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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21 views

Existence of projectives in the category of torsion abelian groups

Consider the category of torsion abelian groups. This category doesn't have enough projectives by the following argument. Suppose $C_2$ (cyclic group of order 2) is the homomorphic image of a ...
0
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1answer
20 views

Monoidal categories in which $\mathrm{Aut}(X \otimes Y) \cong \mathrm{Aut}(X) \sqcup \mathrm{Aut}(Y).$

I'm looking for examples of monoidal categories $\mathbf{C}$ such that one of the following two statements holds. For all objects $X$ and $Y$ of $\mathbf{C},$ $\mathrm{Aut}(X \otimes Y) \cong ...
1
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1answer
26 views

Any object in a locally noetherian Grothendieck category has a noetherian subobject

If $\mathcal{A}$ is a locally noetherian Grothendieck category, is that straightforward the fact that any object $M$ in $\mathcal{A}$ has a noetherian subobject?
2
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1answer
51 views

tensor product and commutation, category theoretical argument

It is a well-known fact that taking direct limits commutes with tensor products in the following sense: Let $I$ be a directed set and suppose for every $i\in I$ we have modules $M_i,N_i$ over a ring ...
3
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2answers
53 views

Are there any disadvantages to working in the category of k-spaces as opposed to Top?

Unlike the category Top of topological spaces with continuous maps as the arrows, the full subcategory of compactly generated spaces (k-spaces) is Cartesian closed. It seems like a very nice ...
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22 views

Slicing a subset of CAT over an object of an object of said subset [on hold]

Say we are in a finite subset of CAT, so our objects are categories. Is there a way (at least somewhat alike) to slice this subset of CAT over an object inside of our categories (the objects of the ...
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1answer
44 views

Isomorphisms in category theory

I'm having trouble understanding isomorphisms. E.g. in the category Posets which Awodey defines to be the category with posets as objects and monotone functions as arrows, he explains that bijective ...
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1answer
38 views

Suppose we want to get a groupoid of mathematical structures in this way, not just a set. How can we do it?

(I ignore size issues in this question. Write $\mathrm{Universe}$ for the set of all things.) We often define sets of object by asserting that: Every object can be fractured into pieces. For all ...
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0answers
55 views

what is the name of this operation?

Let's say I have some map $f$ that'll take a tuple with element types $A, B$ and $C$ to some type $T$ $f (x,y,z): A \times B \times C \rightarrow T$ and then say we have a map $g$, that takes an ...
5
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1answer
52 views

Equivalence of categories…

I was proving that: (i) $F: \mathcal{C} \rightarrow \mathcal{D}$ is and equivalence of categories; (ii) $F: \mathcal{C} \rightarrow \mathcal{D}$ is full, faithful and essentially surjective; are ...
2
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1answer
72 views

$p:E\to B$ is fibration then $p^*:map(X,E)\to map(X,B)$ is fibration as well.

$p:E\to B$ is fibration then for $X$ being completely generated weakly Hausdorff space $p^*:map(X,E)\to map(X,B)$ is fibration as well. We'd like to show that for any $Y$ and continuous $f$ and ...
5
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58 views

Why $R^{op}$ used in the definition of “category of $R$-modules”?

Earlier today, I was thinking: "Oh, an $R$-module is just an additive functor $R \rightarrow \mathbf{Ab}.$" Anyway, I had a bit of a read over at nLab, and it says: For any small ...
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1answer
74 views

Homotopy Groups for Categories

With this observation in mind how far are we from defining $\forall \mathcal{C} \ \text{category}\ \pi_1(\mathcal{C})$? Let me be more clear. Let be $n$ the following category $0 \rightarrow 1 ...
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2answers
42 views

Zero Morphisms in a Category

STATEMENT: This is taken from Robert Ash's,Basic Abstract Algebra. Let us call $0$ the zero object in an arbitrary category. And let us denote $0_{AB}$ the zero morphism from an object $A$ in the ...
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1answer
42 views

Gpd as a presheaf category

I wonder if there exists a way to see the category of groupoids Gpd as (isomorphic to, or maybe just equivalent to) a presheaf category (valued in Set) ? Thanks
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25 views

Applying the functor $H_*$ to the inclusion sequence $A\rightarrow B\rightarrow C$

Does applying the functor $H_*$ to the sequence of inclusions $A\rightarrow B\rightarrow C$ induce a map $\phi_3: H_*(B)\rightarrow H_*(C )$, such that if $\phi_1:H_*(A)\rightarrow H_*(B)$, and ...
3
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1answer
75 views

Why is this not a category?

