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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cokernel pairs left adjoint to equalizers

Given a category $ C $ which has both cokernel pairs and equalizers, how can I see that the functor $ C^\downarrow\longrightarrow C^{\downarrow\downarrow} $, which takes every arrow in $ C $ to its ...
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1answer
21 views

Proving naturality of an isomorphism in MacLane's CWM.

I am going through the adjuntions chapter of MacLane's CWM. I will follow his notation; it would be very difficult to try to describe all of the notation here. So excuse me for not explaining ...
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1answer
18 views

Basic localizers contain adjoint functors

I'm struggling with a property of basic localizers, namely that adjoint functors are weak equivalences. Recall that a basic localizer $\mathcal W$ is a subset/subclass of the arrows of the category ...
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2answers
32 views

What is a zero morphism in an abelian category

I am trying to familiarize myself with some basic category theory and I am getting confused with what a $0$-morphism is. If we are in category of say $k$-vector spaces then I am guessing ...
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1answer
30 views

Arrow between endofunctors over a symmetric monoidal category.

Consider an arrow between the categories of endofunctors over two symmetric monoidal (SM) categories $\mathcal{C}$ and $\mathcal{D}$ $$a:End(\mathcal{C}) \rightarrow End(\mathcal{D})$$ It is a ...
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2answers
65 views

Which constructions on a category are still interesting for a groupoid?

By a groupoid, I mean a (small) category in which every morphism is an isomorphism. It looks to me that constructions on a category like the opposite ("dual") category or products and (co-)limits ...
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2answers
198 views

Basic definition of Inverse Limit in sheaf theory / schemes

I read the book "Algebraic Geometry" by U. Görtz and whenever limits are involved I struggle for an understanding. The application of limits is mostly very basic, though; but I'm new to the concept of ...
3
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1answer
76 views

Is a homomorphism expected to be a (structure-preserving) map?

Is a homomorphism a special type of morphism, namely a structure-preserving map? For a morphism (of a category), it is clear that we can't always expect that a morphism is necessarily a ...
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39 views

Category of Presheaves on a small category $C$ is locally cartesian closed

I'm trying to fill in the details of the proof and need the following result: $Set^{C^{op}}/P\simeq Set^{D^{op}}$, where $D$ is the category of elements of $P$. The objects are pairs $(x,C)$ with $C ∈ ...
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54 views

Generators and relations as a functor [on hold]

Make “generators and relations” into a functor. What is its left adjoint? [Bergman] How could one do this?
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1answer
148 views

Why is there apparently no general notion of structure-homomorphism?

In model theory, one typically defines only embeddings of structures and isomorphisms, but I haven't seen a definition of general structure homomorphisms. Is there some particular reason behind that? ...
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1answer
24 views

Cylinder with bases collapsed to a point.

The problem, although arising from some deeper facts, is quite simple. I would like to visualise the quotient space $A$ given by the cylinder $I\times S^{1}$ ($S^{1}$ is the circle in $\mathbb{R}^{2}$ ...
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3answers
55 views

Showing hom-sets are disjoint in a morphism category

Let $\mathcal{C}$ be a category. We can construct a category of morphisms $\mathcal{M}$ by letting the objects be morphisms in $\mathcal{C}$ and morphisms be appropriate pairs of morphisms that give ...
3
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1answer
167 views

Algebraic topology and homotopy in category theory

I've heard many times that for an algebraic topologist two spaces which are homotopy equivalent are essentially the same. But when the topological space is contractible then it is equivalent to a ...
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1answer
82 views

Is there a category of categories?

My question is quite simple, I would like to know if we can define the category of the categories, unlike Cat which is the category of the small categories. By the way, are there any particular reason ...
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26 views

exercise involving exactness

I have been stuck on this exercise for a little while. We are in abelian category. How do I show that $0 \rightarrow A \rightarrow B$ is exact if and only if $f: A \rightarrow B$ is a monomorphism? ...
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1answer
45 views

equivalence in category

First gives some definitions, and then the property that I am confused. $A$, $B$ are both $R$-module, and $C$, $D$ an (additive) abelian group, consider the category $M(A,B)$ whose objects are all ...
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1answer
23 views

Query on a simple exercise involving representations of functors.

