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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45 views

Let $T$ be the set of full binary trees. In what way $T^7 \cong T$?

I was reading the slides of a talk by Tom Leinster. I have trouble understanding the last line of page 17 (pages 1-15 are irrelevant and can be skipped). Could someone please explain it to me? If I ...
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1answer
54 views

What am I working with? [Inferring a theory in Category Theory using associativity of Cartesian Product]

In the category of sets, there is a "natural isomorphism," given three sets $A$, $B$, and $C$, from the set $(A \times B) \times C$ to the set $A \times (B \times C)$, where $\times$ is a Cartesian ...
4
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1answer
101 views

When $N \to M \otimes_R N$ is not an embedding.

Can someone please provide an example of the following (or tell me why such an example doesn't exist): Let $R$ be a (not necessarily commutative) ring, $M$ an $R$-$R$-bimodule containing a copy of ...
2
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1answer
58 views

Intuition on the Representable Functor

Given a locally small category C, and an object $C$, the functor: \begin{equation} \mbox{Hom}_\textbf{C}(C,-):\textbf{C} \longrightarrow \textbf{Sets} \end{equation} that sends objects to hom-sets ...
3
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1answer
60 views

Adjoint squares

I'm reading Mac Lane's Categories for the Working Mathematician and I'm having some trouble with exercise 5 in part IV.7. To avoid introducing adjoint squares I will only formulate the question in ...
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0answers
33 views

coherence of inverses in 2-groupoids

Given a 2-groupoid $G$, two objects $a,b$, and two 1-morphisms $f,g:a\rightarrow b$ and a 2-morphism $\alpha : f \rightarrow g$, is it the case that there always exists a 2-morphism $\beta : f^{-1} ...
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1answer
66 views

Left Kan extension of a $\mathsf{Set}$-valued finite-product-preserving functor

I've been told that the following is true: Proposition. Consider $\mathcal A,\mathcal B$ small categories with finite products and $j\colon \mathcal A \to \mathcal B$ preserving them. Then for any ...
4
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1answer
109 views

Fiber bundles with category morphisms as fibers

Given a total space $E = M \times F$ of a fiber bundle where $M$ is a smooth manifold and $F$ is the fiber. The fiber $F_x$ corresponding to the point $x \in M$ is the set of morphisms between objects ...
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1answer
46 views

Categorical Products Question

Im currently reading about categorical products in categories. The product of topological spaces etc, the product of graphs, but my stupid question is, if a categorical product is unique? Edit: Also ...
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1answer
73 views

Category of Sets w/ 17 Elements: There does not exist a direct product? (Lots of questions here)

I'm having a pretty hard time with this. I'm asked to show that, in the category of sets with exactly 17 elements, no two objects have either a direct product nor direct sum. Part of me doesn't even ...
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3answers
425 views

Definition of Category

In Spanier's book of algebraic topology, there is a definition of "categories" which entails "a class of objects". I realize that the vagueness of the concept of "class of objects" is exactly used ...
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1answer
73 views

Terminological conventions regarding group actions

Suppose: $G$ is a group $T$ is a Lawvere theory ("algebraic theory") and $X$ is a $T$-algebra What conventions surround the phrase: "action of $G$ on $X$"? In particular, does this mean: a ...
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1answer
55 views

What is a superfluous epimorphism?

In the definition of projective cover, the term superfluous epimorphism is used. Let $\mathcal{C}$ be a category and $X$ an object in $\mathcal{C}$. A projective cover is a pair $(P,p)$, with $P$ ...
2
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0answers
29 views

Unorthodox definition of semi-abelian category

I recently stumbled upon the book Derived Functors in Functional Analysis by Wengenroth. In it, he defines semi-abelian categories quite differently from the nlab: An $\mathsf{Ab}$-category ...
4
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1answer
71 views

Equivalence of definitions of “group object” using the Yoneda lemma

I have just started learning category theory and I am trying to get an understanding of how to think about the Yoneda lemma. Obvious applications are clear to me (Yoneda embedding is full and ...
2
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2answers
96 views

Confused about the Arrow Category

If we use the definition of the Arrow category and the notation from here. $$ \require{AMScd} \begin{CD} A @>h>> C \\ @VVfV @VVgV \\ B @>k>> D \end{CD} $$ I think I can understand ...
4
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2answers
91 views

Coproducts in $\mathsf{Grp}$

The limits and colimits in the category of abelian groups are as nice as can be, since products and equalizers are the same as in the category of sets. In the category of groups, however, the ...
3
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1answer
36 views

Notions of free (and/or cofree) Hopf algebras?

