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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2
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2answers
51 views

Associativity of arrow composition counter example?

I'm trying to achieve a working understanding of category theory. One of the problems I'm having is that many of the concepts seem too straight-forward or obvious so it's hard to see why they're ...
2
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3answers
57 views

Under what conditions do left adjoints preserve products?

Are there any results concerning under what conditions a left adjoint preserves products?
1
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1answer
35 views

Fibre products and induced short exact sequences in abelian categories

Assume we have an abelian category which has fibre products. Let $f:X\to Z$ and $g:Y\to Z$ be two morphisms and let $(W,p,q)$ be their fibre product with $p:W\to X$, $q:W\to Y$. If the category is a ...
3
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0answers
39 views

Unwinding descent via Barr-Beck

Let $f: U \rightarrow X$ be a faithfully flat morphism of nice schemes (quasiseparated, quasicompact, and anything else I might have forgotten). One can understand descent in quasicoherent sheaves ...
1
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1answer
66 views

Pullbacks and pushouts in the category of graphs

Let $\textbf{Grph}$ be the category of simple, undirected graphs without loops, together with graph homomorphisms. Note that there need not be any homomorphisms between two graphs, for instance ...
2
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1answer
124 views

Category Theory

I have two problems and I need some help or ideias on how to solve them. Suppose I have the following Category: the objects are structures (X, Cn) i) X is any set ii) Cn is a map from the power ...
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0answers
74 views

Quotient morphisms in topological subconstructs

Let $(\mathscr{A},U)$ be a topological construct (that is, a concrete category $U\colon\mathscr{A}\to Set$ where every structured sink has a unique final lift) and let $\mathscr{B}\subset\mathscr{A}$ ...
0
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1answer
58 views

References about “algebras over monoids”

Please, could someone point me any reference (with a bit of details) about "algebras over monoids" (in the sense of Schwede & Shipley, Algebras and modules in monoidal model categories)? Thank you ...
3
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2answers
83 views

In $\mathsf{Rel}$, are any two objects isomoprhic?

My knowledge of categories is rather basic, and I was just trying to find out what are isomoprhisms in $\mathsf{Rel}$ where objects are sets and morphisms are relations. As far as I got, an ...
0
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1answer
31 views

Is it always the case that a free construction satisfies this universal property?

this might be a stupid question, but I'm not sure if this is true (at least in some class of cases). Let $F : \mathcal{C} \rightarrow \mathcal{D}$ be left adjoint to an inclusion $\mathcal{D} ...
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0answers
60 views

Monoidal Categories

For a monoidal category $\mathcal{C}$ with $\alpha_{a,b,c}: a \otimes (b \otimes c) \rightarrow (a \otimes b) \otimes c$, $\rho_a : a \otimes 1 \rightarrow a$, and $\lambda_a: 1 \otimes a ...
0
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2answers
44 views

Why is every category in which all (even large) limits and colimits exist thin?

The wiki page on complete category states: The existence of all limits (even when J is a proper class) is too strong to be practically relevant. Any category with this property is necessarily a ...
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1answer
44 views

Example of endofunctor in Cat that is not a 2-functor.

Is there a good example of an endofunctor $\def\Cat{\operatorname{Cat}}\Cat \to \Cat$ (seeing $\Cat$ just as category) that is not a 2-functor?
3
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0answers
55 views

When do (universal-algebraic) varieties have (enough) injectives?

Following Zhen Lin's suggestion in my previous question, I ask this question separately. Suppose $\mathcal{V}$ is a variety in the sense of universal algebra. considered as a category. When does ...
3
votes
2answers
68 views

Are varieties cocomplete?

Consider a variety $\mathcal{V}$ in a sense of universal algebra, i.e. algebras of some fixed signatures described by a set of identities. Then $\mathcal{V}$ can be thought of as a category with ...
1
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0answers
56 views

Set and category theory

I have difficulties with the notions of set and category theory. I don't understand the difference between: a set, a collection and a universe! Although I understand why the collection of all sets ...
1
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1answer
23 views

Is there a special name or any research on Cartesian compact closed categories?

As per the title. I can't find anything about the combination of the two, and such categories interest me. Does anyone know of any such categories?
5
votes
1answer
123 views

Uniqueness of the long exact sequence in homology

A few days ago colleagues of mine and I listened to a talk about spectral sequences and one "application" of them was the proof that any short exact sequence (s.e.s.) $$0 \to A \xrightarrow{f} B ...
2
votes
0answers
52 views

Does the five lemma hold true for Lie algebras?

According to wikipedia, the Five Lemma is true in Abelian categories. But the category of Lie algebras is not Abelian. Then is the Five Lemma still true for Lie Algebras?
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0answers
98 views

Is this a general structure for constructs?

Here a construct is a category where the objects are sets and the morphisms are structure preserving functions. Common examples are groups, graphs and topological spaces. As far as I can see there is ...
1
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2answers
49 views

Showing that the product group of $G$ and $H$ satisfies the universal property for coproducts in the category of abelian groups $\mathbf{Ab}$

I'm working on another problem of Aluffi's Algebra. Given the category $\mathbf{Ab}$ of abelian groups, the problem is to show that for any two groups $G$ and $H$ the product group $G\times H$ ...
4
votes
1answer
46 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
votes
1answer
58 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
34 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
votes
1answer
61 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
votes
2answers
77 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 ...
1
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1answer
57 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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0answers
64 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 ...
6
votes
1answer
65 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 ...
3
votes
3answers
124 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 compactly 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
votes
1answer
68 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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votes
1answer
85 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 ...
0
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2answers
62 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
45 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
0
votes
1answer
35 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 ...
2
votes
1answer
90 views

Functorial cofibrant replacement does not have to be fibration?

I'm new to model category theory, and I find myself confused about the different meanings of cofibrant replacement in literature. The usual definition is that we assign to every object $X$ in our ...
3
votes
1answer
80 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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0answers
33 views

tensor product and composition (in monoidal category) [closed]

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
votes
2answers
45 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 ...
13
votes
2answers
256 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
votes
0answers
58 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
votes
1answer
154 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
60 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 \\ ...
2
votes
1answer
63 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
votes
2answers
64 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
votes
1answer
102 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
53 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
66 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
73 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 ...
4
votes
2answers
68 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: ...