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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309 views

Prerequisites to category theory

I am trying to delve into category theory but my math background is quite limited. What books would be recommended to get me up to speed with what is needed to grasp the concepts?
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1answer
80 views

A nice example of a functor naturally ismorphic to Stone-functor.

I want to explain the natural transformation with an example involving the Stone functor, but, I can't think of any non-trivial one. Does any one have one?
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1answer
109 views

Applications of Concrete Categories in Computer Science

From Wiki: "In mathematics, a concrete category is a category that is equipped with a faithful functor to the category of sets. This functor makes it possible to think of the objects of the category ...
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5answers
618 views

Question about the definition of a category

I am confused about the definition of a category given in the Wikipedia article on Category theory: It seems to me that the structure being described (the "arrows" between objects in some class) is ...
5
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1answer
98 views

Adjoint for functor involving Boolean rings

Let $R$ be a a commutative ring with a unit element, then one can associate to $R$ a Boolean ring $B(R)$, as in this text by Bergman, last line of page 594. (I guess this is a very classical thing. ...
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3answers
147 views

Does $\mathbb{Z}$ with subtraction as morphisms, forms a category?

Let object of the category be integers and for objects $a$ and $b$, define: $$ \operatorname{hom}\left(a, b\right) = \left\{ f \in \mathbb{Z} | a - f = b\right\} $$ Then composition of $f \in ...
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2answers
236 views

How to prove that monos are injective?

Let $A$ and $B$ be non-empty sets, and let $f\,:\,A\rightarrow B$ be a function. $ \color{darkred}{\bf Theorem}$: The function $f$ is injective if and only if $f\circ g=f\circ h$ implies $g=h$ ...
5
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3answers
204 views

Why the morphisms of vector spaces, over different fields is not interesting?

Suppose $V_\mathbb{F_V}$ and $W_\mathbb{F_W}$ are two vector spaces over fields $\mathbb{F}_V$ and $\mathbb{F}_W$. Then a homomorphism of these vector spaces consists of maps $f:V\rightarrow W$ and ...
3
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1answer
140 views

2-object-categories as algebraic structures

Categories with exactly one object are in 1:1 correspondence with the well-known algebraic structures called monoids. Is there a similar correspondence for categories with exactly two objects? ...
2
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1answer
50 views

Colimits of cosimplicial rings

The forgetful functor from the category of rings to the category of sets does not preserve arbitrary colimits. But it does preserve filtered colimits. For example, the colimit of a sequence of rings ...
2
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0answers
59 views

Quivers with a binary operation on the arrows

A set with an arbitrary binary operation is called a magma. A set of dots with a set of arrows between them is called a quiver. A category is a quiver with a binary operation on the arrows obeying ...
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2answers
343 views

Tensor product of monoids and arbitrary algebraic structures

Let $C$ be the category of algebraic structures of a certain type and let us denote by $|~|$ the underlying functor $C \to \mathsf{Set}$. For $M,N \in C$ we have a functor $\mathrm{BiHom}(M,N;-) : C ...
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1answer
136 views

Is this category essentially small?

Let $\mathcal C$ be the category of finite dimensional $\mathbb C$-vector spaces $(V, \phi_V)$ where $\phi_V \colon V \to V$ is a linear map. A morphism $f \colon (V , \phi_V) \to (W , \phi_W)$ in ...
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1answer
100 views

Existence of a Coend in a Monoidal Category Part II

I asked a very similar question but it was suggested that I rephrase and re-ask it. Let $B$ be a monoidal category with multiplication $\Box$. Let $P$ be a category and let $T \colon P^\mathrm{op} ...
3
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2answers
144 views

Which graph products are categorical products?

There is a whole bunch of definitions of graph products, but only one of them - the tensor product - is the categorical product in the (standard) category of graphs with graph homomorphisms. I'd ...
3
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0answers
247 views

On the bounded derived category of a finite dimensional algebra with finite global dimension

Let $A$ be a finite dimensional $k$-algebra with finite global dimension. How can I prove that the category $D^b(A)$ (bounded derived category of the category of left finitely generated $A$-modules) ...
4
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2answers
186 views

Do categories like Grp, Mon, etc. have right adjoints to their forgetful functors?

