nLab
walking equivalence

Contents

Idea

The walking equivalence (as in “walking structure”) or free-standing equivalence is the 2-category (in fact a (2,1)-category) which ‘represents’ equivalences in a 2-category. It is a categorification of the free-standing isomorphism, though not the only one: the walking adjoint equivalence is another.

Roughly speaking, it is the minimal 2-category which contains a 1-arrow ff with an inverse-up-to-isomorphism gg, that is to say, which contains in addition to gg a 2-isomorphism between gfg f and the identity, and a 2-isomorphism between fgf g and the identity.

The walking semi-strict equivalence is the same except that either fgf g or gfg f is required to be equal on the nose to the identity.

Definitions

Definition

Let F 1F_{\leq 1} be the free category on the directed graph with exactly two objects 00 and 11, an arrow i:01i: 0 \rightarrow 1, and an arrow i 1:10i^{-1}: 1 \rightarrow 0. Let FF be the free strict 2-category on the 2-truncated reflexive globular set whose 1-truncation is the underlying reflexive directed graph of F 1F_{\leq 1}, and which in addition has a 2-arrow ι 0:i 1iid(0)\iota_{0}: i^{-1} \circ i \rightarrow id(0), a 2-arrow ι 0 1:id(0)i 1i\iota_{0}^{-1}: id(0) \rightarrow i^{-1} \circ i, a 2-arrow ι 1:ii 1id(1)\iota_{1}: i \circ i^{-1} \rightarrow id(1), and a 2-arrow ι 1 1:ii 1id(1)\iota_{1}^{-1}: i \circ i^{-1} \rightarrow id(1).

The free-standing equivalence is the quotient \mathcal{E} of FF by the relation on 2-arrows generated by forcing the equations ι 0 1ι 0=id\iota_{0}^{-1} \circ \iota_{0} = id, ι 0ι 0 1=id\iota_{0} \circ \iota_{0}^{-1} = id, ι 1 1ι 1=id\iota_{1}^{-1} \circ \iota_{1} = id, and ι 1ι 1 1=id\iota_{1} \circ \iota_{1}^{-1} = id to hold.

Remark

The arrow i:01i: 0 \rightarrow 1 is an equivalence, whose inverse-up-to-isomorphism is the arrow i 1:10i^{-1}:1 \rightarrow 0.

Remark

The 2-arrows ι 0\iota_{0} and ι 1\iota_{1} are 2-isomorphisms.

Remark

The free-standing equivalence is a (2,1)-category, that is, all its 2-morphisms are invertible.

Definition

Let F 1F_{\leq 1} be as in Definition . Let FF be the free strict 2-category on the 2-truncated reflexive globular set whose 1-truncation is the underlying reflexive directed graph of F 1F_{\leq 1}, and which in addition has a 2-arrow ι 1:ii 1id(1)\iota_{1}: i \circ i^{-1} \rightarrow id(1), and a 2-arrow ι 1 1:ii 1id(1)\iota_{1}^{-1}: i \circ i^{-1} \rightarrow id(1).

The free-standing semi-strict equivalence is the quotient semi\mathcal{E}_{semi} of FF by the relation on 1-arrows which forces that i 1i=id(0)i^{-1} \circ i = id(0), and by the relation on 2-arrows which forces that ι 1 1ι 1=id\iota_{1}^{-1} \circ \iota_{1} = id, and ι 1ι 1 1=id\iota_{1} \circ \iota_{1}^{-1} = id.

Remark

The free-standing semi-strict equivalence is the quotient of the free-standing equivalence by the relation on 1-arrows which identifies i 1ii^{-1} \circ i and id(0)id(0), and which identifies ι 0\iota_{0} and ι 1\iota^{-1} with id(id(0))id\left(id(0)\right).

Representing of equivalences

The 2-category \mathcal{E} is the model for all equivalences in all 2-categories. In other words, any equivalence in a 2-category 𝒜\mathcal{A} is just a 2-functor from \mathcal{E}:

Proposition

Let 𝒜\mathcal{A} be a 2-category (weak or strict). Let \mathcal{E} denote the free-standing equivalence. Let ff be a 1-arrow of 𝒜\mathcal{A} which is an equivalence, the equivalence being exhibited by a 1-arrow f 1f^{-1}, a 2-isomorphism ϕ 0:f 1fid\phi_{0}: f^{-1} \circ f \rightarrow id, and a 2-isomorphism ϕ 1:ff 1id\phi_{1}: f \circ f^{-1} \rightarrow id. Then there is a unique 2-functor F:𝒜F: \mathcal{E} \rightarrow \mathcal{A} such that the arrow i:01i:0 \rightarrow 1 of \mathcal{I} maps under FF to ff, such that the arrow i 1:10i^{-1}: 1 \rightarrow 0 of \mathcal{I} maps under FF to f 1f^{-1}, such that ι 0\iota_{0} maps under FF to ϕ 0\phi_{0}, and such that ι 1\iota_{1} maps under FF to ϕ 1\phi_{1}.

