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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In what kinds of categories is a monic epi an isomorphism?

Is there a general description of categories $\mathscr{C}$ in which all monic epis are actually isomorphisms? In general, monic epis need not be isomorphisms. For example, in the ...
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1answer
34 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
40 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 ...
2
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1answer
68 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 ...
4
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1answer
71 views

Not every over-under-category is cocomplete

Something is wrong between me and Hirschhorn: point 3 of this result (in the book Model categories and their localizations): 7.6.4. Homotopy in undercategories and overcategories. Theorem ...
7
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2answers
250 views

Example of a functor which preserves all small limits but has no left adjoint

The General Adjoint Functor Theorem (Category Theory) states that for a locally small and complete category $D$, a functor $G\colon D \to C$ has a left adjoint if and only if $G$ preserves all small ...
2
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2answers
94 views

Is there a simple way of visualising the direct limit of the cyclic subgroups of a group?

By way of background to this question, I am interested in the properties of direct limits. They are usually defined in terms that assume there is an underlying directed poset, but according to ...
4
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3answers
109 views

Limits of Topological Vector Spaces

Let $X, Y_1, Y_2, \cdots$ be a sequence of topological vector spaces, and let $f_n : X \to Y_n$ be a sequence of continuous linear maps. Define the product space $\mathcal Y_N := Y_1 \times \cdots ...
1
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1answer
63 views

Preferred way to write elements of the direct sum of vector spaces

Suppose $V$ and $W$ are vector spaces over the same field and $V\oplus W$ is their direct sum. Reading through the literature I found essentially two ways of writing elements of $V\oplus W$. 1.) We ...
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0answers
25 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 ...
2
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1answer
151 views

Proof of $C^{A+B} \cong C^{A} \times C^{B}$ using only UMP and definitions of products and exponential objects

I need to show that in any category with binary products and coproducts, the following holds: $C^{A+B} \cong C^{A} \times C^{B}$ using only universal mapping properties and definitions of ...
6
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1answer
97 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 ...
0
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1answer
35 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 ...
11
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1answer
873 views

Does May's version of groupoid Seifert-van Kampen need path connectivity as a hypothesis?

May's A Concise Course in Algebraic Topology gives the following statement of the Seifert-van Kampen theorem for fundamental groupoids $\Pi(X)$ (section 2.7): Theorem (van Kampen). Let ...
1
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1answer
67 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
27 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" ...
1
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1answer
34 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 ...
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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} $ ...
4
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1answer
61 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 ...
2
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3answers
63 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
20 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 ...
4
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1answer
172 views

Limits and colimits in the category of schemes

What is the smallest category enlarging the category of schemes over a field $k$ which is: Complete? Cocomplete? Admits a cogenerator? generator? I admit there is some overlap with my previous ...
10
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1answer
681 views

On limits, schemes and Spec functor

I have several related questions: Do there exist colimits in the category of schemes? If not, do there exist just direct limits? Do there exist limits? If not, do there exist just inverse limits? ...
3
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1answer
194 views

Do pullbacks commute with filtered limits in this sense?

Let $A_n, B_n, C_n$ be directed systems in some abelian category. Denote by $A \times_C B$ the fibre product of $A$ and $B$ over $C$. Is it true that $(\varprojlim A_n) \times_{\varprojlim C_n} ...
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0answers
165 views

Direct limits and pullbacks

Suppose we have three directed sequences of $C^*$-algebras, say $(A_n,\varphi_n)$,$(B_n,\psi_n)$ and $(C_n,\theta_n)$ and $*$-homomorphisms $\alpha_n:A_n\rightarrow C_n$ and $\beta_n:B_n\rightarrow ...
5
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1answer
59 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
43 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 $\mathsf C$. ...
5
votes
2answers
134 views

A proof that right adjoints preserve limits?

Assume that categories $\mathscr{B}$ and $\mathscr{C}$ have all limits of shape $\mathbf{J}$. Then there's a slick proof that if $G\colon \mathscr{C} \to \mathscr{B}$ is a right adjoint, $G \circ ...
5
votes
1answer
105 views

Filtered colimits commute with forgetful functors

In many cases, filtered colimits commute with forgetful functors to $\mathbf{Set}$, for example with $\mathbf{CRing} \to \mathbf{Set}$ or $R-\mathbf{Mod} \to \mathbf{Set}$. Is there a slick way of ...
0
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1answer
25 views

Compact objects in Ind-categories

Assume a category $\mathcal{D}$ admits small filtrant inductive limits. Then call an object $Y$ of $\mathcal{D}$ compact, if $\hom_{\mathcal{D}}(Y,\cdot )$ commutes with these small filtrant inductive ...
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39 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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35 views

Category theory,split idempotents,Set [on hold]

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
185 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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0answers
47 views

What is the pullback of a central extension?

Suppose we have three objects $A,B,C$ of an (abelian) category $\mathbf{C}$ and a short exact sequence $ 0\to A \to B \to C \to 0 $ such that $B$ is a central extension of $C$ by $A$ ($im(A\to ...
1
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1answer
61 views

Regarding constructing the $I$-adic completion of a ring

Let $R$ be a commutative ring and let $I \subset R$ be an ideal. For $n \geq m$, let $\varphi_{m,n}$ denote the canonical ring homomorphism $R / I^n \to R / I^m$. Let $J = \cap_n I^n$. Then $R / J$ ...
0
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1answer
488 views

How to understand the “create limit”?

I find it is hard to understand the "create limit." (You can find it in Mac Lane's Categories for the working mathematician, P112; there it defines: "A functor $V:A→x$ creates limits for a functor ...
1
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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 ...
6
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1answer
39 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 ...
5
votes
1answer
47 views

An “identity” functor $f:\mathbf{Rel} \to \mathbf{Rel}^{OP}$

Looking at the category $\mathbf{Rel}$ and its opposite, I would like to know if there is something I'd call identity functor, $f:\mathbf{Rel} \to \mathbf{Rel}^{OP}$ that sends a set to itself and ...
1
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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 ...
3
votes
1answer
91 views
+200

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 ...
4
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1answer
95 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 ...
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0answers
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} ...
2
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1answer
51 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 ...
19
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1answer
401 views

Taking the automorphism group of a group is not functorial.

Once upon a time I proved that there is no functorial 'association' $$F:\ \mathbf{Grp}\ \longrightarrow\ \mathbf{Grp}:\ G\ \longmapsto\ \operatorname{Aut}(G).$$ A few days ago I casually mentioned ...
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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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 $P$ ...
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0answers
67 views

Branches of Category Theory [closed]

I found Category Theory a very interesting subject, although I have much to learn about it. Should I decide, in the future, to undertake research in the field, which would be the typical branches of ...
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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 ...
7
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2answers
165 views

Expand $\binom{xy}{n}$ in terms of $\binom{x}{k}$'s and $\binom{y}{k}$'s

Motivated by this question, I want to find a complete set of relations for the ring of integer-valued polynomials, where the generators are the polynomials $\binom{x}{n}$ for $n\in \mathbb{N}$. The ...