Self-explanatory. Just looking for the general tendency for those "algebraic objects," (sorry can't come up with a better term) because no one seems to talk about any such adjoints. Also could use a ...
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1answer
140 views

Right adjoint to forgetful functor from “dynamical system” digraph

Question about "dynamical systems," as Lawvere/Schnauel calls them in their baby book (ie digraph w exactly 1 arrow out of each point). What would a "chaotic" dynamical system be? In the book's ...
3
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1answer
81 views

Existence of a Coend in a Monoidal Category

Let $B$ be a monoidal category with multiplication $\Box$. Let $P$ be a category and let $T \colon P^\mathrm{op} \to B$ and $S \colon P \to B$ be functors. MacLane [CWM, p226] says that these two ...
5
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1answer
121 views

What would be an interesting example of a Co-algebra with a base category other than Set?

In most or perhaps all the examples of a co-algebra that I have seen, the properties of sets as the base category was used, like the existence of products and co-product and Cartesian closeness. Does ...
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0answers
73 views

What do you call this 2-morphism like object?

Take the category-like picture below: $g$ is not a 2-morphism in terms of category theory because the arrows it connects ($f$ and $g'$) do not have the same type (right?). What would you call it? ...
3
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0answers
123 views

Has the notion of a unique factorization category been defined and studied?

I am using a this notion in a paper that I am writing. The notion is that $(\mathcal{C},\otimes, I)$ (which we will just call $\mathcal{C}$) is a symmetric tensor category, with a unit $I$. Then a ...
6
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1answer
105 views

Categories with limits for large diagrams

This question was originally asked by Paul Slevin, but it was deleted before I had the chance to answer. It's actually quite subtle, so I thought it would be worth reposting it here. Consider the ...
3
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1answer
89 views

Formalising Statements in Category Theory with Regards to Universes (with an example of the end of a functor)

I am currently in the process of changing the way I think about category theory, by adopting the notion of Grothendiek universe and trying not to think of proper classes. Think of a statement such ...
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1answer
413 views

Gaining insight into the Inverse Image Sheaf

Let $f: X \rightarrow Y$ be a continuous map of topological spaces and let $G$ be a sheaf of sets on $Y$. I am trying to understand the definition of the inverse image sheaf $f^{-1}G$ on $X$. This is ...
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2answers
60 views

Category formulas without explicit specifying of objects

Consider the following example: $C$ is a category each of whose Hom-sets is partially ordered. Let $f$, $g$, and $h$ are morphisms of this category. Consider the formula: $g\circ f \ge h$. Intuition ...
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3answers
196 views

confusion over the use of universes in category theory

Fix a universe $U$ of sets. I take it we do this to stop accidentally talking about collections which are not sets. MacLane says that a small category is one where the objects and morphisms are both ...
3
votes
1answer
250 views

What is the Tarski–Grothendieck set theory about?

The wikipedia article on Tarski-Grothendieck set theory states: "[Tarski's axiom] also implies the existence of inaccessible cardinals, thanks to which the ontology of TG is much richer than that ...
5
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0answers
99 views

What is special about simplices, circles, paths and cubes?

There are some ubiquitous families of graphs — the complete graphs (or simplices) $K_n$, the circle graphs $C_n$, the path graphs $P_n$, and the hypercube graphs $Q_n$ — that intuitively ...
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1answer
263 views

Why does the definition of homotopy cartesian involve factorisations

Setup: A diagram $$\require{AMScd} \begin{CD} X @>>> Y\\ @VVV @VV{f}V\\ U @>>> V \end{CD}$$ in a (proper) model category is called homotopy cartesian if there exists a ...
2
votes
2answers
82 views

The existence of ends of functors.

Let $\mathcal {C}$ be a small category. In MacLane's book we have a theorem: If $\mathcal X$ is small complete and $\mathcal C$ is small, then every functor $S \colon \mathcal C^{\text op} \times ...
11
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1answer
752 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 ...
4
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1answer
212 views

Why are these two functors isomorphic?