Proof

Immediate from the definitions.

Proposition

Let 𝒜\mathcal{A} be a 2-category (weak or strict). Let semi\mathcal{E}_{semi} denote the free-standing semi-strict equivalence. Let ff be a 1-arrow of 𝒜\mathcal{A} which is a semi-strict equivalence?, the equivalence being exhibited by a 1-arrow f 1f^{-1} and a 2-isomorphism ϕ 1:ff 1id\phi_{1}: f \circ f^{-1} \rightarrow id. Then there is a unique 2-functor F: semi𝒜F: \mathcal{E}_{semi} \rightarrow \mathcal{A} such that the arrow i:01i:0 \rightarrow 1 of \mathcal{I} maps under FF to ff, and such that ι 1\iota_{1} maps under FF to ϕ\phi.

Proof

Immediate from the definitions.

Structured interval

Let semi\mathcal{E}_{semi} denote the free-standing semi-strict equivalence. We shall view it as an interval object equipped with all the structures required for Corollary XV.6 and Corollary XV.7 of Williamson2011.

Throughout, we shall denote the category of strict 2-categories by 2Cat, and denote the final object of 2Cat by 11.

Notation

We denote by i 0i_{0} (resp. i 1i_{1}) the functor 1 semi1 \rightarrow \mathcal{E}_{semi} which picks out the object 00 (resp. 11) of semi\mathcal{E}_{semi}.

Notation

We denote by pp the canonical functor semi1\mathcal{E}_{semi} \rightarrow 1. It is immediate that it defines a contraction structure on ( semi,i 0,i 1)\left(\mathcal{E}_{semi}, i_{0}, i_{1} \right) in the sense of VI.6 of Williamson2011.

Notation

We denote by vv the functor semi semi\mathcal{E}_{semi} \rightarrow \mathcal{E}_{semi} determined by ii 1i \mapsto i^{-1}, i 1ii^{-1} \mapsto i, ι 1,ι 1 1id(id(0))\iota_{1}, \iota_{1}^{-1} \mapsto id\left(id(0)\right). It defines an involution structure on ( semi,i 0,i 1)\left(\mathcal{E}_{semi}, i_{0}, i_{1} \right) in the sense of VI.10 of Williamson2011.

Since 11 is a final object of 2Cat, it is immediate that vv is compatible with the contraction structure pp of Notation in the sense of VI.12 of Williamson2011.

Notation

Let

be a co-cartesian square in 2Cat.

Explicitly, let F 1F_{\leq 1} be the free category on the directed graph with exactly three objects 00, 11, and 22, and with non-identity arrows r 1:01r_1: 0 \rightarrow 1, r 0:12r_0: 1 \rightarrow 2, and r 1 1:10r_1^{-1}: 1 \rightarrow 0, r 0 1:21r_0^{-1}: 2 \rightarrow 1. Let FF be the free strict 2-category on the 2-truncated reflexive globular set whose 1-truncation is the underlying reflexive directed graph of F 1F_{\leq 1}, and which in addition has 2-arrows ι 1,ι 1 1:r 1r 1 1id(1)\iota_{1}, \iota_{1}^{-1}: r_1 \circ r_1^{-1} \rightarrow id(1) and κ 2,κ 2 1:r 0r 0 1id(2)\kappa_{2}, \kappa_{2}^{-1}: r_0 \circ r_0^{-1} \rightarrow id(2).Then SS can be taken to be the quotient of FF by the relation on 2-arrows which forces ι 0\iota_{0} to be a 2-isomorphism with inverse ι 0 1\iota_{0}^{-1}, and similarly for κ 2\kappa_{2}. The functors r 0r_0 and r 1r_1 can be taken to be functors picking out the equivalences in SS of the same name.

There is a functor s: semiSs: \mathcal{E}_{semi} \rightarrow S which picks out the semi-strict equivalence in SS given by r 0r 1r_{0} \circ r_{1}. It is immediately checked that (S,r 0,r 1,s)\left( S, r_0, r_1, s \right) defines a subdivision structure with respect to ( semi,i 0,i 1)\left( \mathcal{E}_{semi}, i_0, i_1 \right) in the sense of VI.14 of Williamson2011. Moreover, since 11 is a final object, it is immediate that this subdivision structure is compatible with the contraction structure of Notation in the sense of VI.18 of Williamson2011.