I am trying to prove the following fact Let functors $K,K':\mathcal{D}^{\text{op}}\to\mathbf{Set}$ have representations $(r,\phi)$ and $(r',\phi')$ respectively. Prove that to each natural ...
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1answer
112 views

Conglomerate in mathematical literature.

I rememeber I downloaded a pdf in category theory and after talking about classes it defined something called conglomerates, but I can't remember what that is and wikipedia has no article on it. ...
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1answer
50 views

Inequivalent Model Categories

A Model Category is, informally, a category where a "reasonable" notion of homotopy can be developed. I'm curious to know when two model categories are considered equivalent to each other. Thanks for ...
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1answer
46 views

Why is $\mathsf{HTAG}$ (Hausdorff, Topological, Abelian Groups) preabelian?

The category of Hausdorff topological abelian groups are commonly cited as an example of a category which is preabelian, but not abelian. I think one reason that is is not abelian comes from the ...
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1answer
45 views

Tensor Algebra = Universal Property of FORGETFUL FUNCTOR?

Hi there in wiki the tensor algebra is defined w.r.t. the adjoint of the forgetful functor rather than the forgetful functor itself - why so? Besides, does the existence of such algebras for every ...
2
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1answer
17 views

Behaviour of $\operatorname{Ext}$ with left exact sequences.

Maybe is a trivial question but I am not so good in derived functors. Assume we are in the category of abelian groups and we have an exact sequence $$0\longrightarrow A\longrightarrow B\longrightarrow ...
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38 views

2-category as a 2-monad?

It is well known that a category is the same as a monad in the 2-category of spans. So I am wondering if there is a similar statement hold for higher categories: can a bicategory be given as a weak ...
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1answer
48 views

Is the category of these particularly nice spaces cartesian closed?

Is the category of Hausdorff, compactly generated, locally path-connected, semi-locally 1-connected spaces (and continuous maps between them) cartesian closed? If not, in what ways does it fail to be? ...
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6answers
532 views

What does “homomorphism” require that “morphism” doesn't?

I'm starting to learn category theory, but there's one thing I don't get: all morphisms seem to be homomorphisms; the definition seems to be the same. What's the difference between these two? Can you ...
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1answer
60 views

Topological construct

I just started working with some category theory and I would like to understand the link between what I am studying now and what I know about topological spaces. By definition, a construct (in our ...
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1answer
88 views

Foundations book using category theory?

I'm about to embark on a PhD in mathematical biology. My major is in computer science. I would like to acquire a more rigorous understanding of math, which I am going to need to tackle some research ...
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29 views

~ The important use of Frobenius–Schur indicators and Frobenius-Schur exponents ~

I had asked a question on the uses of conjugacy class and centralizer. I had receive various helpful feedback. I appreciate them. Here I like to get some feedback on the Frobenius–Schur indicator. ...
3
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1answer
94 views

Properties Shared by Equivalent Categories

If two categories are equivalent, then if one has products, then so does the other. The proof of this is easy enough so I'm guessing the same result holds for exponentials but I am having trouble ...
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23 views

Underlying functor of tensor product in a closed and symmetric monoidal category.

I will follow, for terminology and notation, G. M. Kelly, Basic Concepts of Enriched Category Theory. For sake of a self-contained exposition, I will try to write here all the needed concepts. Let ...
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0answers
32 views

Module and bimodule categories equivalent to a 2-functor

Let $A$ be a tensor category (i.e. monoidal category). A (right) module category of $A$ is a category $M$ with a coherent action $\mu\colon M\otimes A\to M$. Denote $BA$ be the one-object bicategory ...
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2answers
274 views

how many empty sets are there?

Would I be correct in saying that in the category of sets, the "class of sets that are isomorphic to the empty set is a proper class"? In other words, there are LOTS of initial objects in the ...
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2answers
89 views

Is the categorical product for projective spaces essentially the tensor product?

I wonder whether the categorical product of two projective spaces is essentially given by the tensor product of the underlying vector spaces. Is this at least true for projective Hilbert spaces? One ...
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1answer
113 views

Two topologies over the same set which are carried to each other by a lattice isomorphism induce homeomorphic spaces?

