# Tagged Questions

**5**

votes

**1**answer

276 views

### Not quite adjoint functors

What are standard and/or natural examples of pairs of functors $F:C\leftrightarrows D:G$ and unnatural bijections $\hom_D(Fx,y)\to\hom_C(x,Gy)$ for all $x$ and $y$? Can one do this so that the ...

**12**

votes

**9**answers

1k views

### What are your favorite concrete examples of limits or colimits that you would compute during lunch?

(The title was initially "What are your favorite concrete examples that you would compute on the table during lunch to convince a working mathematician that the notions of limits and colimits are not ...

**4**

votes

**1**answer

210 views

### Example of a non-closed cocomplete symmetric monoidal category

Background
By a cocomplete symmetric monoidal category $C$ I mean a symmetric monoidal category whose underlying category is cocomplete and such that $- \otimes X : C \to C$ is cocontinuous for all ...

**5**

votes

**3**answers

199 views

### two essentially different concretizaions

It is sometimes emphasized that a "concrete category" is not a property of a category $C$, but rather a structure, i.e. a faithful functor from $C$ to $Set$. Thus, When people talk about a concrete ...

**40**

votes

**27**answers

4k views

### Examples where it's useful to know that a mathematical object belongs to some family of objects

For an expository piece I'm writing, it would be useful to have good examples of the following phenomenon:
(1) ${\cal X}$ is a parameterized family of somethings. (Varieties, schemes, manifolds, ...

**3**

votes

**1**answer

319 views

### Examples of “inner products” of parallel morphisms in a dagger category

There is a very interesting abstract notion of the trace of an endomorphism $f : c \to c$ of an object $c$ in a braided monoidal category (although the symmetric case is easier): see, for example, ...

**2**

votes

**1**answer

225 views

### Is there a category with a subobject classifier but which is not finitely complete?

This is a reverse of the question “Is there a finitely complete category with terminal object but NO subobject classifier?” From “An informal introduction to topos theory” by Tom Leinster I learned ...

**12**

votes

**3**answers

1k views

### Is there an additive functor between abelian categories which isn't exact in the middle?

Suppose $F: C\to D$ is an additive functor between abelian categories and that
$$0\to X\xrightarrow f Y\xrightarrow g Z\to 0$$
is and exact sequence in $C$. Does it follow that ...

**1**

vote

**1**answer

1k views

### Classification Problems [closed]

I was thinking about the famous question in philosophy of mathematics: "When are two proofs the same?" and I was wondering if we could somehow "classify" proofs by establishing some sort of functorial ...

**4**

votes

**1**answer

315 views

### Categories with products that preserve quotients

It is well known that in the category of all topological spaces, quotient maps aren't preserved by products (this follows from the simpler fact that $X\times (-):Top\to Top$ doesn't preserve ...

**4**

votes

**0**answers

269 views

### Example of a Grothendieck pretopology satisfying a weak saturation condition

Recall that a singleton Grothendieck pretopology (henceforth 'singleton pretopology') on a category $C$ is a collection of maps $J$ containing the isomorphisms, closed under composition and stable ...

**1**

vote

**2**answers

377 views

### Weakly initial sets - examples and nonexamples

A weakly initial set in a category C is a set of objects I of C such that every object a of C has at least one arrow from an object contained in I.
The question is then, does Fields have a weakly ...

**5**

votes

**9**answers

986 views

### What category without initial object do you care about?

Recently I have been listening to some constructions that have been designed to accommodate categories without an initial object. The speaker has given some idea of a category or two that he cares ...

**1**

vote

**5**answers

601 views

### Is there a finitely complete category with terminal object but NO subobject classifier?

This came up today while thinking about topoi in seminar, as the title suggests my question is;
Is there a finitely complete category with terminal object but NO subobject classifier?
Hopefully ...

**32**

votes

**5**answers

3k views

### What is Yoneda's Lemma a generalization of?

What is Yoneda's Lemma a generalization of?
I am looking for examples that were known before category theory entered the stage resp. can be known by students before they start with category theory.
...

**0**

votes

**1**answer

191 views

### Motivation for Cosuspended Category Axioms

Today I was wondering about the axioms given by Bernhard Keller for Cosuspended Categories.
The axioms of a triangle feel very much like exactness, but not quite. The last axiom about the large ...

**6**

votes

**5**answers

1k views

### Simple show cases for the Yoneda lemma

I've been given a very simple motivating and instructive show case for the Yoneda lemma:
Given the category of graphs and a graph object $G$, seen as a quadruple $(V_G,\ E_G,\ S_G:E\rightarrow V,\ ...

**8**

votes

**5**answers

931 views

### Is a functor which has a left adjoint which is also its right adjoint an equivalence ?

I am looking for a counter-example of two functors F : C -> D and G : D->C such that
1) F is left adjoint to G
2) F is right adjoint to G
3) F is not an equivalence (ie F is not a quasi-inverse of ...

**8**

votes

**2**answers

398 views

### Which commutative rigs arise from a distributive category?

A rig is an algebraic object with multiplication and addition, such that multiplication distributes over addition and addition is commutative. However, instead of requiring that the set forms an ...