The functor q lq_{l} of VI.34 in Williamson2011 is in this case the functor S semiS \rightarrow \mathcal{E}_{semi} which is determined by r 0ir_{0} \mapsto i, r 0 1i 1r_0^{-1} \mapsto i^{-1}, r 1id(0)r_1 \mapsto id(0), r 1id(0)r^{-1} \mapsto id(0), ι 1id\iota_1 \mapsto id, and κ 2ι 1\kappa_2 \mapsto \iota_1. We see then that ( semi,i 0,i 1,p,S,r 0,r 1,s)\left( \mathcal{E}_{semi}, i_0, i_1, p, S, r_0, r_1, s \right) has strictness of left identities in the sense of VI.34 of Williamson2011.

It is similarly the case that ( semi,i 0,i 1,p,S,r 0,r 1,s)\left( \mathcal{E}_{semi}, i_0, i_1, p, S, r_0, r_1, s \right) has strictness of right identities in the sense of VI.34 of Williamson2011.

Remark

Explicitly, semi× semi\mathcal{E}_{semi} \times \mathcal{E}_{semi} can be described as follows. Let F 1F_{\leq 1} be the free category on the directed graph with objects (0,0)(0,0), (0,1)(0,1), (1,0)(1,0), (1,1)(1,1), and with arrows (0,i)(0, i), (0,i 1)(0, i^{-1}), (i,0)(i, 0), (i 1,0)(i^{-1}, 0), (1,i)(1, i), (1,i 1)(1, i^{-1}), (i,1)(i, 1), and (i 1,1)(i^{-1}, 1). Let FF be the free strict 2-category on the 2-truncated reflexive globular set whose 1-truncation is the underlying reflexive directed graph of F 1F_{\leq 1}, and which in addition has 2-arrows (0,ι 1):(0,ii 1)id(0,1)(0, \iota_{1}): (0, i \circ i^{-1}) \rightarrow id(0, 1), (1,ι 1):(1,ii 1)id(1,1)(1, \iota_{1}): (1, i \circ i^{-1}) \rightarrow id(1, 1), (ι 1,0):(ii 1,0)id(1,0)(\iota_{1}, 0): (i \circ i^{-1}, 0) \rightarrow id(1, 0), and (ι 1,1):(ii 1,1)id(1,1)(\iota_{1}, 1): (i \circ i^{-1}, 1) \rightarrow id(1, 1).

Then semi× semi\mathcal{E}_{semi} \times \mathcal{E}_{semi} is the quotient of FF by the relation on 1-arrows which forces (0,i 1i)(0, i^{-1} \circ i) to be equal to (0,0)(0,0), and similarly for (i 1i,0)(i^{-1} \circ i, 0), (1,i 1i)(1, i^{-1} \circ i), and (i 1i,1)(i^{-1} \circ i, 1); and on 2-arrows which forces (0,ι 1)(0, \iota_{1}), (ι 1,0)(\iota_{1}, 0), (1,ι 1)(1, \iota_1) and (ι 1,1)(\iota_1, 1) to be 2-isomorphisms.

Notation

Let Γ ul\Gamma_{ul} be the functor semi× semi semi\mathcal{E}_{semi} \times \mathcal{E}_{semi} \rightarrow \mathcal{E}_{semi} determined by

(0,i),(i,0) i, (0,i 1),(i 1,0) i 1, (1,i),(1,i 1),(i,1),(i 1,1) id(1), (0,ι 1),(ι 1,0) ι 1, (1,ι 1),(ι 1,1) id(id(1)). \begin{aligned} (0, i), (i,0) &\mapsto i, \\ (0, i^{-1}), (i^{-1}, 0) &\mapsto i^{-1}, \\ (1, i), (1, i^{-1}), (i, 1), (i^{-1}, 1) &\mapsto id(1), \\ (0, \iota_{1}), (\iota_{1}, 0) &\mapsto \iota_{1}, \\ (1, \iota_{1}), (\iota_{1}, 1) &\mapsto id\left(id(1)\right). \end{aligned}

Then Γ ul\Gamma_{ul} defines an upper left connection structure with respect to ( semi,i 0,i 1,p)\left( \mathcal{E}_{semi}, i_0, i_1, p \right) in the sense of VI.22 of Williamson2011.

Notation

Let Γ lr\Gamma_{lr} be the functor semi× semi semi\mathcal{E}_{semi} \times \mathcal{E}_{semi} \rightarrow \mathcal{E}_{semi} determined by

(0,i),(i,0),(0,i 1),(i 1,0) id(0), (1,i),(i,1) i, (1,i 1),(i 1,1) i 1, (0,ι 1),(ι 1,0) id(id(0)), (1,ι 1),(ι 1,1) ι 1. \begin{aligned} (0, i), (i,0), (0, i^{-1}), (i^{-1}, 0) &\mapsto id(0), \\ (1, i), (i, 1) &\mapsto i, \\ (1, i^{-1}), (i^{-1}, 1) &\mapsto i^{-1}, \\ (0, \iota_{1}), (\iota_{1}, 0) &\mapsto id\left(id(0)\right), \\ (1, \iota_{1}), (\iota_{1}, 1) &\mapsto \iota_{1}. \end{aligned}

Then Γ lr\Gamma_{lr} defines an lower right connection structure with respect to ( semi,i 0,i 1,p)\left( \mathcal{E}_{semi}, i_0, i_1, p \right) in the sense of VI.24 of Williamson2011.

