I am trying to get an understanding of how to think about the Yoneda lemma. Obvious applications are clear to me (Yoneda embedding is full and ...
2
votes
2answers
90 views

Confused about the Arrow Category

If we use the definition of the Arrow category and the notation from here. $$ \require{AMScd} \begin{CD} A @>h>> C \\ @VVfV @VVgV \\ B @>k>> D \end{CD} $$ I think I can understand ...
4
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2answers
89 views

Coproducts in $\mathsf{Grp}$

The limits and colimits in the category of abelian groups are as nice as can be, since products and equalizers are the same as in the category of sets. In the category of groups, however, the ...
3
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1answer
30 views

Notions of free (and/or cofree) Hopf algebras?

I don't know if this somewhat vague question has a precise answer, but I'd appreciate thoughts and references. Let $k$ be a field. I am looking at Hopf algebras over $k$ by which I mean an algebra ...
2
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2answers
76 views

Rings are $\mathsf{Ab}$-categories with one object. What are commutative rings?

Is there a nice and simple definition of commutative rings that does not use the notion of a commutative monoid object? How, in general, can one "externally" capture the commutative of a set ...
2
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1answer
27 views

Something wrong with proof: left adjoint functor preserves projectives

First a remark, I skipped the hypothesis "left adjoint to an exact functor" on purpose because the sketch of argument I wrote down I didn't use this, at least according to me. I know that there ...
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38 views

On two definitions of the nerve of a simplicial category

Let ${\mathcal C}$ be a simplicial category. Then there are the following two ways of constructing a simplicial set from ${\mathcal C}$: Form the simplicial nerve $\text{N}_\Delta({\mathcal C}) := ...
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65 views

Are subvarieties just full subcategories that happen to be algebraic categories?

Suppose $T$ is a Lawvere theory. Suppose furthermore that $\mathbf{C}$ is a replete full subcategory of $\mathbf{Mod}(T)$ that is equivalent (as a category) to $\mathbf{Mod}(S)$ for some Lawvere ...
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15 views

Explain the compactness relation for elements of dcpos and also in a category if objects

The way below relation is used to define compact elements in a dcpo. Can someone explain compactness and an object way below itself. Also, when we abstract this relation to categories, where the ...
2
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0answers
26 views

Topological reflection of pretopological closure operator

Given a pretopological space $(X,\mbox{cl})$ where $\mbox{cl}$ is a pretopological closure operator. How does one find the topological reflection of $(X,\mbox{cl})$? I know of a way namely by ...