Why need the composite of two monotone functions not be monotone? This is from Rings and Categories of Modules, Anderson., Fuller., page 7
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23 views

tensor product and composition (in monoidal category) [on hold]

For monoidal category, i think that tensor product is parallel relationship for morphisms. Then, If there are $f:A\rightarrow B$, $g:B\rightarrow C$,and $f$,$g$ are in the same category. Can i set up ...
2
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2answers
44 views

Can the underlying set functor corresponding to an algebraic theory always be viewed as a model of that theory?

Let $\mathsf{T}$ denote a Lawvere theory, and let $\mathbf{C}$ denote its category of models in $\mathbf{Set}$. Write $U : \mathbf{C} \rightarrow \mathbf{Set}$ for the underlying set functor. I think ...
11
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2answers
205 views

What does it mean to say that a particular mathematical theory is a foundation for mathematics?

We usually hear that set theory is a foundation for contemporary mathematics. Category theory is also another foundation of maths. There are other theories which deemed to be a foundation for maths. ...
4
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44 views

Equivalent definitions of regular categories?

maybe this is a stupid question, but I could not solve it after some time of meditation. There are four different notions of regular categories: 1) A cartesian category with coequalizers of kernel ...
3
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1answer
153 views

Is $\mathbb{Z}$ the initial rook?

By a rook, let us mean a unital, not-necessarily associative near-ring satisfying $x0=0$. Question. Is $\mathbb{Z}$ the initial object in the category of rooks? (I hope so, since this is my only ...
1
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1answer
47 views

Tensor product of function space and vector space

In one book I found that they treat vector valued functions as tensor product of some function space and vector space. So for example $$ C(\mathbb{R},V) \simeq C(\mathbb{R})\otimes V \\ ...
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33 views

Why are Truth, Conjunction, and Implication called “Negative” fragments of IPL (intuitionisti logic Proposition Logic)?

Someone answered that negative means we are ""Using"" them . But the point is for all of these there is an Introduction rule too. So why call them negative? I don't know whether it's computer ...
4
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1answer
41 views

name this “hybrid” categorical construction

I've found a general categorical construction which I'm not familiar with. Suppose that we have the square shown, with categories $A$, $B_i$, $C$ and functors $F_i$ and $G_i$ such that the diagram ...
4
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1answer
92 views

How to define $A\uparrow B$ with a universal property as well as $A\oplus B$, $A\times B$, $A^B$ in category theory?

In category theory there are definitions for $A\oplus B$, $A\times B$ and $A^B$ via universal properties. I wonder if it is possible to isolate a particular universal property to represent the ...
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2answers
38 views

Exponential objects in $k$-$\mathbf{FDVect}$

In my differential geometry class we've now moved onto algebraic/differential forms and to begin the section we're doing a quick and easy review of dual vector spaces. On a problem sheet I am ...
3
votes
1answer
49 views

Colimits in full subcategory (of all monics) of arrow category

Consider the category $mon(C)$ (objects as monics in $C$) as full subcategory of the $Arrow(C)$ . We know that $Arrow(C)$ is finitely complete and cocomplete, assuming that $C$ has limits as well as ...
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1answer
70 views

Right Ajoints Preserve Limits--alternate proof

I am filling in the details of a proof of this in MacLane, which uses the lim/$\Delta $ adjunction: Suppose $F\dashv G:X\rightleftharpoons A$, let $J$ be an (index) category, and let $T:J\rightarrow ...
3
votes
2answers
46 views

underlying set of direct limit not the direct limit of underlying sets

I am searching for a category $\mathcal C$ defined by a species of structures with morphisms $\Sigma$ (here I mean what is called 'espèce de structure' in Bourbaki Set Theory, chapter IV; put simply: ...
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1answer
48 views

What is a natural exact sequence?

I know what an exact sequence is, but I have searched for the definition of a natural exact sequence, and could not find it. Does "natural" perhaps mean some sort of preservation of structure? I ...
3
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1answer
66 views

Epimorphisms detect structure-preserving maps

Suppose that $U: \mathsf{Top} \to \mathsf{Set}$ is the forgetful functor. I believe that for a topological quotient map $\pi:R \to S$, and a map $\phi:U(S)\to U(T)$, we have that $\phi \circ U(\pi)$ ...
2
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1answer
31 views

Pullback in differential graded algebras

Suppose that $(A, d_A),(B,d_B),(C,d_C)$ are (unbounded) differential graded algebras and that $f:A \to C$ and $g:B \to C$ are homomorphisms of differential graded algebras. What is (or how do we ...
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1answer
58 views

On conglomerates' axiom of choice(Category theory)

There is a requirement of conglomerate(collection of classes) which demands the following property. Axiom of choice for conglomerates: For each surjection between congomerates $f:X\to Y$, there is an ...
3
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1answer
68 views

Do products and/or coproducts exist in the category of closed algebraic sets?