This is a question that I've been turning in my head and haven't been able to come up with a way to proceed to either prove it or construct a counter-example: Let $X$ be a set and $\mathfrak{T}_X$ ...
2
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1answer
57 views

The idea of “generators” for arbitrary categories

Given a partial order $\langle X, \leq \rangle$ and a subset $I \subseteq X$ it is common to consider $I$ as the generators of the set $\{ x \in X: i \leq x \textrm{ for some }i \in I \}$ (i.e., the ...
2
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1answer
58 views

A closed symmetric monoidal category is enriched over itself.

Dear mathstackexchange, I have for some hours been grappling with what should be an easy diagram chase but turns out to be a bit more involved than I would imagine. Let $\mathcal{V}$ be a closed ...
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2answers
49 views

Derivatives on Functors

I'm not even sure if this question makes pedantic sense but is there any way to rigorously define the notion of taking the derivative of a functor?
2
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1answer
35 views

If the functor on presheaf categories given by precomposition by F is ff, is F full? faithful?

Suppose that $F: A \rightarrow B$ is a functor, and $- \circ F: \widehat{B} \rightarrow \widehat{A}$ is the functor on the presheaf categories induced by precomposition. If $- \circ F$ is full and ...
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1answer
206 views

What distinguishes topological spaces from graphs?

Topology would not "work" if one reverted the "direction" in the definition of continuous maps $f$: $$\text{open}(x) \rightarrow \text{open}(f(x))$$ It has to be $$\text{open}(f(x)) \rightarrow ...
3
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0answers
38 views

Existence of a natural transformation

Let $G$ be a group, $S$ a set and $\chi$ an action of $G$ on $S$. This defines a functor $F$ from the group as a single object category to the category of sets. Let $\phi$ be an automorphism of $G$, ...
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2answers
64 views

Preservation of Limit by Hom: Naturality Question.

$\DeclareMathOperator{\Hom}{Hom}$Let $D_{i}$ be a diagram in a category $C$,with d the limit. We have (1) $\lim \Hom_{C}(X,D_{i}) \cong \Hom_{C}(X,d)$. Whenever I see this result used, it is always ...
2
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1answer
25 views

The functor of monoids

I'm studying this book on introductory level category theory and I couldn't solve this exercise: In the first part I've been thinking about the monoid homomorphisms $F: S\to T$ and regarding of ...
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1answer
26 views

Initial F-algebra and its isomorphic arrow proof

A very stupid question from a programmer here. There is a theorem in Pierce's "Basic category theory for computer scientists" that sounds as follows: Let $K$ be a category, and $F$ be an endofunctor ...
3
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1answer
39 views

Why are global elements so special?

I'm starting to study category theory and I don't understand these remarks in this book: I have the following questions: Why does we study the global elements $x:1\to M$, since as we see in ...
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1answer
61 views

Do we lose everything, if the natural transformations in a monad are exactly inverse?

I'm currently explaining monads $$T:{\bf C}\to{\bf C},\hspace{1cm}\eta:1_{\bf C}\to T,\hspace{1cm}\mu:T\circ T\to T,$$ to my brain and the "only" tricky thing are really the identity relations. I ...
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functors on Zero-Object in $_RMod$-category

If I have a functor $F:_RMod\rightarrow _SMod$ that is between the category of $R$-modules to the category of $S$-modules. Can I show that it must be the case that $F(0)=0$. I know that $_RMod$ and ...
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Exact functors in the category of left R-modules - “Fun for the whole family”

The following question has proved troublesome and prompted some deeper questions which I will elaborate on. Our definition of a left exact functor is one which takes exact sequences: $$0 \rightarrow ...
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1answer
41 views

Why is a braided left autonomous category also right autonomous?

In Selinger, P. A survey of graphical languages for monoidal categories (New Structures for Physics, Springer, 2011, 813, 289-233), it is stated that: Lemma 4.17 ([23, Prop. 7.2]). A braided ...
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$\mathbf{Cat}$ the category of the categories is a category

I'm studying this book and I'm trying to prove this assertion the author made: The identity functor is the identity in this category, i.e., for each category $C$, $Id_C:C\to C$ is the identity ...