I don't know if this somewhat vague question has a precise answer, but I'd appreciate thoughts and references. Let $k$ be a field. I am looking at Hopf algebras over $k$ by which I mean an algebra ...
2
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2answers
82 views

Rings are $\mathsf{Ab}$-categories with one object. What are commutative rings?

Is there a nice and simple definition of commutative rings that does not use the notion of a commutative monoid object? How, in general, can one "externally" capture the commutative of a set ...
2
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1answer
27 views

Something wrong with proof: left adjoint functor preserves projectives

First a remark, I skipped the hypothesis "left adjoint to an exact functor" on purpose because the sketch of argument I wrote down I didn't use this, at least according to me. I know that there ...
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0answers
43 views

On two definitions of the nerve of a simplicial category

Let ${\mathcal C}$ be a simplicial category. Then there are the following two ways of constructing a simplicial set from ${\mathcal C}$: Form the simplicial nerve $\text{N}_\Delta({\mathcal C}) := ...
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0answers
15 views

Explain the compactness relation for elements of dcpos and also in a category if objects

The way below relation is used to define compact elements in a dcpo. Can someone explain compactness and an object way below itself. Also, when we abstract this relation to categories, where the ...
2
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0answers
27 views

Topological reflection of pretopological closure operator

Given a pretopological space $(X,\mbox{cl})$ where $\mbox{cl}$ is a pretopological closure operator. How does one find the topological reflection of $(X,\mbox{cl})$? I know of a way namely by ...
1
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1answer
45 views

Tensor power modulo cyclic group action

Let $M$ be some $R$-module and $n \geq 1$ be some positive integer. The cyclic group $\mathbb{Z}/n\mathbb{Z}$, with a chosen generator $t$, acts on $M^{\otimes n}$ via $t(m_1 \otimes \dotsc \otimes ...
4
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1answer
81 views

Do the usual examples of “dimensive” Lawvere theories form an exhaustive list of such theories?

Call a Lawvere theory $T$ dimensive iff, letting $F_T : \mathbf{Set} \rightarrow \mathbf{Mod}(T)$ denote the free functor, we have the following. Every finitely generated $T$-algebra is free. From ...
5
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1answer
85 views

Easiest way to see that $\mathcal{C}$ is cocomplete?

Let $\mathcal{C}$ be a category that has all coproducts and coequalizers. My question is, what is the easiest way to see that $\mathcal{C}$ is cocomplete?
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2answers
40 views

Monotone map on a Preset

Related to my previous question I ran across this in the same book and thought I would try to get some clarity. If $R$ and $S$ are both Presets with relation $\le$ then the function $f$ is a monotone ...
2
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0answers
43 views

Algebra structure on dual to coalgebra

I'm trying to prove the following theorem using braided diagrams: Let $(C,\Delta,\varepsilon)$ be a finite-dimensional coalgebra. There is an algebra structure on $C^*$ given by multiplication ...
5
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2answers
137 views

A proof that right adjoints preserve limits?

Assume that categories $\mathscr{B}$ and $\mathscr{C}$ have all limits of shape $\mathbf{J}$. Then there's a slick proof that if $G\colon \mathscr{C} \to \mathscr{B}$ is a right adjoint, $G \circ ...
7
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3answers
115 views

On the importance of natural transformations

In p. 18 of Categories for the working mathematician (2d ed.), Mac Lane remarks that ..."category" has been defined in order to define "functor" and "functor" has been defined in order to define ...
2
votes
1answer
57 views

Is the category of (pre)sheaves over a singleton isomorphic to the category of sets?

I'm wondering whether $\mathsf{PSh}(\{x\})$ or $\mathsf{Sh}(\{x\})$ are equivalent to the category of sets. For sheaves, since the value on an initial object is a terminal object, It seems like the ...
5
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1answer
115 views

In what kinds of categories is a monic epi an isomorphism?

Is there a general description of categories $\mathscr{C}$ in which all monic epis are actually isomorphisms? In general, monic epis need not be isomorphisms. For example, in the ...
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0answers
31 views

Name (reference?) for lax monoidal functors that are 'full, or surjective, on monoids'?