I was recently reading a note by Mel Hochster where he introduces the category of closed algebraic sets over an algebraically closed field $K$ a ways down on page 6. Out of curiosity, do products ...
2
votes
1answer
58 views

Behaviour of Ext with Hom

Let $K=SO(2,\mathbb{R}), \mathfrak{g}=gl(2,\mathbb{R})$. Let $V$ be a finite dimensional irreducible $(\mathfrak{g},K)$-module, W be a $(\mathfrak{g},K)$ module. I want to prove that ...
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0answers
46 views

Instructive sources for arguing without elements

There is a trend in mathematics towards reasoning without elements if possible (coming from category theory, I presume). I see the appeal and want to learn how to argue avoiding the use of elements, ...
2
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2answers
80 views

co-Yoneda lemma and representable functors

I'm interested in the fact that every presheaf on a category $C$ is a colimit of representable functors. The nLab claims that this can be stated as: $$F \simeq \int^{c \in C} Y(c) \otimes F(c)$$ Where ...
3
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2answers
129 views

Epimorphisms and monomorphisms in algebra

Let $G$ and $H$ be groups. If one has a surjective group homomorphism $f: G \to H$ does there necessarily exist a group homomorphism $g: H \to G$ such that $f \circ g = \text{id}_H$? Similarly $f$ is ...
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1answer
33 views

One of Axioms on arrow's only metacateogory redundant or not

I am still pretty new to category theory. I am reading the Category for working mathematicians. The author defines the axiom of arrows only meta category as followings. (1) Composition (k g)f is ...
2
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1answer
27 views

Covariant Power Set Functor being injective on arrows.

Consider the covariant power set functor $$\mathcal{P}(A\xrightarrow{\ f\ }B) = \mathcal{P}A\xrightarrow{\ \mathcal{P}f\ }\mathcal{P}B$$ where $\mathcal{P}A$ is the power set of $A$, and for ...
3
votes
1answer
61 views

Does monotonicity imply surjectivity?

There is an exercise in "An introduction to Category Theory" by Harold Simmons says: Consider any pair of $\mathcal{Pos}$-arrows $f: S \rightarrow T$ and $g: T \rightarrow S$. Show that when both ...
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1answer
47 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 ...
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33 views

Direct limits and inverse limits commuting

Recently in a reading I was doing, I ran into a fact I do not seem to be able to prove. The claim was that finite direct limits commute with inverse limits. It was also claimed that inverse limits ...
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1answer
36 views

$S$-local objects of presheaves are reflective and characterize local presentability

Let $PSh(\mathcal{C})$ be the category of presheaves on a small category $\mathcal{C}$. Let also $S$ be any fixed set of morphisms in $PSh(\mathcal{C})$. I say that an object $F\in PSh(\mathcal{C})$ ...
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1answer
26 views

Can any object in a Cartesian Closed Category regarded as a binary product?

Suppose there is an object $Z$ in a CCC, can we regard it as $Z \times 1$ where $1$ is the terminal product? Why it is true or otherwise?
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51 views

Are MAGMA-epimorphisms sujective?

The question says it all, but let me recall the definitions. A magma $(X,\cdot)$ is a set with a binary operation $\cdot:X\times X\to X$ (without any further assumptions like associativity). A ...
8
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1answer
161 views

Why are these two definitions of the left-adjoint to $u^p\colon PShv(D)\to PShv(C)$ equivalent?

Suppose $u\colon C\to D$ is a functor between categories. Then there is a functor $$ u^p\colon PShv(D)\to PShv(C) $$ between the associated presheaf categories by precomposition with $u$ as it is ...
5
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1answer
76 views

Counter-example for abelian category that is not concrete

I am trying to figure out a counter-example for abelian category that is not concrete category. Does the category of representations over a quiver work? Thanks,
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1answer
54 views

Right-adjoint to the forgetful functor $R-\mathbf{Alg} \to \mathbf{CRing}$

Does the forgetful functor $U: R-\mathbf{Alg} \to \mathbf{CRing}$ have a right-adjoint? I checked that it commutes with finite colimits but I couldn't guess any other candidate than the tensor ...