Since 11 is a final object of 2Cat, it is immediate that Γ lr\Gamma_{lr} is compatible with pp in the sense of VI.26 of Williamson2011.

Notation

Let Γ ur\Gamma_{ur} be the functor semi× semi semi\mathcal{E}_{semi} \times \mathcal{E}_{semi} \rightarrow \mathcal{E}_{semi} determined by

(i,0) i, (i 1,0) i 1, (0,i),(0,i 1) id(0), (1,i) i 1, (1,i 1) i, (i,1),(i 1,1) id(0), (ι 1,0) ι 1, (0,ι 1) id(id(0)), (1,ι 1) id(id(0)), (ι,1) id(id(0)). \begin{aligned} (i,0) &\mapsto i, \\ (i^{-1}, 0) &\mapsto i^{-1}, \\ (0, i), (0, i^{-1}) &\mapsto id(0), \\ (1, i) &\mapsto i^{-1}, \\ (1, i^{-1}) &\mapsto i, \\ (i, 1), (i^{-1}, 1) &\mapsto id(0), \\ (\iota_{1}, 0) &\mapsto \iota_{1}, \\ (0, \iota_{1}) &\mapsto id\left(id(0)\right), \\ (1, \iota_{1}) &\mapsto id\left(id(0)\right), \\ (\iota, 1) &\mapsto id\left(id(0)\right). \end{aligned}

Then Γ ur\Gamma_{ur} defines an upper right connection structure with respect to ( semi,i 0,i 1,p,v)\left( \mathcal{E}_{semi}, i_0, i_1, p, v \right) in the sense of VI.29 of Williamson2011.

The functor x( semi×s): semi× semi semix \circ \left( \mathcal{E}_{semi} \times s \right) : \mathcal{E}_{semi} \times \mathcal{E}_{semi} \rightarrow \mathcal{E}_{semi} of VI.32 in Williamson2011 is determined by

(i,0),(i,1) i, i 1,0),(i 1,1) i 1, (ι 1,0) ι 1, (ι 1,1) ι 1, (0,i),(0,i 1) id(0), (1,i),(1,i 1) id(1), (0,ι 1) id(id(0)), (1,ι 1) id(id(1)). \begin{aligned} (i, 0), (i,1) &\mapsto i, \\ i^{-1}, 0), (i^{-1}, 1) &\mapsto i^{-1}, \\ (\iota_{1}, 0) &\mapsto \iota_{1}, \\ (\iota_{1}, 1) &\mapsto \iota_{1}, \\ (0, i), (0,i^{-1}) &\mapsto id(0), \\ (1, i), (1, i^{-1}) &\mapsto id(1), \\ (0, \iota_{1}) &\mapsto id\left(id(0)\right), \\ (1, \iota_{1}) &\mapsto id\left(id(1)\right). \end{aligned}

The key observation here is that (1,i),(1,i 1)id(1)(1, i), (1, i^{-1}) \mapsto id(1), which relies on the fact that i 1i=id(0)i^{-1} \circ i = id(0) in semi\mathcal{E}_{semi}.

We deduce that x( semi×s)= semi×px \circ \left( \mathcal{E}_{semi} \times s \right) = \mathcal{E}_{semi} \times p, and thus that Γ lr\Gamma_{lr} and Γ ur\Gamma_{ur} are compatible with (S,r 0,r 1,s)\left(S, r_0, r_1, s \right) in the sense of VI.32 of Williamson2011.

Remark

It follows from the above, Corollary XV.6, and Corollary XV.7 of Williamson2011 that there is a model structure on 2Cat, the category of strict 2-categories, whose fibrations and cofibrations are ‘Hurewicz’ fibrations and cofibrations respectively with respect to the interval object ( semi,i 0,i 1)\left( \mathcal{E}_{semi}, i_0, i_1 \right). See canonical model structure on 2-categories for more.

Remark

All of the structures in this section have an analogue for \mathcal{E}, the free-standing equivalence, as well. All the required compatibilities hold except for one: Γ lr\Gamma_{lr} and Γ ur\Gamma_{ur} are not compatible with (S,r 0,r 1,s)\left(S, r_0, r_1, s \right). This is exactly where the semi-strictness of semi\mathcal{E}_{semi} is needed.

References

Last revised on July 10, 2020 at 05:42:56. See the history of this page for a list of all contributions to it.