A lax monoidal functor $F$ takes monoids to monoids. Is there a name for a lax monoidal functor that is 'full' or surjective with respect to this property? In other words, the functor $F : \mathcal{C} ...
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1answer
58 views

Inverse limit of blow up

Suppose $X_{0} = X$ is a complex space of dimension 2 with divisor $p_{0} \in X_{0}$. We can construct the blow-up, $X_{1}$ of $X$, which comes with a blow-down map $X_{1} \to X_{0}$. Suppose that ...
2
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0answers
37 views

How to speak on limit of sequence categorically? [duplicate]

I was thinking on ways to define limit of a sequence (over the reals, or over a metric space, or even better, over a general topological space) using the categorical limit (final or inicial object of ...
4
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1answer
75 views

Order in writing composed morphisms

When we have a function $f: A \rightarrow B$ between two sets, and we want to explicit that we are applying it to some element $x \in A$, we write $f(x)$. After this, is natural to write $f(g(x))$ ...
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0answers
93 views

Simple question in “Sheaves in geometry and logic”

There's an argument I don't understand in "Sheaves in geometry and logic" by Mac Lane and Moerdijk, that seems a priori easy but I can't see it. Page 174, diagram (10) (involving the power ...
0
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1answer
50 views

Any epi with codomain $P$ is split implies $P$ is projective [duplicate]

I'm struggling to prove that if any epi with codomain $P$ splits, then $P$ is a projective object. The converse direction I proved by factoring the identity to give a right inverse of the pi. How can ...
3
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2answers
62 views

How to show that naturally isomorphic functors preserve the same limits?

I've seen a number of times versions of the claim that if $F$ is naturally isomorphic to $G$, then if $F$ preserves limits of shape $\mathbf{J}$ so does $G$. (Sometimes with "It is easy to show that ...
4
votes
1answer
101 views

Canonical arrow between $\varinjlim _C \varprojlim _D F(C,D)\rightarrow \varprojlim_D \varinjlim _CF(C,D)$ in $\mathsf{Set}$

I'm reading Borceux's Handbook of Categorical Algebra, vol I, section 2.13 on filtered colimits. The author starts by constructing a canonical map $$\varinjlim _C \varprojlim _D F(C,D)\rightarrow ...
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1answer
47 views

When the unit of a universal property is an isomorphism

Let $G \colon \mathbf B \to \mathbf A$ be a functor, and let $A \in \mathbf A$ be an object. A universal arrow from $A$ to $G$ can be described by an isomorphism of functors \begin{equation} \mathbf ...
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0answers
61 views

Relation between categorical operations (limits and co-limits)

Suppose I have a diagram $B \longleftarrow A \longrightarrow C$ in a category, and I execute a push-out operation and get $B \longrightarrow D \longleftarrow C$. If I execute a pull-back over $B ...
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What categorical property do these forgetful functors have in common?

Consider the following examples: The forgetful functor $U_1: \operatorname{Vect}_\mathbb{C} \to \operatorname{Vect}_\mathbb{R}$ The forgetful functor $U_2: \operatorname{Diff}^{\text{or}} \to ...
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1answer
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Any deeper “duality” between non-zero-divisors and units of a ring?

I'm reading Aluffi's algebra book at the moment -- specifically, I'm on the introductory rings/modules chapter. I noticed two interesting pieces of information: in a (not necessarily commutative) ...
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1answer
38 views

Are there different combinatorial species with the same symmetry type?

First off: for my purposes, let $\sf B$ be the category of finite sets with bijections, and ${\sf B}_n$ the subcategory of sets with cardinality $n$, and define a combinatorial species to be a functor ...
6
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3answers
280 views

What is the name for a function whose codomain and domain are equal?

What do we call a function whose domain and co-domain are the same set? Edit: While i expressed my question in terms of functions, domains and codomains, i was actually interested in the most ...
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1answer
48 views

Explanation of proof in Representation Theory: A Homological Point of View

in the book Representation Theory: A Homological Point of View Proposition 3.1.18 Zimmerman proves that a cokernel is a colimit, but I can't understand his proof. He lets $\left((M_i)_{i \in ...
2
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1answer
124 views

Initial elements in Set and identity

By definition for every object there is at least one morphism - identity, so, there must be identity morphism for Set initial object - empty set. But no function can have empty set as codomain, so, ...
10
votes
2answers
186 views

Natural isomorphisms: what is the status now of “the Eilenberg/Mac Lane Thesis”?

The Church/Turing Thesis that we all know and love asserts that every algorithmically computable function (in an informally characterized sense) is in fact recursive/Turing computable/lambda ...
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1answer
51 views

How to call a category with a single morphism between every two objects?

How to call a category where for every pair of objects $A, B$, there is a unique morphism $f\colon A\to B$? (A trivial category?)