# Talk:Equivalence of categories

Is every equivalence between additive categories automatically an additive functor? AxelBoldt 17:24, 26 Jan 2004 (UTC)

I find this hard to believe. Taking a single object, so a ring, doesn't it say more about the multiplicative monoid of the ring determining it, than is likely to be true?

Charles Matthews 17:33, 26 Jan 2004 (UTC)

Yes, but a ring is a preadditive category in our terminology, and for those the conjecture is clearly false; for additive categories it is however true as our own article additive category claims at the very bottom. AxelBoldt 17:36, 26 Jan 2004 (UTC)

## Duality vs. Opposite

The article states:

If a category is equivalent to the dual of another category then one speaks of a duality of categories.

Isn't it more usual to talk of the opposite of a category, rather than its dual? MacLane's CftWM recognises the usage dual, but uses the terminology of opposite instead. I've only seen the term dual category used in informal contexts in recent work. ---- Charles Stewart 11:17, 31 Aug 2004 (UTC)

## Contrast to isomorphism

Shouldn't there be an explanation of the difference between isomorphisms and equivalences? The article on isomorphism says the following:

Isomorphism of categories is a very strong condition and rarely satisfied in practice. Much more important is the notion of equivalence of categories; roughly speaking, for an equivalence of categories we don't require that FG(x) be equal to x, but only isomorphic to x in the category D, and likewise that GF(y) be isomorphic to y in C.

I think that the explanation in the appendix of H. Ehrig's "Fundamentals of algebraic graph transformation" gives a better idea of equivalence than that:

If C and D are isomorphic or equivalent then all “categorical” properties of C are shared by D,and vice versa. If C and D are isomorphic, then we have a bijection between objects and between morphisms of C and D. If they are only equivalent, then there is only a bijection of the corresponding isomorphism classes of objects and morphisms of C and D. However, the cardinalities of corresponding isomorphism classes may be different; for example all sets M with cardinality |M| = n are represented by the set M_n = { 0, ..., n−1 }. Taking the sets M_n (n ∈ N) as objects and all functions between these sets as morphisms, we obtain a category N, which is equivalent – but not isomorphic – to the category FinSets of all finite sets and functions between finite sets.

-- 188.192.81.230 (talk) 15:13, 20 February 2012 (UTC)

## What's that?

If F : C → D is an equivalence of categories, and G1 and G2 are two inverses, then G1 and G2 are naturally isomorphic.

What is that supposed to mean? What's the relation between G1/G2 and F?! -- 188.192.81.230 (talk) 19:33, 6 March 2012 (UTC)

## Slight error?

I refer to these statements made on the page:

• F is a left adjoint of G and both functors are full and faithful.
• G is a right adjoint of F and both functors are full and faithful.

The second appears to be the same statement as the first. Shouldn't this be amended to:

• F is a left adjoint of G and both functors are full and faithful.
• G is a left adjoint of F and both functors are full and faithful.

82.153.112.144 (talk) 07:38, 8 September 2013 (UTC)