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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43 views

Category with pullbacks but not equalizers

Is there an example of a category with pullbacks but not equalizers (i.e. at least one pair of parallel morphisms does not have an equalizer)? Such a category cannot have have the terminal object, it ...
9
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2answers
184 views

What is the best path to learn Category theory and Type thoery?

I have little background in Programming in functional languages and wanted to learn type theory. I started with taking Homotopy type theory class Online videos of Robert Harper. I thought rather ...
2
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1answer
54 views

What does $S^z$ mean for each $z\in\mathbb{C}$?

Let $S$ be a set. What does $S^z$ mean for each $z\in\mathbb{C}$? In Set Theory numbers are sets and for any two sets $A$ and $B$, we define $B^A$ as the set of maps from $A$ to $B$. Well okay, ...
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1answer
35 views

Category defined by a finite commutative diagram

What is the name for a category defined by a finite commutative diagram? Maybe category "induced" by a commutative diagram? or category "defined" by a commutative diagram? Also, what is the exact ...
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1answer
39 views

Equalities in the categories of modules

It is well-known that over a quasi-Frobenius ring $R$ any f.g. right module $M$ is reflexive, in the sense that whenever we take $M^*=Hom_R(M,R_R)$ as a left $R$-module, the modules $M$ and $M^{**}$ ...
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27 views

Catsters Video Question

The first Catsters video on adjunctions has just finished, at this time, describing adjunctions in 2-categorical terms. Basically, the idea is to whisker the adjoint functors and the (co)unit of ...
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2answers
85 views

Why “thin groupoids” are not ubiquitous?

Google search for "thin groupoid" finds surprisingly few (namely 7) pages. But "thin groupoid" is a term to denote an important notation of a groupoid with every loop being the identity. I met it ...
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0answers
25 views

pseudonatural vs natural

By a general result of steve lack's article, I know that there is a nice adjunction between the 2-category of 2-functors (and 2-natural transformations) and the 2-category of 2-functors and ...
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0answers
41 views

2-natural equivalences

This is probably a trivial question. Let D be a 2-category. Consider a model category X. When is it possible to consider a 2-categorical structure in X such that the equivalences of 2-Fun (D, X) are ...
3
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2answers
67 views

Adjoint to forgetful functor from $\textbf{Met}$ to $\textbf{Set}$

As part of an assignment we need to prove that the forgetful functor $U:\textbf{Met}\to\textbf{Set}$ doesn't have a left adjoint (morphisms in $\textbf{Met}$ are contractions). For this, I have ...
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2answers
101 views

Right adjoint of functor from Torsion groups to Groups

For an assignment, we need to show that the inclusion functor $H:\mathsf{\bf Tor}\to \mathsf{\bf Grp}$ has a right adjoint, where $\mathsf{\bf Tor}$ is the category of all torsion groups, with group ...
2
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2answers
105 views

Examples of categories which naturally include End(O) as object

I want examples of categories $\textbf C$ which naturally include $End_{\textbf C}(O)$ as object for objects $O$ in the category. The set of all endomorphims is always a monoid under the composition ...
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2answers
55 views

A category with arbitrary products, but not all limits, or finite limits not commuting with filtered colimits?

I'm interested in finding an example of a locally small category $\mathcal{C}$ having small filtered colimits and arbitrary small products but lacking, either all small limits, or either the ...
1
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1answer
38 views

Quotient Object of Subobjects

A problem (not homework) from CWM: For subobjects $u\leq v$ of an object $a$ in an abelian category $\mathsf A$, define a "quotient" object $v/u$ (to agree with the usual notion in $\mathsf ...
3
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0answers
39 views

Equivalence of Modules

The category of modules over a ring can be viewed as an enriched version of an action of a monoid on a set (see nLab entry). Moreover, if $R$ is a commutative ring, the category of modules over it is ...
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2answers
42 views

Why does the terminal set have exactly one element?

I'm reading Lawvere/Rosebrugh's Sets for Mathematics. He states the axiom of the terminal set: AXIOM: TERMINAL SET There is a set $1$ such that for any set $A$ there is exactly one mapping ...
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1answer
19 views

Meaning of $f=me$ Factorization in Abelian Categories

Propsition 1, part 1 (Maclane, CWM p.199) Let $\mathsf A$ be an abelian category. Then every arrow has a factorization $f=me$, with $m$ monic and $e$ epic; moreover, $$m=\ker (\text{coker} ...
2
votes
2answers
49 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
46 views

Under what conditions do left adjoints preserve products?

Are there any results concerning under what conditions a left adjoint preserves products?
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1answer
28 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
36 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
64 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
101 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
65 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 $\mathscr{B}\subset\mathscr{A}$ ...
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1answer
57 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 ...
2
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2answers
74 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 ...
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1answer
28 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
58 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 ...
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2answers
40 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
41 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
51 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
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2answers
62 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 ...
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51 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 ...
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1answer
20 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
120 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
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0answers
49 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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92 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 ...
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2answers
46 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
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1answer
42 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
55 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 ...
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1answer
29 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
56 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 ...
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2answers
70 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
52 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
60 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 ...
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1answer
57 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 ...
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3answers
120 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 ...
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1answer
64 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
83 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
53 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 ...