Let $A$ be a local noetherian ring, $M$ an $A$-module finitely generated. Let $f$ be an $A$-regular and $M$-regular element (i.e. $f$ is not a zero divisors on $A$ nor on $M$). Then inside the ...
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votes
2answers
195 views

Coproduct of two modules

Suppose that $M$ is an $A$-module, and $N$ is a $B$-module. The coproduct of $A$ and $B$ is $A\otimes_{\mathbb{Z}}B$, and the coproduct of $M$ and $N$ is $M\oplus N$. I was wondering if $M\oplus N$ ...
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1answer
532 views

Infinite coproduct of rings

I just learned from Wikipedia that coproduct of two (commutative) rings is given by tensor product over integers, and that coproduct of a family of rings is given by a "construction analogous to the ...
11
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1answer
464 views

On equivalent definitions of Ext

Let $A$ be an abelian category and $X$, $Y$ two objects of $A$. Let's define Ext in this way: Ext$^i_A(X,Y)$=Hom$_{D(A)}(X[0],Y[i])$ Where $X[0]$ is the complex with all zeros except in degree 0 ...
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0answers
104 views

Left-modules over a bialgebra form a monoidal category

Let $B = (B, \nabla, \eta, \Delta, \epsilon )$ be a bialgebra over a commutative ring $k$. Let $M$ and $N$ be two left $B$-modules. Then the tensor product $M \otimes_k N$ becomes a left $B$-module ...
2
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3answers
113 views

Zero monomorphism

In a category with zero morphisms, can someone think of an example where $A\rightarrow B$ is a zero monomorphism but $A$ is not a zero object? (It is easy to see that $A$ should be a terminal ...
3
votes
1answer
265 views

Sending vector space to dual is a functor

In the category of finite dimensional vector spaces over a field and linear maps between them, the map that sends each space to its dual and linear map to its transpose is a functor, right? But this ...
3
votes
1answer
128 views

Epimorphism and image

Let $\phi:A\rightarrow B$ be a morphism in a category, and $\phi':I\hookrightarrow B$ its image. Intuitively, $\phi$ should be an epimorphism if $\phi'$ is an epimorphism. But I have difficulty ...
12
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1answer
340 views

A strange ring category

Recently I ran across a weird example of a category in Jacobson's Basic Algebra II. The category has, as objects, the class of rings. As morphisms, it uses all ring homomorphisms and ...
5
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1answer
301 views

Categorical definition of tensor product

It is standard to define the tensor product $M\otimes_R N$ of $R$-modules as a universal object of bilinear maps from $M\times N$. Now, suppose that $\mathscr{F}$, $\mathscr{G}$ are sheaves of ...
3
votes
2answers
927 views

Direct and inverse limits of sheaves

Is the direct limit of sheaves a sheaf? Is the inverse limit of sheaves a sheaf? I guess another way of saying it is whether sheafification commutes with direct limit or inverse limit. While we are at ...
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votes
1answer
278 views

Definition of a universal example

I'm not sure how the term is being used here: Let $R$ be a commutative ring and $X_1,\ldots, X_n$ indeterminates over $R$. Set $P = R[X_1, \ldots, X_n]$. Given a ring homomorphism $\phi: R ...
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0answers
101 views

The category of adjoint functors

Can a category structure be defined on the collection $Adj(\mathbf C,\mathbf D)$ of all pairs of adjoint functors $$ (F\colon\mathbf C\to \mathbf D)\dashv (G\colon \mathbf D\to \mathbf C)$$ in such a ...
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3answers
1k views

Cokernels - how to explain or get a good intuition of what they are or might be

When I think about kernels, I have many well-worked examples from group theory, rings and modules - in the earliest stages of dealing with abstract mathematical objects they seem to come up all over ...
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1answer
74 views

Morphisms in a symmetric monoidal closed category.

Let $\mathcal C$ be a symmetric monoidal closed category. This means that every functor $- \otimes B$ has a right adjoint $[B, -]$. Let $I$ be the unit and let $\rho \colon - \otimes I \to 1_{\mathcal ...
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1answer
426 views

The construction of the localization of a category

I was reading the construction of the localization of a category in the book "Methods of homological algebra" of Manin and Gelfand. Let me remind you the definition of the localization of a category: ...
5
votes
1answer
311 views

Adjoint of forgetful functor between category of vector spaces and category of abelian groups

I've just found out about the forgetful functor between the category of vector spaces and the category of abelian groups. It maps a vector space to it's additive abelian group. My question is, is ...
4
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1answer
153 views

Category exponent isomorphism

The book Categories for Working Mathematician - Mac Lane have a exercise described thus: For categories $A$, $B$, and $C$ establish natural isomorphisms $ \displaystyle C^{A \times B} \cong (C^B)^A $. ...