nLab walking equivalence

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 $f$ with an inverse-up-to-isomorphism $g$, that is to say, which contains in addition to $g$ a 2-isomorphism between $g f$ and the identity, and a 2-isomorphism between $f g$ and the identity.

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

Definitions

Definition

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

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

Remark

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

Remark

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

Remark

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

Definition

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

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

Remark

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

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 $\mathcal{A}$ is just a 2-functor from $\mathcal{E}$:

Proposition

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

Proof

Immediate from the definitions.

Proposition

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

Proof

Immediate from the definitions.

Structured interval

Let $\mathcal{E}_{semi}$ 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 $1$.

Notation

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

Notation

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

Notation

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

Since $1$ is a final object of 2Cat, it is immediate that $v$ is compatible with the contraction structure $p$ of Notation in the sense of VI.12 of Williamson2011.

Notation

Let

be a co-cartesian square in 2Cat.

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

There is a functor $s: \mathcal{E}_{semi} \rightarrow S$ which picks out the semi-strict equivalence in $S$ given by $r_{0} \circ r_{1}$. It is immediately checked that $\left( S, r_0, r_1, s \right)$ defines a subdivision structure with respect to $\left( \mathcal{E}_{semi}, i_0, i_1 \right)$ in the sense of VI.14 of Williamson2011. Moreover, since $1$ 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 $q_{l}$ of VI.34 in Williamson2011 is in this case the functor $S \rightarrow \mathcal{E}_{semi}$ which is determined by $r_{0} \mapsto i$, $r_0^{-1} \mapsto i^{-1}$, $r_1 \mapsto id(0)$, $r^{-1} \mapsto id(0)$, $\iota_1 \mapsto id$, and $\kappa_2 \mapsto \iota_1$. We see then that $\left( \mathcal{E}_{semi}, i_0, i_1, p, S, r_0, r_1, s \right)$ has strictness of left identities in the sense of VI.34 of Williamson2011.

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

Remark

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

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

Notation

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

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

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

Notation

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

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

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

Since $1$ is a final object of 2Cat, it is immediate that $\Gamma_{lr}$ is compatible with $p$ in the sense of VI.26 of Williamson2011.

Notation

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

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

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

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

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

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

We deduce that $x \circ \left( \mathcal{E}_{semi} \times s \right) = \mathcal{E}_{semi} \times p$, and thus that $\Gamma_{lr}$ and $\Gamma_{ur}$ are compatible with $\left(S, r_0, r_1, s \right)$ 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 $\left( \mathcal{E}_{semi}, i_0, i_1 \right)$. 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: $\Gamma_{lr}$ and $\Gamma_{ur}$ are not compatible with $\left(S, r_0, r_1, s \right)$. This is exactly where the semi-strictness of $\mathcal{E}_{semi}$ is needed.

References

• Richard Williamson, Cylindrical model structures, DPhil thesis, University of Oxford, 2011. author’s webpage arXiv:1304.0867